Research of Mathematical Economics
No. 1 2011
CONTENTS
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural Areas
…………………………………………………Zhang Dongyang (5)
The Influence of Wealth Gap on China’s Economic Growth
………………………………………………………Guo Yumei (21)
Bundling Sales of Information Goods: Models and Analysis
……………………………………………………Shi Fangning (31)
The Introduction and Applications of RDEU Hypothesis
…………………………………………Wang Bo, Hong Beiyun (54)
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition Survey
……………………………………………………Zhang Luezhao (72)
A Dynamic Model of the Optimal Carbon Tax and Sustainable Development
……………………………………………………Feng Junlong (89)
A Study of Chinese Residential Electricity Demand Elasticities
…………………………………………………Ren Junqiushi (101)
The Factors that Determine a Firm’s Optimal Allocation of Capital Between Research and
Manufacturing
……………………………………Xu Lingjue, Wang Danna (118)
A Quantitative Analysis of the Influences of China’s Inflation to Different
Asset-Holding Classes
………………………………………………………Chen Jun (134)
Analysis of the Mechanism of “Dutch Disease”
……………………………………………………Wang Qizhi (161)
1
数理经济学研究
2011 年第 1 辑（总第 5 辑）
目
录
收入和利率对中国居民消费行为的影响——基于中国城乡统计数据的比较分析
…………………………………………………………………… 张冬洋 (21)
贫富差距对中国经济增长的影响
…………………………………………………………………… 郭豫媚 (47)
信息产品的捆绑销售：模型与分析
…………………………………………………………………… 施芳凝 (31)
REDU 假说的介绍与应用
………………………………………………………… 王
勃、洪蓓芸 (54)
收入目标与劳动供给弹性：来自《中国健康和营养调查》的证据
…………………………………………………………………… 张略钊 (72)
一个最优碳税和可持续发展的动态模型
…………………………………………………………………… 冯焌昽 (89)
中国居民电力需求的弹性探析
……………………………………………………………… 任君秋拾 (101)
决定公司在研究与制造之间资本最优配置的因素
……………………………………………………… 许龄珏、王丹娜 (118)
中国通货膨胀对不同财富持有阶层影响的定量分析
………………………………………………………………… 陈 军 (134)
荷兰病作用机制考察
………………………………………………………………… 王麒植 (161)
2
Research of Mathematical Economics
Organizers:
Economics and Mathematics Double Bachelor Program
Renmin University of China
Academic Adviser:
Yang Ruilong
Liu Yuanchun
Zhang Yu
Liu Rui
Guo Jie
Wang Jinbin
Zheng Xinye
Chen Yanbin
Wang Xianghong
Editorial Committee:
Li Tianyou
Zhang Yanhong
Yu Ze
Cheng Hua
Yu Yihua
Han Song
Instruct Teacher:
Cheng Hua
Editorial Director:
Ren Junqiushi
Xu Lingjue
Zhang Lingyun
Wang Jue
《数理经济学研究》编辑机构
主办单位： 中国人民大学经济学—数学（双学位）实验班
学术顾问： 杨瑞龙 刘元春 张 宇 刘 瑞 郭 杰 王晋斌 郑新业 陈彦斌
编
委： 李天有 章艳红 于 泽 王湘红 程 华 虞义华 韩 松
指导老师： 程 华
编 辑 部： 任君秋拾 许龄珏 张凌云 王 珏
3
4
Research of Mathematical Economics No. 1 2011
The Effect of Income and Interest Rates on the Consuming Behavior of
Chinese Residents----Based on the Comparative Analysis on Statistics
of Chinese Urban and Rural Areas
Zhang Dongyang
(School of Economics, Renmin University of China)
Abstract: This essay analyzes how to expand domestic demand and how to stimulate consumption. Based on the
random walk hypothesis and permanent income hypothesis, consumers are divided into two groups, one with
rational expectations and the other regarded as “short-sighted”. This paper uses ordinary least squares method and
uses data from 2003Q1 - 2008Q2 to estimate the aggregate consumption function of China’s urban and rural areas.
The conclusion is that the number of “short-sighted” consumers in rural areas of China significantly exceeds that
in urban areas. Also, this paper suggests that lowering interest rates has a strong stimulating effect on
consumption growth and verifies the parameters’ stability by hypothesis testing. Last, with regard to the statistics
obtained through econometric model, final conclusions of this essay and certain corresponding policy
recommendations are given out.
Keywords: Permanent Income Hypothesis, Uncertainty, Consumption, Interest Rates
1. Introduction
In 2008, the financial crisis swept the entire world including China, which imposed an
extremely large impact on China’s economy. On the other hand, China has been under reform and
opening-up policy for 30 years until 2008. Along with the ever-deepening reform, Chinese
economy system has gradually turned from traditional planned economy to socialist market
economy, from extensive to intensive. In the support of the three pillars of its development –
investment, domestic demand and foreign trade, the entire national economy follows an orderly
and stable development trend. Hence, in face of such a sudden shock, the problem of how to
expand domestic demand and stimulate China’s economy is put on core agenda once again.
In the traditional researches, Keynes pointed out that national income influences consuming;
5
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural
Areas
Friedman and Modigliani put forward permanent income hypothesis and life-cycle hypothesis
respectively. These theories have been far-reaching. The basis of these theories, interest rates act
as a direct means to influence national income through adjusting saving rates----lowering interest
rates helps to increase the income for people’s consumption----have long been considered as an
effective approach to expand domestic demand.
However, from the “Ninth Five-Year” to the “Eleventh Five-Year”, a lot of scholars, such as
Long Zhiming, Zhou Haoming (2000), think that China’s economic development and consumer
behavior showed some kind of “rebellion” phenomenon: on one hand, the economy experiences
stable and fast growth and national income increases steadily; on the other hand, along with the
ever-deepening reform and opening-up policy, consumer behaviors have changed dramatically,
among which what worthies our particular attention is the fact that in spite of the continuously
increasing saving percentage, which to some extent reflects decreasing marginal consuming
propensity, consumption scale has grown slowly. In fact, it is still too early to draw the
conclusion of whether interest rate adjusting has positive effect on national consumption. Thus,
this paper will explore the effective factors, such as interest rate and income, and their
effectiveness on consumption increase.
Then, the problem is raised: what is the factor that coordinates the consuming behavior with
the trend of economy growth during the transition period of China? Focusing on this
phenomenon, domestic scholars have done a series of researches referring to foreign
consumption theories. For instance, under a unified economic logic framework, Yuan Zhigang
and Song Zheng (1999) conduct a comprehensive analysis of consuming behavior of urban
residents in China in the transition period: based on consumption theories such as Life Cycle
Hypothesis, Permanent Income Hypothesis, Precautionary Saving Theory and Liquidity
Constraint, they try to find out the reason for the variation in urban consumption behavior and the
way to stimulate urban household consumption. Li Yan (1999) makes use of the data from 1978
to 1998 to carry out the empirical analysis which aims to find out the role of interest rates on
household savings. The transfer channel for interest rates’ effect on saving is wealth: the low
level of income and the underdeveloped capital markets are the primary obstacles that restrict the
interest rates’ effect on household saving. That is to say, under the environment of low level of
income and underdeveloped capital markets, wealth value and wealth gains are not sensitive to
6
Research of Mathematical Economics No. 1 2011
interest rates changes. Wang Hongju and Zhang Huilian (2002) put forward the idea of using
modern Western economic theory to analyze the consumption growth in China. They point out
the restriction effect of Random Walk Hypothesis, Precautionary Saving Theory and Liquidity
Constraint on consumption and then analyze the impacts of these factors on China’s household
consumption. They conclude that Liquidity constraints depressed consumer’s current spending;
moreover, low income, uncertainty and liquidity constraints lead to the shortsightedness of the
residents.
However, theoretical research in this regard focused mainly on the analysis from 1999 to
2003 and few in recent years. This situation prompts us to refresh our understanding and study
the problem of consumer behavior both from the perspective of theory and practice, trace the
deep influencing factors behind these consumer behavior and then to give theoretical
explanations as well as pragmatically measures to direct the household consuming behavior.
The second part of this paper is literature review, pointing out the limitation of traditional
consumption theory on explaining consumer behaviors and the feasibility of using Residents
Consumption Excess Sensitivity Hypothesis and Random Walk Hypothesis to explain and
analyze the consumption problems in current China. The third section reviews the theoretical
models based on Random Walk Hypothesis. The fourth part gives out the quantitative results of
consumption functions about China’s urban and rural residents in the transition period with the
2003-2008 quarterly data on China’s urban and rural residents. And the fifth part takes advantage
of the theory and results of quantitative analysis to draw conclusions and policy
recommendations are given.
2.
Literature Review
Consumption function defines the relationship between household consumption expenditure
and the variables that determine the consumption. It was first proposed by Keynes (1936) in the
book “Employment, Interest and Money”. Absolute Income Hypothesis suggests that people’s
current consumption depends on current income and a stable functional relationship exists
between them. With the increase of income, consumption will also increase, but the increase of
consumption is slower than the increase of income: consumption increment in the proportion of
increased income is decreasing ---- that is diminishing marginal consumption propensity, showed
7
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural
Areas
by
c     y, 0    1
Yet, Duesenberry puts forward the Relative Income Hypothesis by negative the conditions
of Absolute Income Hypothesis. In his view, consumers are influenced by their surrounding
environment from time to time. The “model” of consumption is likely to make the low-income
group to get closer to the consumption level of their surrounding people. Hence, he believes that
the lower the income is, the high the marginal consumption propensity will be.
However, both Absolute and Relative Income Hypothesis only take consumers’ current
income into consideration and thus lack of microeconomic foundations. For that matter, within
the framework of neo-classical Friedman and Modigliani extended consumer decision to embrace
inter-stage selection and proposed Permanent Income Hypothesis and Life Cycle Hypothesis,
respectively. They believed that consumers’ decision for current expenditure is determined by
their life income.
Nevertheless, none of these econometric models can escape from Lucas critique. Lucas
(1976) pointed out that no matter how the traditional consumption function fits real data, they
have no value in policy making decisions, because any change in policy will after all alter the
relationship between aggregates in macroeconomic models. Therefore, Hall (1978) proposed a
consumption theory based on rational expectations. His idea is that consumption should follow
the first-order optimal conditions of a typical forward-looking rational consumer, which is also
known as “Euler Equation”. Hence, the Random Walk Model gives out the following equation
assuming the quadratic utility function:
Ct  Ct 1   t  E t  0
(1)
This means that the changes in consumption are unpredictable, that is the current
consumption depends only on the consumption in the last period and independent of any other
economic variables. In view of Hall’s hypothesis, later researchers have done a lot of empirical
analysis but end up with no satisfied results. So Cambell and Mankiw (1990) proposed the idea
that aggregate consumption function does not necessarily satisfy the general random walk. They
put consumers into two categories: one to meet the Permanent Income Hypothesis of rational
expectations and the other to spend their current income which
8
 , also
Research of Mathematical Economics No. 1 2011
known as Excess Sensitivity. The greater
 is, the more sensitive consumption will reflect on
  1 , then consumption depends entirely on the current
  0 , it means the Permanent Income Hypothesis of rational
expectations is verified.
Chinese scholars have made some progress in the research of “random walk” consumption
theory. Wan Guoguang (2001) use the model of Hall and the idea of Campbell and Mankiw
through the data from 1961 to 1998 in China and come to the conclusion that with the
ever-deepening economic reform in China, the consuming behavior of Chinese households has
experienced structural change in the early 1980s. They believe that both the rise in the proportion
of liquidity constrained consumers and the increased uncertainty are responsible for the low
consumption growth rate in China and insufficient domestic demand. Hang Bin and Shen
Chunlan (2004) believe that Excess Sensitivity in the sample period is likely to change due to
policy changes, so they conducted a varying parameters analysis on the consumer sensitivity of
urban residents through data from 1978 to 2002. They come to the results that before 1990
Excess Sensitivity had dropped dramatically, but after 1990, Excess Sensitivity had experienced
stead growth, at about 0.64. Also, the substitution effect of changes in interest rates is greater than
the income effect; that is to say, consumption expenditure will increase along with the rise in
interest rate. Through the analysis of relevant macroeconomic data about rural areas from 1978 to
2003, Zhu Xinkan (2005) concluded that 62.5% of rural household limited by liquidity constraint
are short-sighted. By the analysis of residents’ consumption behaviors on durable and
non-durable goods, Liu Hinquan (2003) found that household savings in China have significant
“precautionary savings” components and future expected income is extraordinary uncertainty.
Zhou Jian through the research into the variable parameter space state model of the consumption
function studied the sensitivity of consuming behavior of rural residents in the transition period.
Empirical results showed that there exists significant “precautionary savings” motivation among
the rural residents. Zhang Xuheng (2007) made use of precautionary saving, Liquidity
Constraints Hypothesis and the dynamic modeling method to compare the characteristics of
consumer behavior in China’s rural areas and urban areas. His empirical results implied that the
proportion of urban residents who were affected by liquidity constraints is greater than that of
rural ones, and rural residents’ responses to uncertainty and real interest rates are more sensitive
than urban people.
9
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural
Areas
However, these models are based on annual statistics and lack research in recent years. This
paper’s main logic is as follows: the Random Walk Hypothesis and the Excess Sensitive
Hypothesis are used as theoretical foundation and the statistics from 2003Q1 to 2008Q2 may be
processed according to the model to solve out China’s consumer function. Then, this paper will
go on to verify the effectiveness of interest rates and income on consumption so as to fill the
bland area in this research field.
3.
Mathematics Model
Campbell and Mankiw (1990) pointed out that consumers can be divided into two types:
one part meets the Hall's permanent income hypothesis of rational expectations, they maximize
the expected utility function:

Et  (1   )  s U (Ct  s )
(2)
s 0
where
 is subjective discount rate, C is the consumption, U satisfies the first derivative to be
greater than 0, the second derivative to be less than 0, and Et is the conditional expectation
which based on the t period information, if the consumption can borrow by interest rates r , then
the optimal first-order conditions as
EtU '  Ct 1  =
1+
U '  Ct 
1 r
(3)
If it is assumed r   , and marginal utility satisfies the conditions of linear, then the
consumption reduces to the random walk, that’s to say Et Ct 1  Ct , or Ct  0, Et  t  0 .
And the other part those who spend their current income, share of
 , also known as Excess
Sensitivity. If the total income of these two types of consumers, respectively Y1t  Y2t , the
equation can be:
Ct  C1t  C2t  Yt  (1   ) t
However, the residual
(4)
 t may be correlated with yt , this formula cannot be directly
used for estimating model. Alternative methods are mainly divided into two types, one is Hall
10
Research of Mathematical Economics No. 1 2011
(1978), which is to see whether lag period consumption and income can predict the current
consumption or not; the other method was proposed by Campbell and Mankiw (1989), by setting
a set of instrumental variables to estimate the equation. This paper uses instrumental variables
method, because any lag and stationary time series may be orthogonal with
 t , so they can be
an effective tool for the candidate instrumental variables. Also a good set of instrumental
variables is also possible to be relevant with yt , so we need to select some of the lag variables
which can predict future income variables. Because in reality, the number of consumption and
income data is closer to log-linear and nonlinear, and therefore we need to adopt all the variables
on the number (with lowercase letters to replace the original variable capital letters), replace the
original regression equation ct  yt  (1   ) t , using the set of instrumental variables to
do regression
ct  X t    ct
(5)
yt  X t    yt
where X t is the set of instrumental variables, with the dimension n  k , k is the number of
instrumental variables. We can get the solution of vector
Re-use the relationship function
to estimate the value of
 ,  ,  from the regression function.
    v  u , which is constituted by k simple equations
 ,  ,u , noted that when k > 3, the equation may have no solution, while
k < 3, there may be many solutions. X t adopted the set of instrumental variables which is no
less than three to avoid multiple solutions in this case. And when k  3 , we use two-stage least
squares to estimate the solution.
4.
The Empirical Results of China's Consumption Function
4.1 Data Selection and Data Processing
Based on the permanent income hypothesis of rational expectations models emphasizing the
microeconomic foundation, the general recommendation is to use quarterly data. In addition, the
annual data model’s spread is too long to avoid the Lucas critique ---- a change in policy will
change the total relationship between macroeconomic models. Monthly data is not a good choice
11
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural
Areas
because there is a big seasonal problem. Therefore, in accordance with established practice of the
total consumption curve model, the paper uses quarterly data for empirical research in China. The
time span is from the 1st quarter, 2003 (2003Q1) to the 2nd quarter, 2008 (2008Q2). We need the
following three types of data: per capita disposable income of urban residents, real per capita
consumption expenditure of urban residents, per capita disposable income, and per capita
consumption expenditure of rural households, rural household, and real interest rates. Data is
collected from China Monthly Economic Indicators and the People's Bank of China website.
(1) Urban per capita actual disposable income.
As the National Bureau of Statistics data is presented quarterly cumulative nominal per
capita consumption expenditure of urban residents, firstly we need to process accumulate data
into quarterly data, that is, the first quarter of each year’s data keeps unchanged, while the other
quarter with this quarter cumulative minus the cumulative data of last quarterly data. Secondly,
we also need to deflate the nominal data the actual data’s deflator. Using the index of disposable
income of urban households is a common approach. However, National Bureau of Statistics did
not give the index of the quarter of the disposable income of China in order to match the data; we
decided to use urban residents’ price index to calculate the deflator. With multiplication of the
quarter’s urban residents with the annulus data, we can get the fixed base quarter price index of
urban residents. The selected base period is December 2002.
(2) Real per capita consumption expenditure of urban residents, real per capita cash income of
rural households, real per capita consumption expenditure of rural households.
In the same way, we handle these data into quarterly data of the urban and rural areas.
(3) Real interest rates.
Since this article takes the quarterly data, and uses the Fisher equation (   r  i ), I use
after-tax deduction of three-month nominal deposit rate minus the current rate of urban consumer
price’s changes, and then real interest rates can be obtained.
Thus, descriptive statistics results of the main variables above in the regression analysis are
in Table 1 below.
12
Research of Mathematical Economics No. 1 2011
Table 1
Descriptive Statistics Results of the Variables
Variables
Mean
Median
Standard
Deviation
Max
Min
Urban per capita actual disposable income
2591.643 2526.235 466.1033 3619.182 1960.499
Rural real per capita cash income
887.749 836.417 279.294 1370.027 421.8753
In- urban per capita actual disposable income
7.8452
7.8345
0.1750
8.1940
7.5810
In- rural per capita actual disposable income
6.7393
6.7291
0.3281
7.2226
6.0447
1.3001
2.8210
15.5045 27.0464 -23.9000
0.6587
0.3493
37.2562 52.6574 -54.6100
Growth rate of urban per capita actual disposable income
Growth rate of rural real per capita cash income
Urban real per capita consumption expenditure
1922.109 1884.996 253.4658 2378.568 1450.421
Rural real per capita consumption expenditure
498.7701 477.9494 115.3829 700.0013 296.4340
In- urban real per capita consumption expenditure
7.5528
7.5417
0.1327
7.7743
7.2796
In- rural real per capita consumption expenditure
6.1856
6.1695
0.2392
6.5511
5.6918
Growth rate of urban real per capita consumption expenditure 1.3015
-0.5442 10.9334 17.2387 -17.5775
Growth rate of rural real per capita consumption expenditure
0.7738
12.216 23.67100 25.2864 -39.0509
Urban real interest rates
0.8291
0.4778
1.2581
3.6633
-1.1525
Rural real interest rates
0.6158
0.4674
1.1467
2.8643
-1.3548
If the random process is non-stationary, then usually, it must be very difficult and imprecise
to use a simple algebraic model to reflect the past and future time series. Therefore, we usually
require that the time series are stationary. Generally we use the ADF test, including the choice of
three parameters: intercept, time trend term, the lag periods. In addition, for the traditional
econometric theory, people think that the overall economic time series are identified as having a
fixed trend, therefore, the general practice is to get rid of overall fixed trend of economics from
fixed-trend model, and then the series are stationary which can be analyzed. However, Nelson
and Plosser (1982) found that the majority of overall economic time series have a stochastic trend,
so I just get rid of the fixed overall trend of economic time series, and did not remove the time
series of the stochastic trend. Then the analysis turns to be a big problem.
In addition to the amendments to the traditional unit root test, Ng and Perron (2001) have
13
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural
Areas
modified information criterion for the selection of the most suitable augmented lag phases, called
the modified AIC, MAIC and modified SIC, MSIC and so on.
In reality, the time series data are often non-stationary processes that cannot be used in the
classical regression model, or will cause regression error (Granger, 1987). However, if stable
relationship exists between the long-term time series of the same order, regression equation
(co-integration) can be made by the classical regression model.
From the above theory, in order to ensure the reliability of estimates, we must firstly
consider the time series of each variable must satisfy a single whole order. Under the modified
AIC, the method used to test is ADF test. Table II shows the variables of the ADF test, ct and
yt are I (1) processes, the consumption of rural areas is I (1) process and rt is a stationary
series.
Use Eviews to do ADF test, with intercept models, with trend models, neither trend nor
intercept models, and also both with the trend and intercept model. Test results are as follow:
14
Research of Mathematical Economics No. 1 2011
Table 2
Variable Unit Root Test Results
Test form
Variable category
p-value
ct (Urban)
ct (Rural)
yt (Urban)
yt (Rural)
ct (Urban)
ct (Urban)
ct (Urban)
ct (Rural)
ct (Rural)
ct (Rural)
yt (Urban)
yt (Urban)
yt (Urban)
yt (Rural)
yt (Rural)
yt (Rural)
rt (Urban)
rt (Urban)
rt (Urban)
rt (Rural)
rt (Rural)
rt (Rural)
0.9562
(0,0,0)
0.6865
(0,0,0)
0.9848
(0,0,0)
0.3566
(0,0,0)
0.0000
(1,1,1)
0.0001
(0,0,1)
0.0000
(1,0,1)
0.0005
(1,1,1)
0.0000
(0,0,1)
0.0000
(1,0,1)
0.0001
(1,1,1)
0.0000
(0,0,1)
0.0000
(1,0,1)
0.0013
(1,1,1)
0.0001
(0,0,1)
0.0000
(1,0,1)
0.0020
(1,1,0)
0.0003
(1,0,0)
0.0014
(0,0,0)
0.0035
(1,1,0)
0.0005
(1,0,0)
0.0006
(0,0,0)
(Intercept, Trend, Lag phases)
Conclusion
Non- stationary
Non- stationary
Stationary
Stationary
Stationary
Stationary
Stationary
Stationary
To prove that the time series is stationary, it must meet the test results of three types of
models (with intercept model, with a trend term model, both a trend and intercept term model)
are stable. So, under the significance level of 0.05, from the p-value of Table 2, we could get
0-order, that all the p-value of ct (urban), ct (rural), yt (urban), yt (rural) are over 0.05. So
every variable is non- stationary. However, after a lag, also from the p-value, every variable can
15
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural
Areas
pass the test, 1-order steady. Therefore equation (6) can be analyzed by quantitative analysis.
4.2 Econometrics Results
The above analysis shows that all of the variables and explanatory variables are stationary
time series of regression model, which meets the requirements to establish the mathematics
model. Thus, we can use a mathematical model which was established to calculate, using Eviews
second-order least squares to get the regression results shown in Table 3:
Table 3
Regression Result
Adjusted
Set of instrumental variables (***)
Category (OLS)
λ(*)
Intercept
θ(*)
R2 (**)
urban
rural
urban
rural
urban
rural
urban
rural
1.71
3.42
0.49
0.50
0.50
-4.84
0.59
0.87
(.43)
(.14)
(.00)
(.00)
(.45)
(.03)
(.00)
(.00)
yt  2 , yt 3 , yt  4
4.59
5.23
0.33
0.48
-4.25
-7.01
0.61
0.83
rt  2 , rt 3 , rt  4
(.15)
(.13)
(.07)
(.00)
(.13)
(.04)
(.00)
(.00)
ct 2 , ct 3 , ct 4
4.77
5.28
0.35
0.48
-4.49
-7.10
0.60
0.83
rt 2 , rt 3 , rt 4
(.13)
(.12)
(.07)
(.00)
(.12)
(.04)
(.00)
(.00)
0.79
5.90
0.50
0.48
-0.79
-6.20
0.64
0.82
(.78)
(.09)
(.00)
(.00)
(.72)
(.04)
(.00)
(.00)
1.90
5.90
0.47
0.48
-1.93
-6.20
0.63
0.82
(.51)
(.09)
(.01)
(.00)
(.42)
(.05)
(.00)
(.00)
None
yt  2 , yt 3 , yt  4 , yt 5 , yt  6
r t  2 , r t 3 , r t  4 , r t 5 , r t  6
ct  2 , ct 3 , ct  4 , ct 5 , ct 6
r t  2 , r t 3 , r t  4 ,, r t 5 , r t 6
Note: (*) Under probability at 5% confidence level coefficient’s The probability levels, that’s t-value.
(**) The probability levels under all regression coefficients are 0, F-statistics of the whole equation.
(***) Instrumental variables set includes constant item, but save them in table.
The first column is a set of selected instrumental variables the second column gives the
intercept of two stage least squares (2SLS) regression results. The third column is the consumers’
16
Research of Mathematical Economics No. 1 2011
income proportion of total income, of which the consumers satisfy "Rule of Thumb". The fourth
column is the interest rate’s intertemporal elasticity of consumption. The last column is adjusted
R 2 , while each column shows the coefficients of the measurement of regression of urban and
rural areas respectively, values in brackets are coefficient under t test’s p-value.
First, no set of instrumental variables’ least squares regression results show that intercept
and
 of the town are not significant, which in line with the results of earlier data analysis in
this article; secondly, with a lag of two to six after the instrumental variables y , r and c, r ,
their least squares regression results of second-order show that urban with intercept and
 are
not significant; finally, containing the lag two to four of the instrumental variables y , r and
c, r , the second least squares regression results of them are more stable.
When considering the income proportion of the consumers satisfying "rule of thumb" is only
 of total income, although in no set the value of the results of the least squares instrumental
variables and the results with lags two to six after the second-order least squares instrumental
variables y , r and c, r of urban and rural areas are nearly equal, due to other poor
regression results, they should be abandoned; Containing the lag two to four of the instrumental
variables y , r and c, r , after using the second-order least squares,
while
 (Urban) is about 0.33,
 (Rural) about 0.48, which means the urban consumers is greater rational expectations
than the consumers in rural areas, more generally, the higher income people get the higher
proportion of rational expectations consumers will be (Campbell and Mankiw, 1989), so in order
to expand consumption, firstly we should aim at increasing rural people's income, followed by
increasing the incomes of the low-income people in urban.
Considering about the interest rate’s intertemporal elasticity, we found that all the results are
showing that urban is lower than rural, while with the lag two to four of the instrumental
variables y , r and c, r
 (Urban) is about -4.3, while  (Rural) is about -7.0, generally
lower interest rates can help to promote growth in consumption; The main reason is reflected in
two aspects. Direct utility is mainly reflected in that lower interest rates reduce the opportunity
cost of future transactions so that people will choose more current consumption, thus stimulate
consumption; From the point of view of indirect utility, as the investment diversifies among
urban residents, the risk can be passed on by other means, compared with single investment in
17
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural
Areas
rural areas, and thus the risk-shift capacity is weak, then a larger intertemporal elasticity appears.
5. Conclusions
Through quantitative analysis of this paper, I think we can improve the level of consumption
of Chinese urban and rural residents from the following aspects:
(1) Lowering interest rates contributes to growth in consumption. Different from previous
researches on consumption which come to the conclusion that consumption is not sensitive to
interest rates, I believe that lowering interest rates is an effective measure of China’s
consumption. Of course, while lower interest rate brings growth to consumption, we also need to
pay attention to the possible cost of the inflation.
(2) It is highly recommended to continue sparing no effect on economic growth and improving
national income levels, thereby stimulating consumption and thus promoting economy growth.
Particularly, in the face of significant "short-sighted" behavior of the rural residents, great efforts
should be made to improve the income level of rural population and poor urban population.
(3) If the "short-sighted" consumer account for the main component of total consumption, and
rapid growth in total income cannot be achieved, then as long as the benefits of consumption
growth are greater than the cost of income redistribution, the redistribution of income may
become a possible way to stimulate consumption demand.
(4) It is suggested to establish a convenient and wholesome credit environment available to the
public. In an incomplete market, residents have a great tendency to be "self-insurance", which is
expected to increase savings in order to alleviate the impact of the uncertainty of income. Thus, a
comprehensive credit market is conducive to smooth their consumption and reduce the
uncertainty of the future, thus contributing to the increase in current consumption. Improving the
credit market and the social security system are the fundamental way of expanding domestic
demand.
(5) The government should focus more on accelerating the establishment of market-oriented
social security system, strengthening the support for social security funds, and maximizing the
possible scope of coverage of the population to eliminate the worries of the residents and to
change the residents of their expected future uncertainty. Because of the imperfection of credit
18
Research of Mathematical Economics No. 1 2011
market in our country, the fees in medical, education and other industries will strengthen the
residents of their precautionary saving motive, thus inhibit the growth of consumption
(6) The government also needs to vigorously promote the development of consumer credit
markets to ease residents of their liquidity constraints. To be more specific, in the present
circumstances, what we need to do is: firstly, to gradually realize mercerization of the interest
rate market, thus to provide both supply and demand sides with a relaxed environment; secondly,
to speed up banking reform and simplify loan procedures, and establish wholesome mechanisms
of internal constraints; thirdly, to gradually improve individual credit system; and finally, to
further improve relevant laws and regulations of consumer credit, thus to maintain the normal
order for the credit economy.
References
杭斌、申春兰, 经济转型期中国城镇居民消费敏感度的变参数分析, 数量经济技术经济研究第 9 期,
2004.
李焰, 关于利率与我国居民储蓄关系的探讨, 经济研究第 11 期, 1999.
龙志和、周浩明, 中国城镇居民预防性储蓄实证研究, 经济研究第 11 期, 2000.
施建淮、朱海, 中国城市居民预防性储蓄及预防性动机强度: 1999-2003, 经济研究第 10 期, 2004.
宋铮, 中国居民储蓄行为研究, 金融研究第 6 期, 1999.
陶长琪、齐亚伟, 转轨时期中国城乡居民预防性储蓄比较研究——中国城乡居民消费的理论框架及实
证研究, 消费经济第 5 期, 2007.
万广华、张茵、牛建高, 流动性约束、不确定性与中国居民消费, 经济研究第 11 期, 2001.
汪红驹、张慧莲, 不确定性和流动性约束对我国居民消费行为的影响, 经济科学第 6 期, 2002.
袁志刚、宋铮, 城镇居民消费行为变异与我国经济增长, 经济研究第 11 期, 1999.
朱信凯, 流动性约束、不确定性与中国农户消费行为分析, 统计研究第 2 期, 2005.
朱春燕、臧旭恒, 预防性储蓄理论——储蓄 (消费) 函数的新进展, 经济研究第 1 期, 2001.
Campbell, J. and Mankiw, G., “Consumption, Income, and Interest Rates: Reinterpreting the Time Series
Evidence”, NBER Working Paper, 1989.
Hall, Robert E., "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and
Evidence", Journal of Political Economy, 1978, 86(6), pp. 971-987.
Hall, R., “Intertemporal Substitution in Consumption”, Journal of Political Economy, 1988, 96(2), pp.
339-357.
19
The Effect of Income and Interest Rates on the Consuming Behavior of Chinese
Residents----Based on the Comparative Analysis on Statistics of Chinese Urban and Rural
Areas
Ng, Serena and Pierre Perron, “Lag Length Selection and the Construction of Unit Root Tests with Good
Size and Power”, Econometrica, 2001, 69(6), pp. 1519-1554.
(Responsible Editor: Xu Lingjue) (Proofreader: Wang Jue)
20
Research of Mathematical Economics No. 1 2011
The Influence of Wealth Gap on China’s Economic Growth
Guo Yumei
(School of Economics, Renmin University of China)
Abstract: With the development of China’s economy, the gap between the rich and the poor is increasingly
severe day by day. People hold different opinions on the influence of the gap. This paper wants to use
Cobb-Douglas production function as the basis of the model and introduces an income distribution coefficient t to
represent Gini coefficient in order to show the positive effect of the gap of wealth. At the same time, we try to
prove the positive effect of the gap through ternary linear regression based on China’s GDP, capital and industrial
income from 2003 to 2008. The positive effect of the gap through ternary linear regression based on China’s GDP,
capital and industrial income from 2003 to 2008 will be proved.
Keywords: The Gap of Wealth, Economic Growth, Reinvestment, Gini Coefficient
1. Introduction
As the reform and opening-up policy goes, China’s economic makes a big difference.
However, the gap of wealth has been widening day by day. China’s GDP reached 34.05 trillion
RMB in 2009 while it is just about 0.36 trillion in 1978, leaping to the third place in the world.
The average real economic growth is over 9.8%, which is a rare occurrence in the development
history. Though the world economy slowed down in 2008, the economic growth of China was
still above 9%. At the same time, as the World Bank said, Gini coefficient of China in 2005 was
over 0.42, which broke through the world cordon. The income of the first 20% people who gets
the highest income is ten times much more than the income of the last 20%.
The gap between rich and poor is a two-edged sword. On one hand, the expansion of the gap
of wealth will contribute to economy. As the rule of diminishing marginal consumption, the gap
will increase savings which will accelerate capital investment. The more capital is invested, the
more rapidly economic will grow. On the other hand, the expansion of economic gap between
rich and poor has inhibition and will cause social problems. Again according to the principle of
21
The Influence of Wealth Gap on China’s Economic Growth
diminishing marginal consumption, the poor consumption ratio is on the increase and rich
consumption ratio is reduced, which will lead to insufficient total consumption demand.
Meanwhile, the gap has caused serious social phenomenon such as anxiety. When overnight and
night laid-off appear frequently, people begin to come up with uncertainty about the future and
are discontent with the gap.
Scholars have different opinions on the influence of wealth gap on China’s economy. Sun
Liping says it is the insufficiency of consumption demand that matters instead of increasing
reinvestment. Combining static and dynamic analysis method, time series data and section data,
Liu Ying thinks about the relationship between income distribution and economic growth of
Beijing based on its Gini coefficient and GDP and she finds the positive correlation. But Xiang
Shujian finds the unemployment rate is the most prominent among the four explanatory variables
of unemployment rate, the first industrial added value of GDP, social relief pension financial
expenditure and average wage growth in relation with Gini coefficient through multiple
regression models. From this, he concludes that if the income distribution is even or not has
nothing to do with economic growth.
However, according to current situation of China’s economic development, a widening gap
between rich and poor influences economic growth dominated by promoting effect in general. As
we all know, China is in and will remain so for a long time of socialist development stage.
Therefore, the effect of capital accumulation and capital reinvestment on economic development
is rather obvious. Since 2005, total capital formation of GDP by expenditure has been above 45%.
Although many scholars believe that the promoting influence of capital is no longer as significant
as it is in 1990s, the negative effect caused by insufficient domestic demand cannot be compared
to the positive effect brought by capital.
Thus, this paper would make some improvements on the basis of past research. First, we
will use Cobb-Douglas production as the basis of the model and introduce an income distribution
coefficient t to represent Gini coefficient. With the rule of diminishing marginal consumption, we
take the first-order of production function to intuitively see how the widening gap influences
economic growth. So far, many researches are based on Solow model which focuses on capital
per unit labor and output per unit labor. The author thinks that capital per unit labor is not
convenient to describe the gap between rich and poor. Capital per unit labor is the average of
capital in labor, which is contradictory to reflect the difference in wealth. Thus, we introduce t
22
Research of Mathematical Economics No. 1 2011
coefficient to make the model simple and intuitive. Second, we will take a ternary linear
regression based on China’s GDP, capital and industrial income from 2003 to 2008 in order to see
the correlation between the gap and economic growth. Instead of word describing, we show the
comprehensive effect of wealth gap in econometrics method.
The structure is as follows: the first part is mainly about the situation and impact of the gap
between rich and poor and the author’s opinions and perspective; in part two, we would build a
simple model to find the function of economic growth involving GDP, capital and the gap in
order to prove that the expansion of the gap of wealth contributes to the reinvestment of capital
and it will promote the economic growth; in part three, we try to prove the positive effect of the
gap through ternary linear regression based on China’s GDP, capital and industrial income from
2003 to 2008 according to the function in the second part; finally, we will comprehensively make
some conclusions and properly make suggestions.
2. Theoretical Model
From the standpoint of capital accumulation, we try to prove that the expansion of the gap
of wealth will contribute to the reinvestment of capital and it will promote the economic growth
in the end.
We would like to build a simple model to represent the economic environment with the view
of that inequality will increase output because Gini coefficient is in direct proportion to the rate
of capital reinvestment and the growth rate. Alien from the previous research, we want to make
some assumptions as below: Firstly, the society is made up of two groups and each group may
have one person or more and we regard each group as a labor unit. But in this paper, we just want
to focus on the gap of wealth between the two groups regardless of the gap within each group.
Secondly, the two groups produce the same product while we don’t distinguish wage income or
capital income. The way to distribute income between the two groups is on the basis of the ratio
of capital each group has. Thirdly, we introduce the t index to represent the gap so that we can
indirectly see the relationship between Gini coefficient and capital reinvestment. Fourthly, loan
market is unavailable and we don’t need to take r into account. Lastly, we assume a concave
consumption function, which indicates that the richer group has higher marginal propensity to
save than the poorer group.
23
The Influence of Wealth Gap on China’s Economic Growth
In the model, we will begin by considering the Cobb-Douglas production function:
Y  K  L ,
where 0    1,0    1 . Since the society is made up of two labor units, we can change the
production function into that:
Y  BK 
And we assume it is a concave returns to scale production function. Also, we can reach that:
Y  Y1  Y2
K  K1  K 2
As we distribute income between the two groups on the basis of the ratio of capital each
group has, we assume that the capital group Ⅰ has is no more than the capital group Ⅱ has. And
the proportion of group Ⅰ’s capital is t so that 0  t  0.5 . Thus,
K1  tK
K 2  (1  t ) K
Y1  tY
Y2  (1  t )Y
If the total capital is K 0 at beginning, the group Ⅰ has K1  tK 0 and group Ⅱ has
K 2  (1  t ) K 0 . So the output or income is distributed as:
Y0  BK 0
Y10  tY0
Y20  (1  t )Y0
At this time, each group begins to decide how much to make consumption and how much to
reinvest. Due to the concave consumption function, the proportion of income group Ⅰ makes
reinvestment is lower than that of group Ⅱ. If we assume that p is the reinvestment proportion, q
is proportion of income to the total income, p is the function of q and Y, p (qY ) , and p (qY )
is an increasing function involving qY . Thus,
24
Research of Mathematical Economics No. 1 2011
K1  p(tY0 )tY0
K 2  p(1  t )Y0 (1  t )Y0
K  K1  K 2  p(tY0 )tY0  p(1  t )Y0 (1  t )Y0
From the production function, we obtain,
Y
K
L


Y
K
L
For L = 2, then L = 0 and
Y
K

Y
K
Thus, when the capital is K 0 , the relationship between reinvestment and output growth is
showed below:
p(tY0 )tY0  p(1  t )Y0 (1  t )Y0
Y
K


Y0
K0
K0
(1)
This is a growth function of t when K 0 is given. In fact, t represents the gap of wealth.
Due to 0  t  0.5 , the gap of wealth is larger when t converges to 0 and smaller when t
converges to 0.5. So we can obtain:
t  G (Gini ) , 0  t  0.5 , G '(Gini )  0
(2)
which means that t is inversely proportional to Gini coefficient and the function is monotonic
decreasing. Then, make derivation of function (1):
 Y 
p ' (tY0 )tY02  p(tY0 )Y0  p ' (1  t )Y0 (1  t )Y02  p(1  t )Y0 Y0

  
K0
 Y0 
'
Here, we assume that p (qY ) is linear,
p(qY )  a * t * Y  b
where a > 0, then,
p ' (qY )  a
Thus,
25
The Influence of Wealth Gap on China’s Economic Growth
 Y 
atY02  p(tY0 )Y0  a(1  t )Y02  p(1  t )Y0 Y0

  
K0
 Y0 
atY02  a(1  t )Y02   p(tY0 )Y0  p(1  t )Y0 Y0 

K0
'


where 0  t  0.5 , then 0  t  1  t . Thus,
atY02  a (1  t )Y02  0
Given that p (qY ) is an increasing function involving qY ,
p(tY0 )  p(1  t )Y0 
p(tY0 )Y0  p(1  t )Y0 Y0  0
With all above, we can conclude that
 Y 
atY02  p(tY0 )Y0  a(1  t )Y02  p(1  t )Y0 Y0

  
0
K0
 Y0 
'
And only when t = 0.5, we can obtain the equation. So, we can learn that
monotonic decreasing about t, which means that the smaller t is, the larger
and (3),
(3)
Y
is
Y0
Y
is and with (2)
Y0
Y
is directly proportional to Gini coefficient. Thus, the expansion of the gap of
Y0
wealth will contribute to economic growth.
3. Empirical Research Based on Chinese Economy
From the simple model in part 2, we can conclude that, theoretically, the expansion of the
gap of wealth would contribute to reinvestment which will result in the growth of economy.
Besides the influence of t index, Y0 has an effect on reinvestment next year. Thus, we’d like to
find the empirical evidence of the relation between the gap and economic growth with the data of
capital, GDP and Gini coefficient.
From the model, we can learn that:
26
Research of Mathematical Economics No. 1 2011
p(tY0 )tY0  p(1  t )Y0 (1  t )Y0
Y
K


Y0
K0
K0
t  G (Gini )
Economic growth rate is a function involving GDP, total capital and Gini coefficient. Thus,
we try to find the relationship between growth rate and Gini coefficient adopting linear regression.
We would take growth rate as the dependent variable and GDP, total capital and Gini coefficient
last year as independent variable.
First of all, we will show the method of calculating Gini coefficient. Firstly, population is
divided into N groups and was labeled A1 , A2 ,
, An by the average wage of each group on
lower-to-upper motion. Then we can get the percentage of the first i group’s total wage named
Yi and the percentage of the first i group’s total population named X i . Y0 and X 0 are equal
to zero. After that, we get the equation of calculating Gini coefficient with its definition.
Graph 1
Lorenz Curve
27
The Influence of Wealth Gap on China’s Economic Growth
1 n 1
 1 1 n
    X i  X i 1 Yi  Yi 1    X i  X i 1  Yi 1 
   X i  X i 1 Yi  Yi 1 
2 i 1  2
2 2 i 1

Gini 

1
1
2
2
Thus, with the wage and population of 19 industries, we can obtain Gini coefficient from
2003 to 2007.
By now, specific date is showed below,
No.
G (growth rate)
K (capital)
Y (GDP)
Gini
1
0.17710944
0.268341054
135822.8
0.472367
2
0.156738594
0.259603788
159878.3
0.46738
3
0.169662816
0.239086846
184937.4
0.461608
4
0.228814633
0.24841952
216314.4
0.457987
5
0.181464375
0.258545672
265810.3
0.450935
The result of linear regression by SAS is that
g = -8.77045 + 0.000018K - 8.23E - 6Y + 19.17453Gini
(0.5984) (7.666E-7)
(3.965E-7) (1.241567)
2
N=5, R  0.99852 , R  0.99408
2
2
From R 2 and R , we can learn that GDP, total capital and Gini coefficient explain
almost all the changes of growth rate. The degree of fitting is high enough to guarantee
credibility of the regression.
By the way, Gini coefficient has a positive correlation with the growth rate that the growth
rate will increase about 20% if Gini coefficient increase 1%, for the t-value is 15.44. So we can
reject H0 at significance level of 5%.
Summarily, we can make a conclusion based on the regression that the gap will contribute to
economic growth.
4. Conclusions
From the paper, we can conclude that:
28
Research of Mathematical Economics No. 1 2011
(1) Based on the model and regression, the widening gap of wealth will contribute to China’s
economic growth in nowadays;
(2) The Gini coefficient in China is about 0.45, which has crossed the international warning line
of 0.4. Therefore, the gap of wealth is a two-edged sword. We cannot blindly negative the gap
between rich and poor. Of course, the increasingly widening gap may cause serious problems
because Chinese regard common prosperity as development goals; meanwhile, complete equality
is not desirable as it will slow growth steps down. Thus, state and local government should take
some measures to maintain the widening gap in a modest and relatively stable level.
From part three, we can learn that the gap in China is about 0.45. According to the UN
organizations, income is absolute average if Gini coefficient below 0.2, while compare average if
0.2-0.3. Gini coefficient of 0.3-0.4 means relatively reasonable and 0.4-0.5 indicates a bigger gap.
And income distribution is extremely bad if Gini coefficient is over 0.5. So, the gap is bigger in
China. Enjoying the fast economic development, the government should take timely measures,
such as tax reform policies, to actively deal with the widening gap and try their best to guarantee
the GINI coefficient around 0.3 to 0.4. Together with the fast development, the government also
needs to stop the reproduction of economy.
However, we wonder if the gap of wealth in China is reasonable. China is a socialist country,
with economic development and common prosperity for a common goal. Given that common
prosperity is established on the basis of development, to realize common prosperity must develop
economy. This will inevitably cause a widening gap between rich and poor. It is easy to
understand that in certain period the richer get richer and the poor get poorer. It is said in Bible
Matthew: a king gives 1000 coins to three people. After a period of time, the first person changes
1000 coins into 5000 coins and the second person gets 2000 coins while the last person is still
holding the 1000 coins. Therefore, the king gets back the 1000 coins and gives it to the first
person. It is the famous Matthew Effect, which seems unfair but is reasonable indeed. For one,
people who have a lot are almost people who have succeeded. They obtain social recognition and
trust, which makes them easier to get more chances. Conversely, people who get little social
recognition and trust have less opportunities. At the same time, rich people are more able to make
big investments. As the effect of capital accumulation is not simply the sum of unit capital, the
expected income of big investment is much higher. This is obviously one of the reasons why the
gap is widening. However, from the angle of efficiency and fairness, these reasons are anyhow
29
The Influence of Wealth Gap on China’s Economic Growth
reasonable.
References
郭熙保, 经济发展: 理论与政策, 中国社会科学出版社, 2000.
李实、赵人伟等, 中国经济改革中的收入分配变动, 管理世界, 1998(1): 43—56.
刘长庚、吕志华, 改革开放以来我国居民边际消费倾向的实证研究, 消费经济, 2005(8): 21-4.
刘红慧, 转型期经济增长与收入分配关系研究, 硕士学位论文.
刘颖, 北京市收入分配与经济增长关系实证研究, 硕士学位论文.
权衡, 收入分配与经济增长的相关性现代分析框架及其争论, 社会科学, 2004 年第 6 期.
王德祥、李建军, 财政分衩、经济增长与外贸依存度----基于 1978-2007 年改革开放 30 年数据的实证
分析, 开放战略探索三十年回归与展望.
向书坚, 我国城镇居民收入分配差距变化定量分析Ⅲ, 当代经济科学, 1998(20): 1-5.
杨俊、张宗益, 收入分配与经济增长近代西方理论与实证考察, 重庆大学学报, 2002, 25(10).
叶礼奇, 基尼系数计算方法, 中国统计, 2003(4).
张汉斌、刘庆华, 浅谈贫富差距对经济增长的影响》, 邢台职业技术学院学报, 2000, 17(2).
朱平, 贫富差距的合理性及其限度, 南京师大学报 (社会科学版), 2001(05).
Alesina A., Perotti R., “Income distribution, political instability and investment”, 1996(06).
Alesina A., Rodrik D., “Distributive politics and economic growth”, 1994.
Barror, “Government spending in a simple model of endogenous growth”, 1990(02).
Barror, “Inequality and Growth in a Panel of Countries”, 2000.
Bourguignon F., “Pareto superiority of unegalitarian equilibria in Stiglitz's model of wealth distribution with
convex saving function”, 1981(49).
Deininger K., Squire L., “Measuring income inequality: A new data-base”, 1996(03).
Deininger K., Squire L., “Inequality and growth: Results from a new data set”, 1998.
Persson T., Tabellini G., “Is inequality harmful for growth? Theory and evidence 1994 development”, 1999.
Romer P. M., “Increasing Returns and Long-Run Growth”, 1986.
Stiglitz J. E., “The distribution of income and wealth among individuals”, 1969.
(Responsible Editor: Xu Lingjue) (Proofreader: Wang Jue)
30
Research of Mathematical Economics No. 1 2011
Bundling Sales of Information Goods: Models and Analysis
Shi Fangning
(School of Economics, Renmin University of China)
Abstract: This paper studies the impact of bundling sales on merchandising of information goods. Our analysis
focuses on seller’s profits and consumer’s surplus, as well as the optimal bundle prices. We consider two common
bundling types: bundling of single-product in amount and bundling of multiproduct. For the two cases, we
develop models to calculate the optimal pricing and its condition, profits and consumer surplus. Our analysis
implies that, under certain situation, bundling sales yields more profits, but harms the consumers’ interest.
Keywords: Information Goods, Bundling Sales, Price Discrimination, Optimal Pricing
1. Introduction
Information good is a type of commodity aiming at spreading information with its main
value derived from the information it contains. In essence, any products which can be digitized
are information goods, such as newspapers and magazines, movies and music. Usually,
information products also contain information service, that is, the seller provides services like
collecting and processing information, or provides a platform where users can share information,
such as search service on web, mailbox and MSN. In this paper, we study only information goods
which can be digitized and traded, like software, newspapers, database, financial information and
research reports and so on, but not information service.
Information goods have several characters which make themselves distinctive from other
commodities. First, values of information goods depend on individual’s preference. Different
consumers value the same information goods bundle differently. For example, individual’s
knowledge, the ability to use the commodity, the necessity of the information for the person,
these factors will all affect individual’s value of the information goods. Consumers do not
consider the costs or the sources of the information but the utility they bring. Second, information
goods have the feature of high fixed costs and low marginal costs. Obviously, it determines that,
31
Bundling Sales of Information Goods: Models and Analysis
to make a lucrative investment, information goods cannot be sold by its marginal price. This
character also means that there exists explicit scale economy, that is, the more the information
goods are produced, the lower the average cost is. So it is more proper to bundle a large number
of information goods than other commodities. On the other hand, this character also leads to
natural monopoly within this industry. Third, information goods have strong network effect.
Growing market penetration or access to the products helps improve their function. For instance,
more users make software spread faster, and also lead to better function.
Bundling sales is a merchandising strategy that is often used by sellers to conduct price
discrimination or enhance their monopoly power. It means sellers provide goods in packages,
which can be the same goods provided in quantity-dependent packages or two even more
different goods provided in a package. Bundling sales is very common in daily life: airline
companies provide both single-way and round-trip tickets; banks offer a series of comprehensive
service which cannot be divided; restaurants usually provides complete dinners which contains
several different dishes; and daily commodities like toothpaste and detergent are sold in small
and large packages.
As for information goods, the cases of bundling sales are also very popular. In this paper, we
divide them into two broad classes: bundling of single-product and bundling of multiproduct.
Bundling of single-product means sellers provide the same goods in quantity-dependent packages.
When providing commodities like internet access and messages, sellers usually charge in this
way. For example, the following tables are price policies of China telecom Beijing Corporation
for 3G cellphones’ charge for Internet tariff and messages1.
1
32
Usually when consumers purchase for the next month’s cell phone services, they will always avoid using the
service exceed the amount contain in the bundle. So we see the price policies as bundling sales, but not two-part
pricing.
Research of Mathematical Economics No. 1 2011
Basic Tariff (per month)
Internet Tariff of the Bundle
(per month)
Tariff for Excessive Amount
5 yuan
30MB
0.005yuan/KB,with upper limit
500 yuan
10 yuan
100MB
20 yuan
200MB
50 yuan
1G
100 yuan
2G
sources: http://bj.ct10000.com/pages/tianyi_taocan5_1.action
Amount of Messages of the Bundle
(per month)
Tariff for Excessive Amount
5 yuan
60
0.10 yuan per message
10 yuan
130
20 yuan
200
50 yuan
300
Basic Tariff (per month)
100 yuan
500
sources: http://bj.ct10000.com/pages/tianyi_taocan5_1.action
Bundling of multiproduct means the seller provides two or more different goods which are
similar to each other in a package. For example, Microsoft sells its office software in bundle;
some websites provide financial information and research reports of stock, time-bargain in
bundles; sales of music discs and newspapers are also cases of multiproduct bundling.
So far, theoretical research for bundling sales concentrated on two aspects. One is to use
bundling to conduct price discrimination and extract more consumer surplus. The phenomenon of
bundling sales was firstly referred to by Burnstein (1960) and Stigler (1980). After that, utilizing
the framework of Stigler, Adams and Yellen (1976) analyzed bundling strategies of the
single-product monopolist in a duple-product market. They compared the three ways of bundling:
pure components strategy (set the single price on each commodity separately), pure bundling
(offer two commodities for sale only in a package), mixed bundling (offer each commodity
separately and a package of both). They illustrated that in most situations, various types of
bundling can be more or less profitable than unbundling sales. Following this, Schmalensee
33
Bundling Sales of Information Goods: Models and Analysis
(1984) extended their model by assuming that the distribution of reservation prices follows a
bivariate normal distribution. He confirms that in the mixed bundling case, it is always more
profitable than unbundling sales if the two goods are complements. Also extending the Adams
and Yellen model, McAfee, McMillan and Whinston (1989) confirmed mixed bundling is often
more profitable than pure bundling, and also provided a general sufficient condition for the
situation when mixed bundling is optimal in a duple-product case. Salinger (1995) developed a
graphical analysis of bundling and illustrates the benefits of bundling depends on the cost of the
individual products, its relative magnitude to the reservation prices and the correlation of
demands between the products. More recently, McCardle, Rajaram and Tang (2007) studied the
impact of bundling products on retail merchandising. They established conditions and insights
under which bundling is profitable, and confirmed again that bundling profitability depends on
individual product demands, bundling costs, and the nature of the relationship between the
demands of the products to be bundled.
The other aspect is to use bundling as a market strategy, so that a monopoly firm can
enhance its monopoly power as well as extend its power to other newly emerged relevant markets.
M.D. Whinston (1990), S. Martin (1999), Carlton and Waldman (2002) have discussed on this
problem elaborately and derived some important results. But in this paper, we do not discuss
effects of bundling in monopoly power, so we don’t talk about these theories.
However, multiproduct of large bundles has received little attention until recent years.
Armstrong (1996) showed that in some certain cases, the optimal pricing in the multiproduct case
can be determined using the techniques typically used in the single-product case. He found that
the optimal bundle price will exclude some low-demand consumers. Bakos and Brynjolfss (1999)
studied the strategy of bundling a large number of information goods. They found that when the
marginal cost of information goods is close to zero, as the number of bundling goods increases,
the deadweight loss per good and the consumer’s surplus per good for a bundle converge to zero,
and the seller’s profit per good increases to its maximum value. That is to say, bundling of large
amount of information goods is profitable. By assuming that consumer types and consumer
valuations are discrete, they also found that, in many cases, consumers can be induced to reveal
information about their valuations through their choices by offering them a menu of bundles at
different prices. Thus the monopolist may gain profit by pursuing a mixed bundling strategy of
offering several bundles, each including a subset of the available information goods.
34
Research of Mathematical Economics No. 1 2011
Our paper differs from the previous work in two aspects. Firstly, in this paper, we consider
two different types of bundling sales, bundling of single-product and multiproduct, in terms of
price discrimination. For single-product case, we refer to Jean Tirol’s (1998) model of optimal
nonlinear pricing and compare the optimal prices, profits, consumers’ surplus and net welfare of
the total society of bundling and unbundling cases. We get one of our results, that is, for the
single-product case, bundling sales can bring more benefits to sellers, but reduce the consumers’
surplus and total welfare of the society in certain circumstances. For the multiproduct case, by
assuming the consumer types and valuations are continuous, we extend Bakos and Brynjolfss’s
(1999) model, which discusses two ways to segment the market and compare them with pure
bundling and unbundling sales. We get the other result that under certain circumstances, price
discrimination through mixed bundling is profitable and extract the consumers’ surplus more
efficiently than pure bundling and unbundling sales. Secondly, our results imply that, in the
single-product case, the effect of bundling is relevant to the difference between the consumers’
types. The bigger the difference is between the consumers’ types, the more consumers’ surplus
the seller can extract by bundling sales. In the multiproduct case, the effect of mixed bundling is
dependent on consumers’ average valuation of products in package and the degree of discount the
seller set. The two bundling patterns can only efficiently segment the market and make profits
when consumers’ average valuation is not too high and the seller set the discount large enough.
This paper is organized as follows. In the second section, we present our model and
analysis of sing-product bundling sales. We use this model to determine the optimal bundle prices
and prices of unbundling sales, seller’s profits and consumers’ surplus. We establish conditions
under which bundling is profitable. In section 3, we present our model and analysis of
multiproduct bundling sales. We discuss the “quality discount” and “quantity discount” of
bundling separately and repeat this analysis. In section 4, we present some intuitive explanations
for the results we get. In the conclusion section, we summarize our key results and provide future
research directions.
2.
Model and Analysis of Single-Product Bundling
When provide commodities like internet tariff and cellphone messages, sellers usually
provide different bundles containing different amount of products according to various consumer
35
Bundling Sales of Information Goods: Models and Analysis
types. Consumers choose the bundles which maximize their utility. In this way, sellers can
segment the market and extract more consumers’ surplus.
Now, we develop a model to illustrate that the bundling strategy is more profitable than
unbundling sales. To simplify the model, we assume that the sellers only provide two bundles
which contains q1 and q 2 amount of goods, aiming at high-demand consumers and
low-demand consumers separately. Consumer types are continuous, denoted by taste parameter
 
 ,    , . Usually the information goods tend to have zero or very low marginal costs of
production, so in our model, we ignore the effect of marginal costs.
First, we assume consumers have the following utility function2:
U  V (q)  T , if they purchase q units;
U  0,
if they purchase nothing.
where V (0)  0 , V (q )  0 and V (q)  0 (that is, in this utility function, consumers have
decreasing marginal utility); T (q ) is the consumers’ total expenditure of the q units of goods;
 is consumer’s taste parameter, differs from one person to another; V (.) is identical for all
 
the consumers. Then, assume taste parameter  follows uniform distribution in  ,  , with
f ( ) being the uniform density. Setting
f ( ) 

 f ( )d  1 ,
it is easy to know that
1
.
 
In this model, we can get the following two propositions in the single-product case:
Proposition 1: For information goods, a single-product bundling sale is more profitable
than unbundling sale.
Proposition 2: As long as  /  is large enough, consumers’ surplus under bundling sale
is less than that under unbundling sale; at the same time, bundling sales reduce net surplus of the
whole society.
Now we present the proof.
2
36
Refered in Jean Tirole, 1994, Industry Organization Theory (7e).
Research of Mathematical Economics No. 1 2011
2.1 Bundling Sales
If the seller provides bundle I with q1 amount of goods and price of T1 to consumers
with taste parameter
   , 0  , and provides bundle II with q 2 amount of goods and price
of T2 to consumers with taste parameter
 
   0 , , in which    / 2 , then the profit of
the seller is:
0


0
1   T1 f ( )d   T2 f ( )d
(1)
The seller will maximize this profit under two kinds of constraints.
The constraint of the first kind requires consumers to be willing to purchase, that is, the
individual rationality constraints.
V (q1 )  T1  0 , if    , 0  ;

V (q2 )  T2  0 , if    0 , .
That is,
V (q1 )  T1  0

 0V (q 2 )  T2  0
(2)
The constraint of the second kind requires that the consumers do not exercise personal
arbitrage. These are known as “incentive-compatibility constraints”. In particular, the
high-demand consumers should not want to consume the low-demand consumers’ bundle and
vice versa.
So,
V (q1 )  T1  V (q2 )  T2 , if    , 0  ;

V (q2 )  T2  V (q1 )  T1 , if    0 , .
From these constrains, we get,
0 
T2  T1
V (q 2 )  V (q1 )
(3)
Substituting for  0 in (1) using the formula (3), we yield the function of profit in term of
37
Bundling Sales of Information Goods: Models and Analysis
T1 , T2 and q1 , q 2 :
 1  T1 f ( )  ( 0   )  T2 f ( )  (   0 )





T2  T1
1  T2  T1
    T1   

  T2 
V (q 2 )  V (q1 ) 
   V (q 2 )  V (q1 )



(4)
T2  T1 2 
1 

T2  T1 

V (q 2 )  V (q1 ) 
  
Differentiating
 1 in respect to T1 and T2 separately, we have
 1
2(T2  T1 ) 
1 
1

   
 2 0   

T 1    
V (q2 )  V (q1 )    
 1
2T2  T1  
1 
1

  
   2 0

T2     V (q 2 )  V (q1 )    

It is obvious that

 1
 0 always satisfies. So combining with the inequality (2), the
T 1
seller will choose
T1  V (q1 ) .
Setting
(5)
 1
 0 , we get
T2
 0   / 2 , T2 
1
V (q 2 )  V (q1 )    V (q1 ) ;
2
(6)
The seller maximizes his profit at this point3. Substituting for T1 , T2 in function (4)
using (5) and (6), we can derive the maximum profit:
2

(  2 ) 2



V
(
q
)

 V (q1 )
max 1
2
4(   )
4(   )
Consumers’ surplus is:
3
38
The second-order condition is satisfied, for
 2 1
2

0
2
T2
    V (q 2 )  V (q1 )


Research of Mathematical Economics No. 1 2011
CS1   V (q1 )  T1   f ( )d   V (q 2 )  T2   f ( )d
0


0
0
 
1

  V (q1 )  V (q1 )  f ( )d   V (q 2 )  (V (q 2 )  V (q1 ))  V (q1 )  f ( )d

0
2


1 
1 2 1 2
1 2 1
1
1 2 

V (q1 )  (  0          0 )  V (q 2 )  (  0   0 )
2
2
2
2
2
2
  


  
1
4  
8  


2
2


2
   V (q1 )    V (q 2 )
2.2 Unbundling Sales
Assume the seller sets price at p for per unit of goods, then T (q ( ))  pq ( ) , and
each consumer chooses q ( ) to maximize his own utility: U (q )  V (q )  pq .
Setting U (q )  V (q )  p  0
,
we
get
p  V (q ) ,
the
profit
is

 2   pq( )  f ( )d .

The seller chooses p to maximize this profit.
Setting

 2
q 
  q  f ( )  pf ( )  d  0 , we can get the optimal value of p .

p
p 

Consumers’ surplus is,




CS 2   (V (q( )  T (q( )) f ( )d   (V (q( ))  pq( )) f ( )d
2.3 Comparison
To simplify the model, let V (q ) 
1  (1  q) 2
, and V (q)  1  q ,
2
(1) If the products are sold in bundles, the profit is,
39
Bundling Sales of Information Goods: Models and Analysis
2

(  2 ) 2
1 
 V (q 2 ) 
 V (q1 )
4(   )
4(   )
2
2
2
2q 2  q 2

(  2 ) 2 2q1  q1




2
2
4(   )
4(   )
The seller choose q1 and q 2 to maximize the profit, that is, q2  1 , q1  0 4. We get
the maximum profit:
2
2

   V (q1 )    V (q 2 )

1

,
 
max  1 
4(   ) 2 8(   )
Consumers’ surplus is,
CS1 
  
1
4  
8  

2
2
2

2

2
1





8    2 16(   )


(2) If the products are sold separately, the adverse demand function is p   (1  q) , the profit
is  2 



p


pqf ( )d   p(1 
Setting
) f ( )d  p 
p 2 (ln   ln  )
.
 
 2
2(ln   ln  ) p
 
 1
 0 , we get p 
,
p
 
2 ln(  /  )
max  2 
 
 
 


.
2 ln(  /  ) 4 ln(  /  ) 4 ln(  /  )
Consumers’ surplus is,

CS 2   (V (q( ))  pq( )) f ( )d 


1
 


( 
2q  q 2
 pq)d
2
1
3 (   )
(   ) 
4
8 ln(  /  )
We do not take the cost of production into consideration, so the values of q1 and q 2 do not affect the result we
derive.
4
40
Research of Mathematical Economics No. 1 2011
(3) Now compare seller’s profit with consumers’ surplus under the two cases.
2(   ) 2

 
Setting  1   2 
.

 0 , we get ln(  /  ) 
2
8(   ) 4 ln(  /  )

2
Rearrange the inequalities, we have
2(1   /  ) 2  ln(  /  )  0 .
Let’s define F ( x)  2(1  x) 2  ln x , x   /  , so
F ( x)  4(1  x) 
Obviously,
F ( x)  0
always
satisfies
1
1
 4x   4 .
x
x
x   / 
for
1
,
2
so
we
have
1
F ( x)  F ( )  0 .
2
That means
 1   2 , so bundling sales is always more profitable than unbundling sales.
Now compare the consumers’ surplus.


1

3   
  (   ) 
  0.
Setting CS1  CS 2 
8 ln(  /  ) 
16(   )  4
2
Rearranging the inequalities, we get

6( /   1) 2  3  / 

2

 4  ln(  /  )  0 .
(7)
Let’s define G( x)  6( x  1) 2  (3x 2  4) ln x , x   /  . Differentiating G ( x ) by
x , we have
G ( x)  12( x  1)  6 x  ln x  (3x 2  4) 
1
4
 9 x  6 x  ln x   12 .
x
x
It’s easy to know that G x  is always less than zero when x  2 . So as long as  / 
is large enough, inequality (7) can be satisfied, which means CS1  CS 2 .
As for the total social surplus
 1  CS1 and  2  CS2 ,
41
Bundling Sales of Information Goods: Models and Analysis
2
2


 
1
3 (   )
 1  CS1   2  CS 2 


 (   ) 
8 ln(  /  )
8(   ) 16(   ) 4 ln(  /  ) 4
2
3
1
1 (   )

 (   ) 
8 ln(  /  )
16(   ) 4
Setting
 1  CS1   2  CS 2  0 , we get
     
2
2( /   1) 2  4 ln  /    /  ln  /   0
Define T ( x)  2( x  1) 2  4 ln x  x 2 ln x , x   /  .
So,
T ( x)  4( x  1) 
4  x2
4
 ln x  2 x  3x  4   ln x  2 x
x
x
T ( x)  1 
4
 2 ln x  0
x2
When x is large enough, we have T ( x )  0 , T ( x )  0 , that is, as long as  /  is
large enough, there is
 1  CS1   2  CS2 .
So, we can know the quantity-dependant bundling of single-product is profitable.
Moreover, as long as the value of  /  is large enough, bundling sales will harm the
consumers’ surplus and even reduce the total surplus of the society.
3.
Model and Analysis of Multiproduct Bundling
In this section, we consider bundling strategies for multiproduct. Bundling of multiproduct
is also a widely-used market strategy. For example, music disc is a kind of bundle which contains
certain amount of songs; newspaper, like music disc, is also composed of various kinds of
information. Bakos and Brynjolfss (1999) illustrated that under the condition that consumer
valuations are independent, if the sellers bundle large amount of information goods together, the
law of large number implies that the total willingness to pay the bundle of consumers can be
similar. By setting the bundling prices, the sellers provide their information goods at the average
willing to pay of consumers. Thus as the number of bundling goods increases, the deadweight
42
Research of Mathematical Economics No. 1 2011
loss per good and the consumers’ surplus per good for a bundle converge to zero, and the seller’s
profit per good increases to its maximum value. Based on this result, they consider two ways for
the seller conducting price discrimination. One of them is “quality discount”, that is, to remove
or degrade a feature that disproportionately affects high-demand consumers. By offering
appropriately-priced bundles with and without this feature, the seller may be able to infer
consumers’ expected valuation for the bundle based on their purchasing decisions. The other way
is “quantity discount”, that is, to leave certain items out of some bundles. For example, the seller
can offer an “economy” bundle at a reduced price that is a subset of “premium” bundle. By
assuming the consumer types and consumer valuations are discrete, Bakos and Brynjolfss found
that, when the type of consumers satisfies certain conditions, the two bundling patterns can
efficiently segment the market and thus make more profit. In this section, we extend Bakos and
Brynjolfss’s model to a continuous case and prove that under certain circumstances, those
bundling strategies can still be more profitable, and extract the consumer’s surplus more
efficiently than pure bundling and unbundling sales.
Suppose consumers’ types change continuously, which is denoted by
 .  follows the
uniform distribution on an interval 0,1. Let f ( ) be the density and f ( )  1 . The higher
the value of
 , the higher the demand of consumers is. Consumers have either a high or a low
valuation for each information good, respectively denoted by vH and v L . Consumers of type
 value each good at vH with probability  ( ) , and at v L with probability 1   ( ) .
Assuming that  ( ) is strictly increasing on the interval 0,1. When
Additionally, we suppose
  0 ,  ( )   0 .
    0 and 1          . Then we can know, when
bundling large amount of goods, consumers of type
 value each good in the bundle at an
  vH  1    vL .
average
If the seller conduct unbundling sales, he may set a high price and selling only to the high
demand consumers separately; that’s to say, he chooses the price v H so that the profit is
 s1      f  d  v H , and the consumers’ surplus is S s1  0 . If not, he may set a low
1
0
price instead and sell to everyone. In this case, the price is v L , profit is  s 2  v L , and the

43
Bundling Sales of Information Goods: Models and Analysis
consumers’ surplus is S s 2  v H  v L 
    f  d .
1
0
If the seller conducts pure bundling sales, the seller’s profit is
 B  max  f  d    vH  1    vL   0vH  1  0 vL 5
1

Consumers’ surplus is:
S B     v H  1    v L  f  d   0 v H  1   0 v L 
1
0
1
 v H  v L     d  0 
 0

Now we will discuss the following two ways of price discrimination.
3.1 Quality Discount
Let the seller offer a “degraded” bundle where the value of goods for the high-valuation
consumer is reduced to
 ( )vH intended for consumers of type  . Assuming that  ( ) is
strictly increasing on the interval
0,1 ,
and  ( )   0 when
  0 ;  ( )  1 when
  1. Also  ( ) satisfies vL   ( )vH ,  ( )  0 .
In this model, we can get the following proposition:
Proposition
   
 0 0 
 0
When
3:

the
two
conditions,
     d  max  , 
1
0
 0 v L

 0 v H
1
2
and
are satisfied, conducting price discrimination by quality discount
will bring the seller more profit and reduce the consumers’ surplus at the same time.
Proof: selecting arbitrary
 0  0,1 , we know that for every  1   0 6, there are two
kinds of constraints.
To satisfy the individual rationality constraints, we have,
5
By differentiate


B
, we get
d B
 1         v H  v L   v L  0 , so set   0 , that
d
means the seller choose to set a low price and sell to everyone.
6
44
Case of
1   0
can be analyzed in the same way.
Research of Mathematical Economics No. 1 2011
p( 0 )   ( 0 )  0 v H  1    0 v L
p( 1 )   ( 1 )  1 vH  1    1 vL .
To satisfy the incentive-compatibility constraints, we have,
 ( 0 )  0 v H  1    0 v L  p 0    ( 0 )  1 v H  1    0 v L  p 1 
 ( 1 )  1 v H  1    1 v L  p 1    ( 1 )  0 v H  1    1 v L  p 0 
The seller will choose p 0  and p 1  as large as possible which satisfies the four
constraints. So we get
p( 1 )   ( 1 )  1 vH  1    1 vL
p( 0 )   ( 0 )  0     1    ( 0 )   1 v H  1    1 v L
 v L   ( 0 )  0 v H     1    ( 0 )  1 v H    1 v L
Because
 1 is chosen arbitrarily, so
p( 0 )  v L   ( 0 )  0 v H   min      ( 0 )  v H    v L 
 0, 0 
Setting T       ( 0 )  v H    v L , we have
T
     v H        0   v H    v L

 2T
     v H  v L   2    v H        0   v H
 2
 0 0   0 v L
 2T


Obviously,
, we can always have
 0 , and if   0  
 2
 0
 0 v H
T
 0.

So, when the condition    
 0 0 
0


 0 v L

is satisfied, the corresponding price
 vH
0
for the bundling is:
p  ( )  v L   ( )  v H    0   ( ) 0 v H   0 v L
 1   0 v L   0 0   ( )       0 v H
45
Bundling Sales of Information Goods: Models and Analysis
The seller’s profit is:
 d    p  f  d   1   0 v L   0 0   ( )       0 v H  f  d
1
1
0
0
 1   0 v L   0 0 v H  v H    ( )       0 d
1
0
Price discrimination must be more profitable for the seller than selling the goods separately
or in bundles. It means the following two inequalities should be satisfied:
 d  max  s1 ,  s2  , and  d   B
This is the case when the following two conditions are met:
     d     (1  
1
1
0
0
0
)d 1   0 
vL
 0 0  1
vH
     d     0 d  1   0  0   2
So,
when
the
1
1
0
0
quality
discount
satisfies
     d  max  , 
1
0
   
 0 0 
 0

 0 v L

 0 v H
1
2
and
, the price discrimination can make bigger profits to the seller.
Then consider the change of consumers’ surplus.
When the seller conduct quality discount, the consumers’ surplus is:


S d       v H  1    v L  p    f  d
1
0
1
1
      v H  1    v L d  1   0 v L   0 0 v H  v H         0 d 


0
0
1
  0 v H  v L     d   0 
 0

So, the change is:
1
1
S d  S B   0 v H  v L     d   0   v H  v L     d  0 
 0

 0

1
  0  1v H     d   0   0
 0

46
Research of Mathematical Economics No. 1 2011
1
1
S d  S s 2   0 v H  v L     d   0   v H  v L    d
0
 0

1
1
  0  1v H    d  v L     d   0   0
0
 0

That means the consumers’ surplus is reduced by bundling sales of quality discount.
Compare total surplus of the society in the three cases, we can know,
Sd   d*  v L  v H         d  v L     d
1
1
0
0
S s 2   s2  S B   B  v L  v H  v L    d
1
0
Obviously, we get S s 2   s 2  S B   B > Sd   d .


*
Now we can know, in the case of price discrimination, the consumers’ surplus is decreasing,
that means, the seller extracts more consumer surplus by quality discount. Additionally, in this
way, bundling sales of quality discount also leads to decreasing of total surplus of the society.
3.2 Quantity Discount
The quantity discount is similar to the quality discount. Suppose the seller can segment the
market according to the consumers’ demand type, and provide an incomplete bundle that
contains a fraction
   of the goods in the full bundle, intended for consumers of type  .
The values of goods for the high-valuation consumer and low-valuation consumer are reduced to
  vH and   v L separately. Assuming that    is strictly increasing, and     0 .
 ( )   0 , when   0 ;  ( )  1 , when   1 .
In this model, we can get the following proposition.
Proposition
   
 0  0 
 0
4:
When
the
two
conditions
     d  max 
1
0
1
, 2 
and
are satisfied, conducting price discrimination by quantity discount will bring
the seller more profit and reduce the consumers’ surplus at the same time.
Proof: Set arbitrary
 0  0,1 , for every  1   0 , to satisfy the individual rationality
constraints, we have,
47
Bundling Sales of Information Goods: Models and Analysis
p( 0 )   ( 0 )   0 v H  1    0   0 v L
p( 1 )   ( 1 )  1 vH  1    1   1 vL
To satisfy the incentive-compatibility constraints, we have,
 ( 0 )  0 v H  1    0   0 v L  p 0    ( 0 )  1 v H  1    0   1 v L  p 1 
 ( 1 )  1 v H  1    1   1 v L  p 1    ( 1 )  0 v H  1    1   0 v L  p 0 
The seller will choose p 0  and p 1  as large as possible which satisfy the four
constraints. So we get,
p( 1 )   ( 1 )  1 vH  1    1   1 vL
p( 0 )   ( 0 )  0     1    ( 0 )   1 v H  1    1 v L
 v L   ( 0 )   0 v H  v L     0    ( 1 )v H  v L   1 
Because
 1 is chosen arbitrarily, we have
p( 0 )  v L   ( 0 )  0 v H  v L   max   0    ( )vH  v L   
 0, 0 
Set T    0        , then,
T
         0       

 2T
       2        0       
 2
 0  0 
T
 2T
 0.
Obviously,
, we can always have
 0 , and if   0  
2


 0
So, when the condition
   
 0  0 
0

is satisfied, the corresponding price for the
bundling is:
p  ( )  v L   ( )   (v H  v L )   ( )   0  0 (v H  v L )
 (v H  v L ) ( )      ( )  0   0 0   v L
The seller’s profit is:
48
Research of Mathematical Economics No. 1 2011
 d    p  f  d
1
0
 v H  v L  0 0  v L  v H  v L         0 d
1
0
 d    p  f  d   1   0 v L   0 0   ( )       0 v H  f  d
1
1
0
0
 1   0 v L   0 0 v H  v H    ( )       0 d
1
0
Price discrimination must be more profitable for the seller than selling the goods separately
or in bundles. It means the following two inequalities should be satisfied:
 d  max  s1 ,  s2  , and  d   B .
This is the case when the following two conditions are met:
     d  ( 0 
1
0
1
vH
vL
)    d 
  0 0   1
0
vH  vL
vH  vL
     d      d  
1
1
0
0
So,
when
the
quantity
discount
0
0
(1   0 )   2
     d  max 
1
satisfies
0
   
 0  0 
0

1
, 2 
and
, the price discrimination can make bigger profits to the seller.
Then consider the change of consumers’ surplus.
When the seller conduct quantity discount, the consumers’ surplus is:


S d       v H  1      v L  p    f  d
1
0
      v H  1      v L  v H  v L          0   0  0   v L d
1
0
1
  0 v H  v L     d   0 
 0

So, the change is:
1
1
S d  S B   0 v H  v L     d   0   v H  v L     d  0 
 0
 0


1
  0  1v H  v L     d   0   0
 0

49
Bundling Sales of Information Goods: Models and Analysis
1
1
S d  S s 2   0 v H  v L     d   0   v H  v L    d
0
 0

  0  1v H    d   0 0 v H  v L   0
1
0
That means the consumer’s surplus is reduced by bundling sales of quantity discount.
Compare total surplus of the society under the three cases, we can know,
Sd   d*  v L  v H  v L      d
1
0
S s 2   s2  S B   B  v L  v H  v L    d
1
0
Obviously, we get S s 2   s 2  S B   B > Sd   d .


*
In the case of quantity discount, just like the quality discount case, the consumers’ surplus is
decreasing, as well as total surplus of the society.
4.
Explanations
In the single-product bundling case, we can explain our results intuitively. Proposition 1 and
Proposition 2 tell us the bigger the difference is between the consumers’ types, the better the
price discrimination is, because the seller can extract more consumers’ surplus by mixed bundling
in our model. This result does not contradict with our intuitive sentiment, since the seller will
have no incentive to conduct price discrimination if the consumers’ valuation is identical and they
have the same preference. But at the same time, bundling sales cause the total welfare of the
society to decrease, which can be seen from the model.
In
unbundling
sales,
consumers
choose
to
purchase
q
goods
so
that
T (q)  p  V (q) . This means a consumer’s marginal purchase equals his marginal utility.
But in the case of bundling sales, we have,
T (q1 )  V (q1 )  V (q1 ) ,
   , 0 
1
T (q 2 )  V (q 2 )   0V (q 2 )  V (q 2 ) ,    0 ,
2

This means the consumer’s marginal utility exceed his marginal purchase, that is, the seller
induces the consumers to purchase certain amount of goods which is lower than the optimal
50
Research of Mathematical Economics No. 1 2011
amount for the society.
We can see, bundling sales for single-product distort the product distribution of the society.
It violates the conditions for distributive efficiency and lowers the net welfare of the whole
society.
In the multiproduct bundling case, the seller use bundles of quality discount and quantity
discount to screen consumers by market segment. These bundling strategies induce consumers to
reveal their characteristics and preferences. They can also be seen as mixed bundling, the essence
of which is also to sort consumers of various types, attract low-demand consumers and extract
consumers’ surplus as much as possible. So, the result in our paper is corresponding with which
have been derived earlier by Adams and Yellen (1976)，Schmalensee (1982)，McAfee et al (1989),
that is, in most cases, mixed bundling is more profitable than pure bundling. Proposition 3 and
Proposition 4 indicate that if  ( ) is small enough, and  ( ) ,    are large enough, price
discrimination is profitable. Intuitively, when the average valuation of consumers is not too high,
the two bundling patterns can efficiently segment the market and make profits if the seller set the
quality discount and quantity discount large enough. This result does not contradict with our
intuitive sentiment, too.
5.
Conclusions
This paper studies the impact of bundling sales on merchandising of information goods.
We put our emphasis on the difference between consumers and pricing strategy of firms since
they are important factors that affect seller’s profits and consumer’s surplus. This paper discusses
only about information goods, but the results we get are just similar with that of common goods.
We consider two familiar bundling types: bundling of single-product in amount and bundling of
multiproduct. In the single-product model, we assume the seller provides two kinds of bundles,
which contain different amount of goods, and the consumers choose the one which maximizes
their utility. Using the model, we compare the seller’s profit, consumers’ surplus and net welfare
of the society of bundling sale and unbundling sale. Our result is that a single-product bundling
sale is more profitable than unbundling sale, but consumers’ surplus and net welfare of the
society decrease. At the same time, the bigger the difference is between the consumers’ types, the
better the price discrimination is. In the multiproduct case, we extend Bakos and Brynjolfss’s
51
Bundling Sales of Information Goods: Models and Analysis
(1999) model and discuss two bundling patterns: quality discount and quantity discount. We
prove that when the consumers’ type is continuous, under certain circumstances, those mixed
bundling strategies can be more profitable, and extract consumer surplus more efficiently than
pure bundling and unbundling sales.
Also, we present the conditions when price discrimination is profitable: When the average
valuation of consumers is not too high, the two bundling patterns can efficiently segment the
market and make profits if the seller set the quality discount and quantity discount large enough.
This paper provides several new avenues for future research. First, for many information goods,
there may exist “usage barrier”, that is, different users may have different costs when using the
same goods. This barrier will surely affect sales of information goods to some extent. Second,
this paper is based on the setting that the seller is a monopolist in the market. If we consider the
existence of competitors, different results may be discovered. Third, most information goods are
“experience goods”, meaning consumers can evaluate the goods only after they have purchased
them. Some consumers who have purchased certain products in the first phase may be unwilling
to buy them again in the second phase. So, if we develop a dynamic model of bundling sales of
information goods, valuable results can also be derived.
In conclusion, we believe that the methods developed in this paper provide a simple and
useful framework to understand and analyze in the context of information goods merchandising.
Bundling of information goods is a problem which calls for further study.
References
曹洪, 2004, 捆绑销售的社会福利分析, 学术研究, 2004 年第 2 期, 40-43.
泰勒尔, 1997, 产业组织理论, 中国人民大学出版社.
Adams, W.J., Yellen, J., 1976, “Commodity bundling and the burden of monopoly”, Quarterly Journal of
Economics, 40, 475-498.
Armstrong, M., 1996, “Multiproduct nonlinear pricing”, Econometrica, 64(1), 57-75.
Bakos, J.Y., Brynjolfsson, E., 1999, “Bundling information goods: Pricing, profits and efficiency”,
Management Science, 45(12), 1613-1630.
Bakos, J.Y., Brynjolfsson, E., 2000, “Bundling and competition on the Internet”, Marketing Science, 19(1),
63-82.
Carlton, D.W., Waldman, M., 2002, “The Strategic Use of Tying to Preserve and Create Market Power in
52
Research of Mathematical Economics No. 1 2011
Evolving Industries”, The RAND Journal of Economics, 33(2), 194-220.
Choi, J.P., 2003, “Bundling new products with old to signal quality, with application to the sequencing of
new products”, International Journal of Industrial Organization, 21, 1179-1200.
Jensen, S., 2008, “Two-part tariffs with quality degradation”, International Journal of Industrial
Organization, 26, 473-489.
Martin, M., 1999, “Strategic and welfare implications of bundling”, Economics Letters, 62, 371-376.
McAfee, R.P., McMillan, J., Whinston, M.D., 1989, “Multiproduct monopoly, commodity bundling, and
correlation of values”, Quarterly Journal of Economics, 103, 371-383.
McCardle, K.F., Rajaram, K., Tang, C.S., “Bundling retail products: Models and analysis”, European
Journal of Operational Research, 177, 1197-1217.
Mirman, L.J., Sibley, D.S., 1980, “Optimal Nonlinear Prices for Multiproduct Monopolies”, The Bell
Journal of Economics, 11(2), 659-670.
Salinger, M.A., 1995, “A Graphical Analysis of Bundling”, The Journal of Business, 68(1), 85-98.
Schmalensee, R., 1982, “Commodity Bundling by Single-Product Monopolies”, Journal of Law and
Economics, 25(1), 67-71.
Schmalensee, R., 1984, “Gaussian demand and commodity bundling”, Journal of Business, 57, S211-S231.
Sibley, D.S., Srinagesh, P., 1997, “Multiproduct Nonlinear Pricing with Multiple Taste Characteristics”, The
RAND Journal of Economics, 28(4), 684-707.
Spence, A.M., 1980, “Multi-Product Quantity-Dependent Prices and Profitability Constraints”, The Review
of Economics Studies, 47(5), 821-841.
Whinston, M.D., 1990, “Tying, Foreclosure, and Exclusion”, The American Economic Review, 80(4),
837-859.
(Responsible Editor: Zhang Lingyun) (Proofreader: Wang Jue)
53
The Introduction and Applications of RDEU Hypothesis
The Introduction and Applications of RDEU Hypothesis
Wang Bo, Hong Beiyun
(School of Economics, Renmin University of China)
Abstract: The passage gives us a brief introduction of the RDEU theory and its two simple applications. At the
end of the passage, the author discusses the disadvantages of the RDEU theory.
Keywords: Expected Utility, Rank-dependent Expected Utility, Nash Equilibrium
1. Preface
The Expected Utility (EU) hypothesis, established by Von Neumann and Morgenstern
(1944), evaluates uncertain prospects according to their expected level of “satisfaction” or
“utility”. The EU hypothesis is the predominant descriptive and normative model of choice under
uncertainty in economics, which provides the analytical underpinnings for many game theories
on the aspects of insurance and financial decisions. However, through numerical simulations and
realistic experiments, many experts found the evidence like Allais paradox，Ellsberg paradox and
Common Ratio effect, showing that individuals may systematically depart from its predictions
and then Quiggin (1982) came up with the rank-dependent expected utility (RDEU) hypothesis.
In the first part of the passage we give a brief introduction of EU hypothesis. The second
part is the introduction of RDEU hypothesis, following with 2 simple applications of it in the
third part. Then we make some simple improvement of RDEU hypothesis mentioned in the last
part of the passage. Most of the mathematics analysis is list in the Appendix.
2. The Introduction of EU
The expected utility theory requires the following axioms:
(1) Completeness: for any two distributions x1 and x2 , either x1  x2 , x1  x2 or x1 ~ x2 .
54
Research of Mathematical Economics No. 1 2011
(2) Transitivity: if x3  x2 , and x2  x1 , then x3  x1 .
(3) Mixture Continuity: if x3  x2  x1 , then there exists the only p (0 < p <1) such that x2 ~
x2 ~ px3  (1  p) x1 .
(4) Independence: for any two distributions
x1 and x2 , x1  x2 if and only if
px1  (1  p) x3  px2  (1  p) x3 for all p (0 < p ≤ 1) and for all x3 .
Under these four axioms, we can find a utility function U (.) that meet: if x1  x2 ,
U ( x1 )  U ( x2 ) ; else if x1  x2 , U ( x1 )  U ( x2 ) .
The formal representation of the objects of choice, and hence of the expected utility
preference function, depends upon the set of possible outcomes. When the outcome set
X  {x1 ,
, xn } is finite, we can represent any probability distribution over this set by its
vector of probabilities P  ( p1 ,
, pn ) (where pi  prob( xi ) ) and the expected utility
preference function takes the form V ( P)  V ( p1 ,
, pn )  Ui pi . When the outcome set
consists of the real line or some interval subset of it, probability distributions can be represented
by their cumulative distribution functions F (.) (where F ( x)  prob(  x) ), and the

expected utility preference function takes the form V ( F )  U ( x) dF ( x) .
In fact, the expected utility hypothesis of behavior towards risk is the hypothesis that the
individual possesses (or acts as if possessing) a “Von Neumann-Morgenstern Utility Function”
U (.) or “Von Neumann-Morgenstern Utility Index” {U i } defined over some set X of
alternative possible outcomes, and when faced with alternative risky prospects or “lotteries” over
these outcomes, will choose the prospect that maximizes the expected value of U (.) or {U i } .
Since the outcomes could be alternative wealth levels, multidimensional commodity bundles,
time streams of consumption, or even non-numerical consequences (such as a trip to Paris), this
approach can be applied to a tremendous variety of situations, and most theoretical researches in
the economics of uncertainty, as well as virtually all applied work in the field (e.g., insurance or
investment decisions) is undertaken in the expected utility framework.
55
The Introduction and Applications of RDEU Hypothesis
3.
The Introduction of RDEU
There is much evidence that individuals may systematically depart from its predictions.
Many people disagree to the fourth axioms as it’s different from the choice of a real person. As a
result, many alternative models developed, among which the Rank-Dependent Expected Utility
(RDEU) model is the most widely used, and arguably the most empirically successful,
generalization of Expected Utility (EU).
Firstly, if the outcome set X  {x1 ,
, xn } is finite, we make that xi is better than x j ,
if i > j. Now we can represent any probability distribution over this set by its vector of
probabilities P  ( p1 ,
, pn ) (where pi  prob( xi ) ).
Secondly, we define a new function-----emotion function  ( p ) :
 (.) :[0,1]  [0,1] ,  (0)  0 ,  (1)  1 ,  '( p)  0
If  "( p )  0 , it means the person has an optimistic emotion; else if  "( p )  0 , it means
the person has a pessimistic emotion.
Usually, we set  ( p )  pr (Quiggin, 1982). As a result, if r > 1, it means optimistic
emotion; else if 0 < r < 1, it means pessimistic emotion; else if r = 1, the decision is not affected
by the emotion, which is the same as the EU theory.
Thirdly, we can define the rank-dependent expected utility.
j
j 1
k 1
k 1
Wi ( p (ji ) )  wi ( p k(i ) )  wi ( pk(i ) )
When the outcome set consists of the real line or some interval subsets of it, probability
distributions can be represented by their cumulative distribution functions F (.) (where
F ( x)  prob(  x) ), and the rank-dependent expected utility preference function takes the

form V ( F )  U ( x)d  ( F ( x )) .
4.
56
The Applications of RDEU
Research of Mathematical Economics No. 1 2011
4.1 The First Application: Battle of the Sexes with Mixed Strategies
In the Battle of the Sexes Game, a husband (A) and wife (B) are planning a vacation. A
prefers mountain locations; B prefers the seaside. Both players prefer a vacation spent together to
one spent apart. The payoffs in the table reflect these preferences.
Graph 1
Battle of the Sexes with Mixed Strategies
Suppose that the spouses in the problem tire of constant bickering about vacations and
decide to let “chance” decide. Specifically, suppose A decides to choose his mountain strategy
with probability p and seaside with probability 1  p . Similarly, suppose B chooses her
mountain strategy with probability q and seaside probability 1  q . Given these probabilities,
the outcomes of the game occur with the following probabilities: mountain-mountain, pq ;
mountain-seaside,
p (1  q ) ;
seaside-mountain,
(1  p ) q ;
and
seaside-seaside,
(1  p)(1  q) .
A’s expected utility is then given by
E (U A )  pq  2  p(1  q)  0  (1  p)q  0  (1  p)(1  q) 1
 1  q  p(3q  1)
Obviously, A’s optimal choice of p depends on B’s probability, q. If q 
1
, utility is
3
57
The Introduction and Applications of RDEU Hypothesis
maximized by choosing p  0 . If q 
expected utility is
1
1
, A should opt for p = 1. And when q  , A’s
3
3
2
no matter what value of p is chosen.
3
For spouse B, expected utility is given by
E (U B )  pq 1  p(1  q)  0  (1  p)q  0  (1  p)(1  q)  2
 2  2 p  q(3 p  2)
When p 
2
2
, B’s expected utility is maximized by choosing q  0 . When p  ,
3
3
utility is maximized by choosing q  1 . And when p 
2
, B’s expected utility is independent
3
of what q she chooses.
According to the analysis above, we get three Nash equilibriums. Two of these we can see
obviously: p  0 , q  0 and p  1 , q  1 represent the joint vacation strategies. And
p
2
1
, q
is a Nash equilibrium with mixed strategies.
3
3
Now, we deal with this problem through the RDEU theory instead of the EU theory.
Many books have given the proof that there exists the Nash equilibrium with mixed
strategies depending on the EU theory if there are finite people and the set of the plots is finite for
each people. The following theorem will show that it is the same depending on the RDEU theory.
Theorem: If the set of players is finite and the set of sections is finite for each player, there
exists a Nash equilibrium depending on the RDEU theory.
The proof will be shown in the Appendix.
We assume that the emotion function of A is wA ( p )  p
wB (q )  q s .
The utility function of A is
58
r
and the emotion function of B is
Research of Mathematical Economics No. 1 2011
VA  ( p r 1  p r )(
2 0 qs
)(
)
0 1 1 qs
 2 p r q s  (1  p r )(1  q s )
 3 pr qs  pr  qs  1
The utility function of B is
VB  ( p r 1  p r )(
qs
)(
)
0 2 1 qs
1 0
 p r q s  2(1  p r )(1  q s )
 3 p r q s  2 p r  2q s  1
As A will maximize his utility, the partial differentiation will be zero:
VA
 3rp r 1q s  rp r 1  rp r 1 (3q s  1)  0
p
1 1
q*  ( ) s
3
As B will maximize his utility, the partial differentiation will be zero:
VB
 3sp r q s 1  2sq s 1  sq s 1 (3 p r  2)  0
q
2 1
p*  ( ) r
3
We can get several conclusions:
2
3
1
1
3
1
(1) p*  ( ) r , q*  ( ) s is a Nash equilibrium with mixed strategies. Obviously, it depends
on r and s.
(2) If the emotion factor will not affect the decision of the husband and wife, which means
r  s  1, the RDEU theory can reach the same result as the EU theory.
(3) As r increases, which means the husband has a strong desire to the seaside, p* will increase.
In this case, if s > 1, which means the wife will consider the decision of the mountain, especially
when s > 1.9, the husband and wife will probably make the same decision to go to the mountain.
(When s > 1.9, q* will be more than 0.56, which means the wife has a probability which is more
than 56% to go to the mountain)
(4) If the wife is disgusted about the mountain, which means s < 1, even when the husband has a
strong desire to the seaside, which means r >> 1, they will probably make a different decision.
(5) The same analysis can be taken on the seaside strategy.
59
The Introduction and Applications of RDEU Hypothesis
Obviously, the RDEU makes the conclusion more clear and precise.
4.2 The Second Application: Insurance Premiums with Moral Hazard
Firstly, we give two theorems.
Theorem 1: If F ( x ) is a distribution function of the random variable  , and
P (  0)  0 , then
 G( x)dF ( x)  G(0) F (0)   G( x)dF ( x)
x0
( G ( x ) is an arbitrary utility function)
x 0
Theorem 2: If F ( x ) is a distribution function of the random variable  , and  ( x ) is a
strictly increasing continuous function: [0,1]→ [0,1], π(0)=0, π(1)=1, then  ( F ( x)) is a
distribution function.
The proof would be shown in the appendix.
We assume that the premium rate is
 , the quantity of the insurance you buy is π, the
wealth you have at the beginning is w0 , the loss is x, the money you can get if the accident
happen is s ( x ) which is a function of x, the utility function of the consumer is u ( x) , the
emotion function of the consumer is w1 ( x) , the utility function of the insurance company is
v ( x ) and the emotion function of the insurance company is w2 ( x) . We also assume that the
behavior of the consumer is measured by a, and the cost of the consumer to behave as a is c ( a ) ,
which is a function of a. Obviously , we can see that c '(a )  0 .
In the paper, we only consider two special conditions.
The first condition :
After signing the contract with the insurance company, the consumer’s behavior will only
have an effect on the quantity of the loss but the probability of the accident.
As consumer’s behavior will not affect the possibility of the accident, we assume that the
possibility of the loss is q , which is a constant, and the distribution function of the loss is
G ( x, a ) .
So the consumers will maximize their utility by
60
Research of Mathematical Economics No. 1 2011
max
 u(w

0
 s( x)  x   )dw1 (G( x, a))  c(a)
x 0
s ( x ),
According to the theorem 1 and 2, it’s the same as
w (1  q)u ( w
max

1
0
 u( w
  ) 
0
 s( x)  x   )dw1 (G( x, a))  c(a)
x 0
s ( x ),
(Notice that s (0)  0 , and G (0, a)  1  q )
To
simplify
the
model,
we
define
the
conditional
distribution
function
F ( x, a)  G ( x, a) / (1  G (0, a)) , which means the possibility of the loss of x in case that you
have known that the incident must happen.
In general,
w(q)  q r , so w(q1q2 )  w(q1 ) w(q2 ) .
As a result,
w1 (G ( x, a))  w1[ F ( x, a)(1  G(0, a))]
 w1 ( F ( x, a)) w1(1  G (0, a))  w1 (q) w1 ( F ( x, a))
In this case, the model can be simplified as
w (1  q)u ( w
max

1
0
  )  w1 (q)
 u(w
 s( x)  x   )dw1 ( F ( x, a))  c(a)
 u(w
 s( x)  x   )dw1 ( F ( x, a))  c(a)
0
x 0
s ( x ),
So the insurance model is
w (1  q)u ( w
max

1
0
  )  w1 (q)
s.t.
0
x 0
s ( x ),
w2 (1  q)  w2 ( q)
 v(  s( x))dw ( F ( x, a))  v(0)
2
x 0
a  argmaxw1 (1  q)u( w0   ) 
w1 ( q)
 u( w
0
 s( x)  x   ) dw 1( F ( x, a))  c( a)
x 0
In the second equation, v (0) means the minimum utility the company will suffer. If the
company gets a utility less than v (0) , he will quit the market. By assuming that the insurance
market is completely competitive, “  ” can be replaced by “=”.
The third equation means that the consumer will maximize its utility by choose a, so we can
61
The Introduction and Applications of RDEU Hypothesis
make a partial differentiation on a. According to Mirrlees (1974) and Holmstrom (1979), the third
equation can be replaced by
w1 (q)  u(w0  s( x)  x   )
x 0
w1 '( F ( x, a)) f ( x, a)
dx  c '(a)
a
(To simplify the model, we assume that F ( x, a ) has a density function f ( x, a ) here)
So the insurance model should be
w (1  q )u ( w
max

1
0
  )  w1 ( q)
 u(w
0
w2 (1  q)  w2 ( q)
s.t.
 s( x)  x   ) dw1 ( F ( x, a))  c( a)
x 0
s ( x ),
 v(  s( x))dw ( F ( x, a))  v(0)
2
x 0
w1 ( q)
 u( w
0
 s( x)  x   )
x 0
w1 '( F ( x, a )) f ( x, a )
dx  c '(a )
a
As a result, the Lagrangian function is
 u(w
L( s ( x),  )  w1 (1  q)u ( w0   )  w1 ( q)
0
 s( x)  x   ) dw1 ( F ( x, a))  c( a)
x 0
 [ w2 (1  q)  w2 ( q)
 v(  s( x))dw ( F ( x, a))]
2
x 0
 [ w1 ( q)
 u( w
0
 s( x)  x   )
x 0
w1 '( F ( x, a)) f ( x, a)
dx  c '( a )]
a
 is
The first order differentiation condition of
 w1 (1  q)u '( w0   )   w1 (q)
 u '(w
0
 s ( x)  x   )dw1 ( F ( x, a )) 
x 0
 w2 (1  q)   w2 (q)  v '(  s( x))dw2 ( F ( x, a )) 
x 0
 w1 (q)  u '( w0  s ( x)  x   )
x 0
(1)
w1 '( F ( x, a )) f ( x, a )
dx
a
The first order differentiation condition of s(x) is
w1 (q)u '( w0  s ( x)  x   ) w1 '( F ( x, a)) f ( x, a)
 w2 (q)v '(  s( x)) w2 '( F ( x, a)) f ( x, a)
w '( F ( x, a)) f ( x, a)
  w1 (q)u '( w0  s ( x)  x   ) 1
0
a
62
(2)
Research of Mathematical Economics No. 1 2011
(Notice that it depends on calculus of variations，the proof will be shown Appendix)
Combining with the equation (2), we can simplify the equation (1) as
u '( w0   )  
We can see that the insurance premium
is a constant, so that
w2 (1  q)
w1 (1  q )
 is irrelevant with the loss x, which means 
 and  are in an inverse proportion, which we can get as a same result
through the EU theory.
By simplifying it, we can get that
u '( w0  s ( x)  x   ) w2 (q ) w2 '( F ( x, a ))


v '(  s ( x))
w1 (q ) w1 '( F ( x, a )) 1   f a ( x, a )   w1 ''( F ( x, a )) F ( x, a )
a
f ( x, a )
w1 '( F ( x, a ))
In the case of EU theory, if we give some assumption, we can get a relationship of s(x) and x.
However, it’s much harder in the RDEU case, as we don’t know whether the consumer and the
company are optimistic or pessimistic. If we assume that the company and the consumer have a
same emotion function,
u '( w0  s ( x)  x   )


f
(
x
,
a
)
w
''( F ( x, a ))
v '(  s ( x))
1  a
 1
Fa ( x, a )
f ( x, a )
w1 '( F ( x, a ))
As
w1 ''( F ( x, a))
and Fa ( x, a) exist here, it’s so hard to analyze. As a result, the
w1 '( F ( x, a))
conclusion we get through EU cannot be applied now. We cannot get an accurate relationship of
s(x) and x here.
The second condition:
After signing the contract with the insurance company, the consumer’s behavior will only
have an effect on the probability of the accident but the quantity of the loss.
As consumer’s behavior will not affect the quantity of the loss, we assume that the
possibility of the accident is q (a ) , which is a function of a. Obviously, q '(a)  0 . The
distribution function G ( x, a ) is equals to q(a) F ( x) .
Similar to the condition 1, the insurance model should be
63
The Introduction and Applications of RDEU Hypothesis
w (1  q(a))u ( w
max

1
0
  )  w1 (q (a ))
s.t.
 u(w
0
 s ( x)  x   )dw1 ( F ( x))  c(a )
x 0
s ( x ),
w2 (1  q( a))  w2 ( q( a))
 v(  s( x))dw ( F ( x))  v(0)
2
x 0
 w1 '(1  q(a))q '(a)u ( w0   )  w1 '(q(a))q '(a)
 u(w
 s ( x)  x   )dw1 ( F ( x))  c '(a)
0
x 0
The Lagrangian function is
w (1  q(a))u ( w
max

1
0
  )  w1 (q(a))
 u(w
0
 s( x)  x   )dw1 ( F ( x))  c(a)
x 0
s ( x ),
 [ w2 (1  q(a))  w2 (q(a))
 v(  s( x))dw ( F ( x))  v(0)]
2
x 0
+[ w1 '(1  q(a))q '(a)u ( w0   )  w1 '(q(a))q '(a)
 u(w
0
 s( x)  x   ) dw1 ( F ( x))  c '(a)]
x 0
The first order differentiation condition of
 is
 w1 (1  q(a))u '( w0   )   w1 (q(a))  u '( w0  s( x)  x   )dw1 ( F ( x))
x 0
[ w2 (1  q(a))   w2 (q(a))  v '(  s( x))dw2 ( F ( x))]
(3)
x 0
 [ w1 '(1  q(a))q '(a)u '( w0   )   w1 '(q(a))q '(a)
 u '(w
0
 s( x)  x   )dw1 ( F ( x))]  0
x 0
The first order differentiation condition of s ( x ) is
w1 (q(a))u '( w0  s( x)  x   ) w1 '( F ( x)) f ( x)
 w2 (q(a))v '(  s( x)) w2 '( F ( x)) f ( x)
(4)
+  w1 '(q(a))q '(a)u '( w0  s( x)  x   ) w1 '( F ( x)) f ( x)  0
Combining with the equation (4), we can simplify the equation (3) as
u '( w0   ) 
 w2 (1  q (a ))
w1 (1  q (a ))   w1 '(1  q (a ))q '(a )
We can see that the insurance premium
is a constant, so that
 and  are in an inverse proportion, which we can get as a same result
through the EU theory.
By simplifying it, we can get that
64
 is irrelevant with the loss x, which means 
Research of Mathematical Economics No. 1 2011
u '( w0  s ( x)  x   ) w2 '( F ( x)) w2 (q (a ))


w
'(
q
v '(  s ( x))
w1 '( F ( x)) w1 (q (a )) 1   1 (a )) q '(a )
w1 (q (a ))
Similarly to the condition 1, we can get nothing through this equation, however, if we
assume that the consumer and the company have a same emotion function, which means
w1  w2 , then we can see that
u '( w0  s ( x)  x   ) w2 (q (a ))


v '(  s ( x))
w1 (q(a)) 1   w1 '(q (a )) q '(a)
w1 (q (a ))
The right side is irrelevant with x. by assuming that the company is risk neutral, we can get
that v '(.) is constant, as a result, u '( w0  s( x)  x   ) should be constant, so that
s ( x)  x is constant. We can get a same result through the EU theory.
5.
An Improvement of RDEU
5.1 The Violations of Monotonicity
Violations of monotonicity come from an experiment. People can pick up the ball in a box
while the revenue and the probability had been shown in the table. People were asked to choose
C or D.
Here is the table:
90% White
6% Red
1% Green
1% Blue
2% Yellow
A
0
45000
30000
-15000
-15000
B
0
45000
45000
-15000
-10000
C
D
White
Red
Green
Yellow
90%
6%
1%
3%
0
45000
30000
-15000
90%
7%
1%
2%
0
45000
-15000
-10000
We can see that the two tables are really the same and table 2 only combines the choice of
table 1. In the table 1, B is obviously a better choice. Similarly, D is better than C. But in fact,
65
The Introduction and Applications of RDEU Hypothesis
58% of the people choose C, which can’t be interpreted by the EU theory.
Suppose we have n results x1 ,
, xn . If i > j, then xi is not worse than x j . The
probability of xi is pi . We break up xi into xi1 and xi 2 , and the probability is pi1 and
pi 2 . According to the RDEU theory, the changing utility in the violation is
wi1U ( xi1 )  wi 2U ( xi 2 )  wU
i ( xi )
 wi1U ( xi )  wi 2U ( xi )  wU
i ( xi )
 ( wi1  wi 2  wi )U ( xi )
wi1  wi 2
  ( p1  p2  ...  pi 1  pi1  pi 2 )   ( p1  p2  ...  pi 1  pi1 ) 
 ( p1  p2  ...  pi 1  pi1 )   ( p1  p2  ...  pi 1 )
  ( p1  p2  ...  pi 1  pi1  pi 2 )   ( p1  p2  ...  pi 1 )
  ( p1  p2  ...  pi 1  pi )   ( p1  p2  ...  pi 1 )
 wi
So the choice will similarly not change according to the RDEU theory although we have
spitted the event.
5.2 An Improvement of RDEU
Now, we give an improvement of RDEU so that it can explain the violations of
monotonicity.
In fact, we make our decision by the conditional possibility instead of the objective
possibility. So we can correct the RDEU theory in the following ways:
w1   ( p1 )
j 1
pj
w j  (1   wi ) (
)
j 1
i 1
1   pi
i 1
It means that you select a choice in case that you can’t choose a better one.
In this case,
66
Research of Mathematical Economics No. 1 2011
wi1  wi 2
i 1
j 1
pj
 (1   w j ) ( i 1 )  (1   wi  wi1 ) (
)
j 1
j 1
i 1
1   pi
1   pi  pi1
pi1
i 1
i 1
i 1
p
 (1   w j ) ( i 1i )  wi
j 1
1  p j
j 1
And the violation also changes w j , when j > i. As a result, the choice may change, which
means that now violations of monotonicity can be interpreted by RDEU depending on the
conditional possibility.
In fact, if we consider the application 1 through RDEU depending on the conditional
possibility, the answer will not change.
Proof: In the application 1, n = 2, which means there are only two outcomes x1 and x2 ,
and the probability is p1 and p2 .
Assume x1 is better than x2 . Notice that:
w1   ( p1 )
w2   (
p2
)(1  w1 )  1  w1
1  p1
As p1  p 2  1 , we get a same result as before.
So the answer of the first application needn’t to be changed.
6. Further Research
We give an improvement of RDEU theory which depends on the conditional possibility at
the end of the passage. Further research is to put this model into a continuous way so that it can
be applied in more situations.
67
The Introduction and Applications of RDEU Hypothesis
Appendix
The Proof of Theorem
Firstly, we will give out the meanings of the symbols.
We define a finite simultaneous game as a tuple {N ,{Si },{U i }} , where N is a set of
players, and for each player Si is a set of plots and U i is the utility function.
For each person i ( i  N ), he can make several choices {x1 ,
(i )
, xm (i ) } , the probability
mi
that he chooses x j
(k )
is p j
(k )
(i )
1
(i )
mi
. Si = { p ,..., p
|  p (ji )  1, p (ji )  0} , which means the
j 1
plots set of person i.
mn
m1
U i ( p, w)  
 u (x ,
j1 1
jn 1
i
1
, xn )W1 ( p (1)
j1 )
Wn ( p (jnn ) ) , which means the utility of the
person i depending on the RDEU theory.
j
j 1
k 1
k 1
Wi ( p (ji ) )  wi ( p k(i ) )  wi ( pk(i ) )
where wi means the emotion function of person i.
Now, I will give you the proof of the theorem.
Proof: We denote that
Fi ( pi )  { p*i | U i ( p*i , pi )  max U i ( pi , pi )}
pi
n
F ( p)   Fi ( pi ) , S =[0,1], t   mi
i 1
(1) St is a compact convex set.
t
Then F (.) is a reflection : S  S . As S is a compact convex set, S is a compact
t
t
convex set too.
(2) F ( p )   .
U i ( pi , pi ) is a continuous function and S is a closed set. So U i can be maximized at
68
Research of Mathematical Economics No. 1 2011
S , which means that Fi ( pi )   . As a result, Fi ( pi )   .
(3) F ( p ) is a convex set.
Ui ( pi , pi )   c jWj ( p(ji ) )
, where c j is irrelevant with pi , which can be seen as a
constant here. Obviously, c j is nonnegative here. We denote A  {i | ci  max c j } .
j
We can see that if q  {q1 ,
, qmi }  Fi ( p i ) , then q j equals zero if j  A . Or,
U i (q, pi )  U i (q* , pi ) , where i*  A ( A   ,see (2)), q*  (0,0,..., qi*  1,0,...,0) ,
which is in contrast with the definition of Fi ( pi ) .
If p  Fi ( pi ) , then
U i ( p, p i )   c jW j ( p (ji ) )   c jW j ( p (ji ) )   c jW j ( p (ji ) )
jA
jA
(i )
Notice that if j  A , p j  0 ; if j  A , c j  ci* . So
U i ( p, pi )   c jW j ( p (ji ) )   c jW j ( p (ji ) )
jA
jA
  ci*W j ( p )   ci*W j ( p (ji ) )  ci*  W j ( p (ji ) )  ci*
(i )
j
jA
jA
  Fi ( pi )
Notice that if j  A ,  j   j  0 , so  j   j  (1   )  j  0 ; if j  A , c j  ci* .
As a result,
U i ( , pi )   c jWi ( j )   c jWi ( j )
jA
jA
  ci*Wi ( j )   ci*Wi ( j )   ci*Wi ( j )  ci*
jA
So,
j A
  Fi ( pi ) , from which we can know that F ( p ) is a convex set.
(4) F is upper semi-continuous.
69
The Introduction and Applications of RDEU Hypothesis
Assume
n
n
 n   ,    ( n  ),   F n
as  i'
set
n
U i ( i ,  ni )  U i ( i' ,  ni )
n
we get
 i'
U i ( i ,   i )  U i ( i' ,   i )
so i  i  Fi (  i )
As a result,
  F .
According to Kakutani Theorem,  , s.t.  F , so
 is a Nash equilibrium.
The Proof of Theorem 1
Proof:
{x : x  0}  {x : x  0}  {x : x  0} {x : x  0}  {x : x  0}  

 G( x)dF ( x)   G( x)dF ( x)   G( x)dF ( x)
x0
x 0
x 0
 G( x)dF ( x)  G(0) P(  0)  G(0) P(  0)
( P(  0)  0)
x 0
The Proof of Theorem 2
Proof:
F ( x) is a distribution function
 (1) lim F ( x)  0
x 
(2) lim F ( x)  1
x 
(3)x0 , lim F ( x)  F ( x0 )
x  x0
(4)F ( x) is increasing
 (x) is a continuous function
 lim  ( F ( x))   ( lim F ( x))   (0)  0
(1)
lim  ( F ( x))   ( lim F ( x))   (1)  1
(2)
x 
x 
70
x 
x 
Research of Mathematical Economics No. 1 2011
  0,   0, if y  F ( x0 )   , then  (y)- (F(x 0 ))  
x0 , lim F ( x)  F ( x0 )
x  x0
1  0, if x  x0  1 , then F ( x)  F ( x0 )  
and then  (F ( x))   (F ( x0 ))  
so lim  ( F ( x))   ( F ( x0 )) (3)
x  x0
x>y,F ( x)  F ( y ),so  ( F ( x))   ( F ( y ))
according to (1)(2)(3)(4),  (F ( x)) is a distribution function
(4)
The Proof of Equation 2
Proof:
x2
A( f )   L( x, f , f ')dx
x1
if A( f ) is max imized at f 0
L d L

F dx f '
this is The Euler  Lagrange equation
then
if we put it in the equation (2), we can get the result
References
何凯洁, 基于信息修正的非期望效用模型和保险市场均衡问题研究.
钟桦, 道德风险下的最优保险模型研究.
Nicolson, Intermediate microeconomics 9th edition.
Mark, J.M., “Expected utility hypothesis”.
Matthew, J.R., “Risk aversion in RDEU”.
Xiong Guoqiang, n-person, “Non-expected Utility Games Model and Its Application Based on RDEU
Theory”.
(Responsible Editor: Xu Lingjue) (Proofreader: Ren Junqiushi)
71
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
Income Targets and Labor Supply Elasticities: Evidence from China
Health and Nutrition Survey
Zhang Luezhao
(School of Economics, Renmin University of China)
Abstract: This article extends the work on labor supply theory of behavioral economics. It proposes some new
views of the model of behavioral labor supply theory, for example the effects on labor supply elasticity that are
made by income target. It estimates the elasticities of two kinds of labor working in urban informal sector and
compares them. The samples are derived from China Health and Nutrition Survey. The empirical results confirm
the predictions of the model. However, the article doesn’t avoid the “division bias”, which should be improved in
the future.
1. Introduction
In recent decades, labor economics has developed a lot. In labor economics, labor supply
theory can be divided into two branches: the neo-classical labor supply theory and behavioral
labor supply theory.
1.1 Neo-classical Theory of Labor Supply
When neo-classical labor supply theory is used to analyze labor supply behavior, consumer
theory is applied to labor economic. Here, an individual has two kinds of special commodities,
which can give him utility: consumption and leisure. Individuals’ time in a day is limited, which
can be used for work to get income for consuming, or for leisure. Individual consumption is the
opportunity cost of leisure he gives up. The opportunity cost of leisure is the consumption he
gives up. Labor market price (wage) w is the opportunity cost of leisure time monetization, said:
for each additional unit of leisure time, one should give up w units of consumption.
In this way, the problem faced by individuals is, when a budget constraint is given, how to
allocate time between consumption and leisure resources to achieve their utilities maximization.
Labor supply can be regarded as the opposite of leisure demand. The core of this analysis is:
72
Research of Mathematical Economics No. 1 2011
labor supply responses to different wages.
It is generally believed that the increase of labor market prices (wages) would reduce the
demand for leisure, increasing labor supply, which is called substitution effect, that is, if wage
increases, workers will substitute leisure time with labor, and thus labor supply increases. The
decrease of labor market prices (wages) will increase the demand for leisure, reducing labor
supply, that is, if wage decreases, more workers will replace the labor with leisure. This is the
substitution effect. In addition, when leisure is a normal good and income increases, people will
have more leisure time and consumption increases, which accordingly reduces the labor supply.
This is the income effect.
The individual's response to wage changes depends on the relative size of the two kinds of
effects. If the income effect is greater than the substitution effect, in the labor market, wage
growth will reduce labor supply. If the income effect is smaller than the substitution effect, wage
growth will increase labor supply. For a family, the individual’s goal is to achieve the family's
overall joint utility maximization. But different members face different wage levels. At this point,
each family member’s labor supply not only relies on their own wage, but also on other family
members’ wages.
1.2 Theory of Behavioral Labor Supply
In recent years, the behavioral economics penetrates into the various fields of economics
where the original theory is modified or refined. After the introduction of psychological and
behavioral perspective, labor economics has resulted in the behavioral labor economics.
The standard neo-classical labor supply theory predicts that: the increase in wages will
enable workers with more labor, when the wages are low, the wage increase will allow workers to
substitute leisure with labor, where income effect is greater than substitution effect. So, we can
get an upward-sloping labor supply curve. A lot of empirical research evidences show that it is
true for the participation of the marginal changes in labor supply. However, the daily wages and
daily labor supply don’t show a clear positive correlation. On the contrary, some studies have
shown that intertemporal wage elasticity is negative, indicating that the higher daily wage leads
to less labor supply.
For example, Camerer et al and Farber respectively studied the relationship between daily
income and daily working hours in New York City taxi drivers. They took regression of daily
73
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
working time and the daily wage and found that taxi drivers cross-elasticity of substitution is
negative, but, according to neo-classical labor supply theory, this cross-elasticity of substitution
should be positive. Neo-classical theory of labor supply usually attributes this negative
cross-elasticity of substitution in labor supply to the measurement errors. But it is not a good
explanation, since Camerer et al and Farber have conducted a calibration of the measurement
errors.
And different from the assumption in the neo-classical economic theory that individual is
fully rational, behavioral economics theory suggests that a person’s rationality is limited. And,
compared with the final result, the individual may pay more attention to the decision-making
process and the relative results. For example, during wage negotiations, workers tend to focus on
the average wage levels of the group he stays in, rather than the equilibrium wage level in the
labor market.
Specifically in the labor supply, the difference between neo-classical labor supply theory
and behavioral labor supply theory is the introduction of three concepts:
Loss Aversion: The hate an individual has for one unit of loss is stronger than the happiness
an individual has for one unit of revenue.
Income Target: workers may not make labor supply decisions based on the utility
maximization in entire life cycle, but in a relatively short period of time, such as daily or weekly.
In this short period, workers consider achieving this income target.
Reference Dependent Preferences: This is the greatest difference between neo-classical
theory and the behavioral economics. Neo-classical theory suggests that people's preferences are
stable. The reason why people's behavior will change is that the constraints faced by the
individual have changed. And behavioral economics think people’s preferences depend on their
reference point. Individuals evaluate the income change not according to how much income they
earned, but according to the gap between the goals and their income. When the income is less
than this goal, the individual will regard it as the loss and failure, and when income is higher than
this goal, the individual will regard it as income and success.
1.3 Main Points
Lewis proposed dual economic structure in the study about economic development of
developing countries. He divided a country's economy into the traditional economic sector in
74
Research of Mathematical Economics No. 1 2011
rural areas and modern sector in urban areas. Along with economic development, the surplus
labor of the traditional sector will continue to shift to the modern sector of the economy, this
process will last until surplus labor of the traditional sector was completely absorbed by the
modern sector of the economy. Further, Todaro proposed that the city's modern economic sector
can also be divided into two parts. One is the formal sector, where wages are higher and the
working hours are relatively fixed. Another part is informal sector, where wages are low, work is
unstable, working hours are relatively flexible. Workers can choose the time of labor freely there.
Labor force shifting from rural to urban areas can generally only be employed in the informal
sector. Few workers have access to the city's formal sector employment.
This article describes China's labor supply issues under the framework of the behavioral
theory of labor supply. I discussed when the labor supply elasticity is positive and when it is
negative. This article also discusses the impacts on labor supply elasticity made by daily income
target, which didn’t exist in previous research.
Since those who works in a company taking 8-hour per day is clearly not free to decide their
own working hours, so this article will mainly focus on working hours of those in the informal
sector.
This paper examined changes of working time in the family business and domestic service
as wage increases from 1993 to 2006 in China's 10 provinces. And, the samples will be separated
into two categories: urban and rural according to registration residence type.
Section 2 describes the model of this article that will be used, including the form of
theoretical analysis and specific econometrical forms. Section 3 is a description of the statistics
and empirical analysis. Section 4 gives the summary and conclusions of this article.
2. Labor Supply Model
In our model, labor is an existence which can lead to two results. On one hand, the increase
of the labor will increase workers’ labor income, allowing workers to consume more goods and
buy more services, which enable workers to obtain a higher utility. On the other hand, the
increase in labor would make workers physically and psychologically feel more tired and weary,
increasing the disutility of workers. Workers’ disutility’s increase with labor time’s increase and
the marginal utility of leisure’s increase are the both sides of a coin in the neo-classical labor
75
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
supply theory.
We denote v( wl  y  r ) to represent the utility workers get from labor income where
w is an individual’s wage, l is his daily working time, y is his non-labor income and r is his
income target. According to the assumption of behavioral economics, individuals’ sensitivity to
income changes is decreasing. Relative to the reference point, i.e. our income target, regardless of
income increase or decrease, its marginal effect is diminishing. This is a very different point
between the neo-classical theory and behavioral economics. In the neo-classical theory, for a
risk-averse individual, the increase in the marginal utility of income must be diminishing, the
revenue for the loss of marginal disutility must be increasing at any point. Thus, the form of
utility function our model used is different from that in the neo-classical theory. Neo-classical
theory uses Von Neumann-Morgan Stein utility function, which is smooth, increasing and
concave everywhere (Figure 1), while in behavioral economics, the individual's utility is
described with Kahneman-Tversky utility function-a function which is smooth and increasing
everywhere except the reference point. That is to say, on the left of reference point it is a convex
function, and on the right side of the reference point it is a concave function (Figure 2). This also
indicates that the individual’s marginal utility of wealth increases before reaching the reference
point, and only after a reference point his marginal utility of wealth decreases.
Figure 1
76
Von Neumann - Morgan Stein Utility Function
Research of Mathematical Economics No. 1 2011
Figure 2
Kahneman-Tversky Utility Function
To simplify our analysis, we can assume that in the long run, individuals’ daily income must
be higher than his income target. (This is well understood. If a person simply cannot achieve his
daily revenue target, he might wonder whether the daily revenue target is set too high to achieve.
So he will decrease his daily income goals.) Thus, in our analysis, after the individual’s work in a
single day, he must be on the right side of the reference point in the utility function. We consider
the utility function on the right side of the reference point. The utility function is increasing, so
the derivative of the utility function v '( wl  y  r ) is greater than 0. And marginal utility is
decreasing with the increase in l, derivative of v '( wl  y  r ) is less than 0. For a further
derivative of v "( wl  y  r ) , what will happen? In fact, it can’t always equal to 0. If it equals to
0, v "( wl  y  r ) will be a negative constant, then the marginal utility eventually becomes
negative, which is incompatible with common sense. So v "'( wl  y  r ) will not equal to 0
(whether it is greater than 0 or less than 0 has respective meaning). I will discuss it in the
appendix. And, we believe that in normal circumstances, v "'( wl  y  r ) is greater than 0.
Let us consider the disutility of labor. We use c (l ) to indicate the disutility labor produced,
l is an individual’s working hours in a day. Obviously, c (l ) are incremental as l increases, that
is, the marginal disutility is positive, so the first derivative of c (l ) , i.e. c '(l ) is positive.
Marginal disutility is also increasing, because, with the increase in labor time, the uncomfort
77
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
individual felt would increase faster and faster. For example, after one hour’s work, the disutility
that one more minute makes is so small that we cannot feel. However, when we finish 20 hours
of work, one more minute work will make us tired and no longer willing to continue to work,
which shows marginal disutility is increasing. So the derivative of c '(l ) , i.e. c ''(l ) is greater
than 0. To simplify our analysis, here we assume that the increasing rate of marginal disutility is
constant, and thus c ''(l ) is a positive constant, and c '(l )  lc "(l ) . So the derivative of c ''(l )
will be 0. In short, c (l ) is an increasing and convex function.
To sum up, we can list the properties of utility function and the disutility function as
follows.
v '( wl  y  r )  0
v "( wl  y  r )  0
v "'( wl  y  r )  0
c '(l )  0
c "(l )  0
c "'(l )  0
c '(l )  lc "(l )
Next, we define the value function of an individual as u  v( wl  y  r )  c(l ) . Then the
individual faces the following question when he makes labor decision:
Max u  v( wl  y  r )  c(l )
subject to
0  l  24
Because when l is large, the increasing speed of marginal disutility will be faster than the
increasing speed of marginal utility, thus the optimal solution to the planning problem L must be
taken in time to 0  l  24 . Then, first-order condition is:
u
0
L
i.e. wv' ( wL  y  r )  c' ( L)
(1)
This is an implicit function where L is the dependent variable and w, r is the independent
variables. We can get worker's labor supply function as follows:
78
Research of Mathematical Economics No. 1 2011
L  L( w, r )
(2)
2.1 Worker's Labor Supply Elasticity
The partial derivative of L with respect to w may be:
L v '( wL  y  r )  wLv "( wL  y  r )

w
c "( L)  w2v "( wL  y  r )
So labor supply elasticity is:
e
wL wv '( wL  y  r )  w2 Lv "(wL  y  r )

Lw
Lc "( L)  w2 Lv "(wL  y  r )
(3)
We can deform equation (3) like:
e
wv '( wL  y  r )
v "( wL  y  r )
[1  wL
]
2
Lc "( L)  w Lv "( wL  y  r )
v '( wL  y  r )
So the elasticity is
e
where   
wv '( wL  y  r )
(1  wL )
Lc "( L)  w2 Lv "( wL  y  r )
(4)
v "( wL  y  r )
is the coefficient of absolute risk aversion. It evaluates the
v '( wL  y  r )
individual’s risk preferences and level of risk preferences. For a risk aversive individual, his
  0 . The more the individual averts risk, the larger  will be.
By the properties of v( wl  y  r ) and c (l ) which are described earlier, we can tell
wv '( wL  y  r )
 0 . Then the sign of elasticity will depend on whether
Lc "( L)  w2 Lv "(wL  y  r )
1  wL  is positive or negative. This can explain why at lower income levels labor supply
elasticity is likely to be negative. Because if  is very large, that is, the individual is extremely
averse to risk, even if w is very small, elasticity will still be negative. That is, even wages are low,
relatively higher wages will lead to reduction in working hours.
2.2 Changes of Labor Supply Elasticity Resulting From the Changes of Income Targets
In equation (3), the partial derivative of e with respect to r is:
79
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
e
w2 Lv "'( wL  y  r )c '( L)c "( L)

r c '( L)  w2 Lv "( wL  y  r )  2 [ w2v "( wL  y  r )  c "( L)]


(5)
Find specific derivation in Appendix 1.
Since
2
c '( L)  w2 Lv "(wL  y  r )   0
w2 Lc '( L)c "( L)  0 , the sign of
v "'( wL  y  r )  0 , we have
,
w2v "(wL  y  r )  c "( L)  0
,
e
will depend on the sign of v "'( wL  y  r ) . If
r
e
e
 0 . Here, we
 0 . If v "'( wL  y  r )  0 , we have
r
r
assume that v "'( wL  y  r )  0 (reasons are presented in Appendix 2). Then, e decreases
when r increases. Therefore, we can get the following conclusions:
When the labor supply elasticity is positive, the higher the income target is, the smaller the
labor supply elasticity will be.
When the labor supply elasticity is negative, the higher the income target is, the greater the
labor supply elasticity will be.
For these conclusions, we can explain them like this. There is no doubt that if wages remain
unchanged, the higher the income target, the longer the working time. In fact, on both sides of
equation (2), the partial derivative of L with respect to r is
Obviously,
L
wv "( wL  y  r )
 2
.
r w v "( wL  y  r )  c "( L)
L
 0.
r
For the two kinds of people, the ones with high income targets work longer, the ones with
low income target work shorter. With an additional unit of wage, income of people who work
longer will increase much more, but income of people who work shorter will increase only a
little.
If their labor supply elasticity is positive, the workers working longer can achieve utility
maximization by increasing working time only a little, while the workers working shorter hours
have to increase much more work before they can obtain additional utility maximization.
If their labor supply elasticity is negative, the workers working longer are able to reduce
80
Research of Mathematical Economics No. 1 2011
much more labor to obtain utility maximization, while the workers working shorter time can’t
reduce too much labor in order to obtain utility maximization.
2.3 Econometric Model Used in this Article
To estimate the elasticity of labor supply, we use the following form of measurement
equation:
yit   0  1Wageit   2Typeit *Wageit   it
Since it is difficult to get the data of income target, I choose the variable household
registration type as its proxy variable. y it is the daily working hours of individual i, Wage it
is the hourly wage, and Typeit is the individual’s household registration type.  it is a
stochastic error term. This is a panel equation so t denotes the time.
and
 1 represents the elasticity
 2 represents the effect on elasticity made by household registration type.
3.
Empirical Analysis
3.1 Data Source
Chinese CDC and the North Carolina University conducted China health and nutrition
survey (CHNS) in year 1989, 1991, 1993, 1997, 2000, 2004, 2006. A total of 19,000 samples
were taken from China's nine provinces (Guangxi, Guizhou, Heilongjiang, Henan, Hubei, Hunan,
Jiangsu, Liaoning and Shandong). This detailed survey, covers family relations, property, work,
income, health, diet and so on.
This article uses this statistics. For the research’s need, I will restrict the samples to those in
the urban informal sector employment whose jobs are family and individual business,
commercial, household workers, workers in informal enterprises, carpenters, nurses, tailors,
family clinics, repairers of electrical appliances and so on. All of them are the so-called
“self-employers”. The common feature of these workers is that they are able to freely decide their
own working hours. Considering the limitation of years, we choose the samples from year 1993,
1997, 2000, 2004 and 2006. Because of this restriction, the amount of samples satisfying our
requirements is reduced to 633.
81
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
3.2 Statistics Description
Table 1
Variables and their means
Vaviables
Unit
Means
Labor time
Hourly wage
Hours per day
Yuan per hour
0 represents urban and
1 represents rural
yuan
8.95577
4.649300
Standard
deviation
2.62497
11.43677
0.396524
7755.387
Household registeration type
Annual income
Min value
Max value
1
0.41
18
176.22
0.48956
0
1
8980
835.668
98400
In Table 1 we can find that the average working time of self-employer is 8.95577 hours per
day, nearly 1 hours more than the usual employee. Average hourly wage is 4.6493000 yuan per
day, minimum value is 0.41, maximum value is 176.22 and standard deviation is 11.43667, which
show that polarization among the self-employee is rather severe, income of minimum level can
only cover the basic need and the maximum level exceeds the social average level. In our
samples, 39.6524% of the workers is rural and the other is urban. The average income is
7755.787 yuan, which is lower than the average wage of all the workers.
3.3 Empirical Results
Table 2
Empirical Results
Independent variables
Coefficent
Standard deviation
p-value
Ln (hourly wage)
-0.1829
0.02225
<0.0001
Interactive item
0.069357
0.0294
0.0189
Constant
2.258651
0.1616
<0.0001
F-value: 2.04
<0.0001
Joint significance test
R2
0.6357
From Table 2 we can find that two independent variables explain the labor supply hour
significantly. We can find working time will decrease by 0.1829% if the wage increases by 1%.
This shows that labor supply elasticity of urban workers is -0.1829. The coefficient of interactive
item shows that labor supply elasticity of rural labor is 0.069357 less than urban labor.
According to the estimation result, labor supply elasticity is likely to be negative though the
82
Research of Mathematical Economics No. 1 2011
hourly wage level of self-employer is low. It supports that some people’s labor supply elasticity is
likely to be negative although their wage level is low.
Furthermore, labor supply elasticity of rural labor is significantly smaller than that of urban
labor. Combined with the assumption we mentioned before that the income target of rural labor is
comparatively low, the estimation results support the second conclusion of the model: if the labor
supply elasticity is negative, the higher the income target, the larger the labor supply elasticity.
4. Summary and Conclusions
This article has done a certain amount of expansions for the behavioral labor supply theory.
It discusses another reason why the labor supply elasticity is positive or negative, i.e. individual's
coefficient of absolute risk aversion. This article discusses how the income target affects the size
of labor supply elasticity based on Kahneman-Tversky utility function and theory of reference
dependent preferences. In addition, when doing empirical research, this paper examines
performances of the two different labor forces in the urban informal sector of China according to
the division of household registration type, which have never been done before.
For empirical results, this paper examines the labor supply elasticity in urban informal
sector and found that the elasticity is negative, which is contrary to neo-classical labor supply
theory, but consistent with the predictions of behavioral labor supply theory. In addition, the
article also compares the sizes of the elasticities of urban labor supply and rural labor supply in
informal sector, verifying the forecast of the model about the effects on elasticity made by the
income target.
This article still has some shortcomings, which are reflected in the following aspects. (1)
Assumptions of the model are too strict, especially in which increasing rate of marginal disutility
is constant. This assumption needs further empirical examination. (2) For the regression results in
Table 2, some coefficients need to be explained more. (3) There is a measurement error. The
hourly wage in this article is computed by dividing annual income by annual working hours
where there may be a "division bias”. For these shortcomings, I will make further improvements
in the future, for example, to discuss how the labor supply model will change when increasing
rate of marginal disutility is non-constant. Also, I will use more detailed "field data", or use
instrumental variables to estimate the individual's hourly wage, and thus avoid the "division
83
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
bias".
Despite above shortcomings, the expansion of the behavioral labor supply theory is worth
considering. And it shows it is necessary and meaningful to divide the labor in the informal sector
into two types and conduct a comparative study.
84
Research of Mathematical Economics No. 1 2011
Appendix 1
The Derivation of
e

r
[c "( L)
e
r
L 2
L
L
 w v "( wL  y  r )  w2 Lv "'(w  1)][c '( L)  w2 Lv "( wL  y  r )]
r
r
r
2
2
c '( L)  w Lv "( wL  y  r ) 
L
L
L
[c " (L )  w2 v " (w L y r)  2 w L "v' ( w 1 ) ] [c ' L(2 ) w L v " (wL
r
r
r

2
2
c ' ( L ) w L v" ( wL  y  )r

w2 v" ( w L
L
y )r 'c( L) 
r
2
L
w L"v( w L y ) r" (c ) L 2
r
2
2
c ' ( L ) w L v" ( wL  y  )r
w "L' (v w
L
y r) ]
L
y) 'r( c) . L(
r
Since Lc "( L)  c '( L) , the first two items of numerator offset. Differentiate equation (2),
L
wv "( wL  y  r )
 2
can be obtained. Substitute it into equation (6), we can get:
r w v "( wL  y  r )  c "( L)
e
w2 Lv "'( wL  y  r )c '( L)c "( L)

r c '( L)  w2 Lv "( wL  y  r )  2 [ w2v "( wL  y  r )  c "( L)]


Appendix 2
About v "'( wl  y  r )
We know that, after the reference point, v '( wl  y  r ) is a decreasing function and
85
w 1)
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
always positive, and v "'( wl  y  r ) is its second derivative.
Therefore, we consider a general function f ( x ) and its second derivative f "( x ) .
If f ( x ) is a decreasing function and it must always be greater than 0, when x is large
enough, the image is shown as Figure 3. This is a convex function, so f "( x )  0 . Therefore,
v "'( wl  y  r )  0 is a good and reasonable assumption.
Another evidence comes from the precautionary savings effect (see Kimball, 1989) in
macroeconomics. In macroeconomics, the premise of the existence of precautionary saving effect
is that the third derivative of utility function is positive. In real life, precautionary saving effect
obviously exists and that shows v "'( wl  y  r )  0
Figure 3
f ( x)
is a decreasing function and it must always be greater than 0
References
Zhou Yean, Zhang Quan, “Advances in Researches on Labor Supply”, Teaching and Research, 2006(2):
55-61.
86
Research of Mathematical Economics No. 1 2011
Ronald G. Ehrenberg , Robert S. Smith, Modern Labor Economics Theory and Public Policy, China Renmin
University Press, 2007.
Colin F.Camerer, George Loewenstein, Matthew Rabin, Advances in Behavioral Economics, Princeton
University Press and Russell Sage Foundation, 2004.
Liu Fengliang, Zhou Yean, Chen Yanbin, Behavioral Economics: Theory and Extension, China Economic
Press, 2008.
Kahneman, Daniel, Amos Tversky, "Prospect Theory: An Analysis of Decision under Risk", Econometrica,
1979(XLVII): 263-91.
Gary S. Becker, A Treatise on the Family, Harvard University Press, 1981.
Robert E. Lucas, Jr., Leonard A. Rapping, “Real Wages, Employment, and Inflation”, The Journal of
Political Economy, 1979 Sep.-Oct., Vol. 77, No. 5: 721-754.
Camerer, Colin, Linda Babcock, George Loewenstein, Richard Thaler (1997), "Labor Supply of New York
City Cabdrivers: One Day at a Time", Quarterly Journal of Economics, 1997, 112(2): 407-41.
Farber, Henry S., "Is Tomorrow Another Day? The Labor Supply of New York Cab Drivers", NBER
Working Paper 9706, 2003.
Fehr, Ernst, Lorenz Goette, "Do Workers Work More when Wages are High? Evidence from a Randomized
Field Experiment", IEW Working Paper 144, 2002.
Oettinger, Gerald S., “An Empirical Analysis of the Daily Labor Supply of Stadium Vendors”, J.P.E. 107,
1999 April: 36-92.
Lorenz Goette, David Huffman, “Reference-Dependent Preferences and the Allocation of Effort over Time:
Evidence from Natural Experiments with Bike Messengers”, Working Paper of University of Zurich and IZA,
Bonn.
M. Anne Hill, “Female Labor Supply in Japan: Implications of the Informal Sector for Labor Force
Participation and Hours of Work”, The Journal of Human Resources, 1989 Winter, Vol. 24, No. 1: 143-161.
John W. Pratt, “Risk Aversion in the Small and in the Large”, Econometrica, 1964 Jan.-Apr., Vol. 32, No. 1/2:
122-136.
Mark Huggett, Sandra Ospina, “Aggregate Precautionary Savings: When is the Third Derivative Irrelevant?”,
Journal of Monetary Economics, 2001(48): 373–396.
Miles S. Kimball, N. Gregory Mankiw, “Precautionary Saving and the Timing of Taxes”, The Journal of
Political Economy, 1989 Aug., Vol. 97, No. 4: 863-879.
W.A. Lewis, “Economic Development with Unlimited Supply of Labor”, The Manchester School, 1954, Vol.
22, No. 2: 139-91.
Michael P. Todaro, “A Model of Labor Migration and Urban Unemployment in Less Development
Countries”, American Economic Review, 1969, Vol.59: 138-148.
87
Income Targets and Labor Supply Elasticities: Evidence from China Health and Nutrition
Survey
(Responsible Editor: Xu Lingjue) (Proofreader: Ren Junqiushi)
88
Research of Mathematical Economics No. 1 2011
A Dynamic Model of the Optimal Carbon Tax and Sustainable
Development
Feng Junlong
(School of Economics, Renmin University of China)
Abstract: With the development of the industrialized society, global warming caused by the increasing emission
of greenhouse gases has become a threat to the whole human beings. Under this pressure, governments discuss
every possible solution, among which levying carbon tax is one of the most considered way. In a view of
sustainable development, this paper discusses the optimal tax rate within a dynamic frame. Besides, the optimal
reforestation subsidy and the relationship between this subsidy and forest stock are also studied as forest is a
social common capital absorbing CO2, which can be either felled for present profit or reforested for future welfare.
Theoretically, this paper is a development of the Ramsey dynamic model, while for the reality significance this
paper provides some positive ideas for the policy making.
Keywords: Carbon Tax, Reforestation Subsidy, Imputed Price
1. Introduction
Nowadays, global warming has become a threat to the whole mankind. One of the
fundamental reasons is the emission of greenhouse gases, among which CO2 is the first to blame.
However, as a sign of modern industry, the emission of CO2 is a result of the economic
development, and even the utility of consumers increases with CO2 emission since almost all
consumption is connected with it. Therefore we have a trade-off in the decision of CO2 emission.
Meanwhile, forests provide woods as a vital capital for an industrialized economy. However,
on the other hand, forests make great contribution to CO2 absorbing. According to a report from
IPPC (1991a, 1992, 1996a, 2001a, 2002), if we keep the rate of felling trees and releasing CO2,
the average global temperature by the end of 21 century will be 3-6 degrees higher than that
before the Industry Revolution. So, we also meet a tradeoff between felling trees and
reforestation.
89
A Dynamic Model of the Optimal Carbon Tax and Sustainable Development
After the Copenhagen Conference in 2009, to levy carbon tax has become a hot topic
worldwide. Then it’s important to decide the optimal tax rate which will exactly cover the
negative externality for firms and make no dead-weight loss. And as reforestation calls for money,
part of the tax can be transformed to be subsidies for it.
Therefore, this paper is dedicated to the study of the optimal carbon tax rate and the optimal
subsidies, as well as the relationship between subsidies and the forest stock. In this field, Keeling
(1983), Takahashi et al. (1980) and Uzawa (2005) have developed some fundamental models. But
these models took only CO2 into consideration and the factors like forests were treated as
exogenous. And for simplicity, some of them only discussed discrete cases of two periods. For
forests as a social common capital, Solow (1974a, b), Sen (1982), Page (1997) made great
contribution to solving the very problem within a dynamic framework.
This paper will develop a dynamic model in long run equilibrium. Forests as a social
common capital will be treated endogenous. The Ramsey-Cass-Koopmans dynamic model is
applied and developed here to work out the optimal tax rate as well as the subsidies. This paper is
within a framework of Uzawa’s theory of social common capital, while some evolutions are
made so that the model is much more realistic.
The rest of the paper is organized as follows: divided into 3 parts, section 2 presents basic
model, where the propositions to be proposed are formally derived and proved; section 3 presents
the main results of this paper and my concluding remarks.
2. The Dynamic Model
2.1 Consumers1
First, we study the consumers’ behavior. To simplify the problem, we discuss a case where
utility can be aggregated as a whole. We consume that there is only one goods in the market and
the price is given. Then, we derive the utility function,
u  u(c, ac ,Vw ,Vc )
(1)
where c is consumption， ac is the CO2 emission in consumers’ life, Vw is forestry stock and
1
In this paper, to focus mainly on the target problems, we omit discuss of constrains and the first order
conditions for consumers and producers. Nonetheless, the analysis is already set within a frame of equilibrium, so
the result derived will satisfy the omitted conditions automatically.
90
Research of Mathematical Economics No. 1 2011
Vc is CO2 stock.
Obviously, in reality, the utility function should be independent for (c, ac ) , Vw , Vc . The
reason is that in a short period, the stock of either CO2 or forests don’t affect consumers’ daily
choice. Therefore, the utility function can be
u(c, ac ,Vw ,Vc )   (Vc ) (Vw )u(c, ac )
where  (Vc ) and
(2)
 (Vw ) are the extent to which the consumers are affected by a marginal
increase of CO2 and forests, which can be referred to as the impact index. Therefore they satisfy
the following conditions:
 (Vc )  0,  ' (Vc )  0,  '' (Vc )  0
(3)
 (Vw )  0,  ' (Vw )  0,  '' (Vw )  0
(4)
Meanwhile, we assume that u (c, ac ) is neo-classical.
To simplify the discussion, we introduce the impact coefficient. We define
 (Vc )  
 ' (Vc )
 ' (Vw )
, (Vw ) 
 (Vc )
 (Vw )
as the impact coefficient which means the relative rate of the marginal change in the impact index
due to the marginal change in CO2 or the forest stock (Uzawa, 2005). From the definition, we
have
 (Vc )  0, '(Vc )  0
 (Vw )  0, '(Vw )  0
2.2 Producers
Here, we divide the producers into two sectors. One is all the industries connected with
woods, including felling trees and reforestation. As forestry is social common capital, we assume
there is only one designer in this sector to decide the amount to fell and the amount for
reforestation. The other is the firms producing the only goods as mentioned in 2.1. And we
assume this market is competitive.
2.2.1 Forestry
91
A Dynamic Model of the Optimal Carbon Tax and Sustainable Development
Before we come to the problem of decision making, an important fact should be introduced.
Since trees can regenerate by themselves, it’s necessary to introduce a regenerating function
 (Vw ) , then we have  '' (Vw )  0 , and there exist V w ,Vw , such that  (V w )   (Vw )  0 .
Now if we add felling and reforestation to the model, a dynamic function for the forest stock
can be:

Vw   (Vw )  YVw  X
(5)
where YVw is the amount of reforestation and X is the amount felled.
We assume the cost for felling trees is connected with the stock of the trees and the amount
to be felled, so the cost function for felling trees C ( X , Vw ) satisfies the following conditions:
C ( X ,Vw )  0, C X ( X ,Vw )  0, CVw ( X ,Vw )  0
CXVw  0, CXX , CVwVw  0, CXX CVwVw  CXVw 2  0
(6)
And the cost for reforestation is also connected with the stock as well as the amount to be
reforested, so the cost function for reforestation B (YVw , Vw ) satisfies the following conditions:
B(YVw ,Vw )  0, BYV (YVw ,Vw )  0, BVw (YVw ,Vw )  0
w
BYV Vw  0, BYV YV , BVwVw  0, BYV YV BVwVw  BYV Vw 2  0
w
w
w
w
w
w
(7)
Therefore the profit of this sector, to be denoted as  w , is derived as:
 w  pw X  B(YV ,Vw )  C ( X ,Vw )
w
(8)
2.2.2 Other Firms
Now we’ll turn to the firms producing the only goods for consumers. As we have assumed,
the price pg is given. And we assume CO2 emission per product of each firm equals2, then we
have:
   pg x   a p
where   is denoted to be the
2
92
(9)
 th firm’s profit, a p is the CO2 emission by firms,  is
In reality, this assumption is satisfied. We only convert the product into carbon unit.
Research of Mathematical Economics No. 1 2011
carbon tax per unit of emission.
As for the whole sector,
 g  

(10)
Therefore, the profit of this sector can be derived as
   g   w   ( pg x   a p )  pw X  B(YVw ,Vw )  C ( X ,Vw )

(11)
2.3 The Imputed Price and Social Optimum
Now, as we have discussed both sides of the market, we can study the optimal choices of
consumers and firms. But before that, it’s necessary to introduce a new concept, i.e. the imputed
price. The imputed price means the marginal welfare change due to marginal change of CO2 or
forest. Denote the imputed price of CO2 and forest to be  ,  . Therefore, the optimal carbon tax
and reforestation subsidies should be equal to these factors’ imputed price, so that they could
reflect the very welfare change and cause no dead weight loss.
And for simplicity, we assume that in an equilibrium condition, the society’s utility
maximization and the profit maximization of producers could be reached together, and they are
connected by the following equation:
  u
where
(12)
 is the inverse of marginal utility of income (as assumed, equals profit). So if we
convert the imputed price to discuss utility, we have:
  
(13)
  
(14)
Now, as there are two aspects to discuss, the optimal carbon tax together with the optimal
subsidies for reforestation, and the relationship between the subsidies and forest stock, we’ll
study a symmetric problem from the utility maximization and profit maximization. As mentioned
above, such two maximizations are equal and only two sides of a unified equilibrium. To discuss
them separately is only a convenient way to solving the problems.
2.3.1 Social Utility Maximization
In order to achieve sustainable development, we maximize the following Ramsey utility
93
A Dynamic Model of the Optimal Carbon Tax and Sustainable Development
function U :

U   ut e t dt
(15)
0
We should guarantee each ut is maximized and the integral converges.
The increase of CO2 in the atmosphere satisfies the following equation:

Vc  a  Vw
(16)
a  ac   ap
(17)
where

Therefore we derive the imputed utility function at time t:
H t   (Vc ) (Vw )ut (c, ac )  t ( (Vw )  YVw t  X t )  t (at  Vwt )
(18)
The Hamiltonian Equation should be

t  t  

 t   t 
H t
Vwt
(19)
H t
Vct
(20)
with the transversality conditions:
lim tVwt e t dt
(21)
lim tVct e t dt
(22)
t 
t 


The steady state should be     0 3.
t
t
From (19) (20) (13) (14) (11) (2), we derive
* 
1  (Vc )  (Vw )
(

)
   ' 

* 
3
 (Vc )


Here, the realistic meaning of the steady state can be interpreted that policies are relatively steady and not
flexible with time.
94
(23)
(24)
Research of Mathematical Economics No. 1 2011
Then it’s easy to derive the expressions of the optimal carbon tax and the optimal
reforestation subsidies:
 v* 
 v* 
 (Vc ) v


(25)
1  (Vc )
(
  (Vw )) v
'
 

(26)
Here we have the following conclusions:
Proposition 1: The optimal carbon tax should be proportional to firms’ profit. It’s also in
direct ratio with people’s CO2 coefficient and is the inverse of utility discount rate.
Proposition 2: Firms should take the responsibility of reforestation, which is proportional to
profit and affected by the coefficient of CO2 and forestry.
2.3.2 Producer’s Profit Maximization
Now we’ll turn to the other side of the market to study the dynamic relationship between
subsidies and forests stock. First, we construct a profit maximizing problem.
Similarly we have

P   t e t dt
(27)
0
where P is denoted to be the total profit with sustainable development.
Then maximize the profit of each time t with the converging conditions. From (11) we get
H t   ( pgt x t   a pt )  pwt X t  B(YVw t ,Vwt )  C ( X t ,Vwt )   t ( (Vwt )  YVw t  X t )  t (at  Vwt )

(28)
Take partial differential of X t , YVw t and set them to be 0，we get
pw  C Xt   t
(29)
BYwt   t
(30)
In the mean time, with the Hamiltonian Equation, we have

 t  [   '(Vwt )] t  [CV ( X t ,Vwt )  BV (YV ,Vwt )]  t
w
w
(31)
w
with the transversality conditions being
95
A Dynamic Model of the Optimal Carbon Tax and Sustainable Development
lim  tVwt e t  0
(32)
t 
To simplify the model, we’ll fix  t at first and make it changeable after we analyze the
steady state characters. So here it’s a given parameter at time t. Then we’ll focus on a dynamic
optimum time-path to be denoted as ( X t , YVwt , Vwt ) and we need to find a positive continuous
time-path of the imputed price  t . With the differential equations (31) and (5), under the
constraints of (29) (30), we’ll derive the following phase diagram of Vw and  4:

B
A'
C
E
*
B'
C'
A
O
Vw
Vw0
Figure 1
The Phase Diagram of
Vw
and

From figure 1, we can get the following conclusions:
Proposition 3: If the price of woods increases, curve AA’ shifts upward with a slight upward
shift of curve BB’. Therefore, the long run equilibrium forest stock will decrease which calls for
an increase of reforestation subsidies.
4
96
The derivation of the diagram is proved in detail in the index.
Research of Mathematical Economics No. 1 2011
Proposition 4: If time discount rate increases, curve BB’ shifts downward leading to a
decrease in both long run forest stock and reforestation subsidies.
Proposition 5: If carbon tax increases, curve BB’ shifts upward so that the long run forest
stock and reforestation subsidies increase.
3. Conclusions
This paper develops some basic models for sustainable development. In a dynamic view, the
optimal carbon tax together with the optimal subsidies for reforestation, and the relationship
between the subsidies and forest stock are discussed. Through analysis, the paper gives five
conclusions rearranged as follows:
Proposition 1: The optimal carbon tax should be proportional to firms’ profit. It’s also in
direct ratio with people’s CO2 coefficient and is the inverse of utility discount rate.
Proposition 2: Firms should take the responsibility of reforestation, which is proportional
to profit and affected by the coefficient of CO2 and forestry.
Proposition 3: If the price of woods increases, curve AA’ shifts upward with a slight
upward shift of curve BB’. Therefore, the long run equilibrium forest stock will decrease which
calls for an increase of reforestation subsidies.
Proposition 4: If time discount rate increases, curve BB’ shifts downward leading to a
decrease in both long run forest stock and reforestation subsidies.
Proposition 5: If carbon tax increases, curve BB’ shifts upward so that the long run forest
stock and reforestation subsidies increase.
Nevertheless, further work should be done in order to perfect the model. The optimal tax
rate and subsidies derived in this paper are connected with some physical and psychological
parameters like
 (Vc ) ,  (Vw ) and  . To get the average level of these parameters, some
statistic work is required.
97
A Dynamic Model of the Optimal Carbon Tax and Sustainable Development
Appendix
The Derivation of Figure 1
First, take differential of (29) and (30):
dX  [
C Xvw
]dVw 
C XX
dYVw  [
BYV Vw
w
BYV YV
w
1
1
dpw 
d
C XX
C XX
1
]dVw 
BYV YV
w
w
d
(33)
(34)
w
With the consumption of (6) and (7), we derive
YVw
YVw
X
X
X
 0,
 0,
 0,
 0,
0
Vw
p w

Vw

The conditions of steady state are
 (Vw0 )  YV *  X *  0
w
[   '(Vwt 0 )] t 0  [CVw ( X t * ,Vwt 0 )  BVw (YVw * ,Vwt 0 )]  t  0
(35)
(36)
Take differential of (35) and (36) (  t is given), and with (33) (34), we get
0 
 1

 dp
 11 12   dVw   C XX
 w 

  *    C
  d 
  21  22   d   XVw


 C XX  
0
where
11 
C Xvw
C XX
12 

BYV Vw
w
BYV YV
w
w
1
1

0
C XX BYV YV
w
98
 '  0
w
（37）
Research of Mathematical Economics No. 1 2011
 21 
C XX CVwVw  C XVw 2
C XX
 22  

BVwVw BYV YV  BYV Vw 2
w
w
w
C XVw
C XX

w
BYV YV
BYV Vw
w
BYV YV
w
  ''  0
w
 (   ')  0
w
It’s obvious that
  11 22  12  21  0
Now, we’ll discuss the relationship between  and Vw .
(1) (

d
)    11  0
dVw Vw 0
12

Then, V  0 is an upward sloping curve denoted to be AA’. On the right side of AA’,
w


Vw  0 while on the left side of AA’, Vw  0 .
(2) (

d
)    22  0
dVw  0
 21


Then,   0 is a downward sloping curve denoted to be BB’. Above BB’,   0 while

below BB’,   0 .
Therefore E is a saddle point and only one pair of solution paths, CE and C’E exists.
References
Aghion, P. and Howitt, P., “A Model of Growth through Creative Destruction”, Econometrica, 60, 1992, pp.
323-352.
Keeling, C.D., “The Global Carbon Cycle: What We Know from Atmospheric, Bio-Spheric, and Oceanic
Observations”, Proceedings of Carbon Dioxide Research, Science and Consensus, United States Department of
Energy, Washington, DC, Ⅱ, 1983, pp. 3-62.
Page, T., “Sustainability and the Problem of Valuation”, in R. Costanza (ed.), Ecological Economics: The
Science and Management of Sustainability, New York: Columbia University Press, 1997, pp. 58-74.
Ramsey, F.P., “A Mathematical Theory of Saving”, Economic Journal, 38, 1928 pp. 543-559.
99
A Dynamic Model of the Optimal Carbon Tax and Sustainable Development
Sen, A.K., “Approaches to the Choice of Discounted Rate for Social Benefit-Cost Analysis”, in R.C. Lind,
K,J. Arrow, and G.R. Cores. Discounting for Time and Risk in Energy Policy, Baltimore: Johns Hopkins
University Press, 1982, pp. 325-353.
Solow, R. M., “Intergenerational Equity and Exhaustible Resources”, Review of Economic Studies, 41, 1974a,
pp. 29-45.
Solow, R. M., “The Economics of Resources, or the Resources of Economics.” American Economic Review,
64, 1974b, pp. 1-14.
Uzawa, H., Economic Analysis of Social Common Capital, Cambridge University Press, 2005, pp. 49-57,
257-292.
Uzawa, H., “Imputed Price of Greenhouse Gases and Land Forests”, Renewable Energy 3, 1992a, pp.
499-511.
(Responsible Editor: Zhang Lingyun) (Proofreader: Xu Lingjue)
100
Research of Mathematical Economics No. 1 2011
A Study of Chinese Residential Electricity Demand Elasticities
Ren Junqiushi
(School of Economics, Renmin University of China)
Abstract: As energy plays a more and more significant role in the economy, electric power as an extremely
important kind of energy has received more and more attention. This paper measures short-term residential
electricity demand elasticities of Beijing as the representative of China in two ways: econometric analysis (using
the ELES model) and terminal-energy-need method (adopting a double-log demand function). Both methods
suggest that the income elasticity of Chinese residential power demand is about 0.5, while the self-price elasticity
is around -0.3. Further, this paper put forward approaches to estimate long-term elasticities by improving these
two methods. The findings of this research can help better inform public policy makers and regulators about the
responsiveness of residential electricity consumers to price and income changes, as well as scholars who would
like to research more in-depth.
Keywords: Residential Electricity Demand, Income Elasticity, Price Elasticity, Econometric Analysis,
Terminal-Energy-Need Method
1. Introduction
1.1 Research Meaning and Significance
Energy resources have served as a strong constraint for long-term economic development of
China, and even of the whole world. As the United Nations Millennium Development Goals
Report of 2009 noted, for the realization of the world’s economic sustainable development, every
country must try to change its economic development approach so that the globe could break this
constraint.
Among all the energy resources, electricity occupies a ponderable status. In fact, the electric
power, which is an important kind of fuel as well as a clean secondary source, has had a
far-ranging appliance in different sectors of society. Moreover, in recent years, Chinese residents’
electricity consumption has been continuously increasing and occupying a significant weight in
101
A Study of Chinese Residential Electricity Demand Elasticities
the total consumption of electric power. Between 2005 and 2007, the annual consumption of
residents hits 282.48 billion degrees, 325.16 billion degrees and 358.40 billion degrees
respectively, with an average annual growth rate of 12.7%; in the next few years, the average
annual growth rate has maintained a high bit at 10%, and residents’ electricity consumption
weighed 12.86%, 10.04% and 12.53% of the total electricity consumption respectively.1 Thus,
taking into account the short-term requirement of supply-demand balance and the long-run
energy substitutability, we need to analyze more in-depth the demand, especially the demand of
residents, for power.
Currently, the abroad has already had many studies exploring residential electric power
demand, but few researches have considered residential electricity demand in Asian countries,
especially in the developing ones. As a matter of fact, most researches about Asian electricity
market have focused on the industrial demand merely. This paper will study the electricity
demand of Chinese residents based on the representative city Beijing, thus fill up this blank.
In the study of residential electric power demand, a very important and fundamental subject
is to measure demand elasticity. Theoretically, elasticity is not only a basic economic concept but
also an indispensible factor in demand description; practically, no matter what we aim
at—understanding Chinese electricity market, judging whether it would be effective to control
residents’ power using by adjusting the price of power, or estimating how much more burden is
put on residents when electricity price increases a certain amount—we cannot reach our goal
without studying elasticity first. Therefore, this paper is meaningful when it puts forth effort to
measure the electric power demand elasticity of Chinese residents.
1.2 Overview
In retrospect, many models and theories have been proposed to research the demand of
electricity. The most commonly used methods in studying residents’ electric power demand are
econometric analysis, terminal-energy-need method and artificial intelligence.
Econometric analysis aims at describing the interdependent or causal relationships among
different economic variances according to a set of simultaneous equations derived from certain
economic theory and hypothesis. If we could set up simultaneous multi-equation econometric
model which would describe the relationships between electric power demand and relevant
1
Statistics are obtained from http://www.cpnn.com.cn/.
102
Research of Mathematical Economics No. 1 2011
economic variables accurately, we would be able to estimate the elasticity of power demand
accordingly. (See T. R. Lakshmanan and William Anderson, 1980; Xavier Labandeira, José M.
Labeaga and Xiral López-Otero, 2009)
The basic thought of terminal-energy-need estimation is to figure out the consumption of
fuel through terminal equipment. As for residents, we can regard equipment such as lighting,
cooking, home appliances (television, refrigerator and washer), heating and air conditioners as
terminators, thus estimate resident power consumption based on their universality rates,
occupation coefficient and fuel outgo rates, etc. (Tsuyoshi Ueno, Ryo Inada, Osamu Saeki and
Kiichiro Tsuji, 2005). This thought averts from sophisticated mathematical deduce; moreover, it
is able to resolve problems such as the inevitable (sometimes rather large) deviation from reality
caused by econometric models, which endows it a higher precision when being used to estimate
elasticity.
A famous representative of artificial intelligence—artificial neural network has been
successfully applied to analyzing short-term electric power market, and valued by scholars at the
early part of 1990s because of its outstanding study function, ability to handle the non-linear
relation between input and output variables and shorter time needed in the calculation process
(see Abdallah Al-Shenri, 1999). However, to make use of an artificial intelligence tool to steer
demand analysis, we need a great deal of training data, and the accuracy would be seriously
compromised when historical data was limited; furthermore, this tool lacks the ability to study
uncertainty or to handle misty information.
Therefore, this paper will abandon artificial intelligence tool, but make use of the two
previous methods: econometric analysis (using the ELES model) and terminal-energy-need
approach (adopting a double-log demand function), according to data between 1999 and 2008 got
from Beijing Statistical Yearbook (compiled by Beijing Bureau of Statistics and National Bureau
of Statistics Survey Office in Beijing, and published by China Statistics Press), to estimate
short-term income elasticity and self-price elasticity of residential electricity demand in Beijing,
arrive at a conclusion and then analyze this conclusion.
This paper is structured as follows: section 1 is an introduction; section 2 is the application
of econometric analysis method, in which the specific ELES model is introduced to estimate
short-term elasticity of demand; section 3 is based on the thought of terminal energy needs, in
which a double-log demand function is built to re-estimate elasticity, and the results are
103
A Study of Chinese Residential Electricity Demand Elasticities
compared with those of the ELES model; section 4 includes conclusions and analysis; section 5
discusses some probable directions of future development, in which the two ways described in
this paper are improved so that they can be used to estimate long -term elasticities of demand.
2.
Extended Linear Expenditure System Model
2.1 Development of Model
Beginning from the 1980s, scholars have developed and applied the Almost Ideal Demand
System (AIDS):
n
i  i  ij log p j  i log
j 1
M
(i  1, 2,
p
, n)
(1)
where i is the consumption expenditure of the ith commodity;  i 、  ij and i are
parameters needed estimation; p j is the price of the jth good; M represents the total
expenditure; p is the general price index, and is defined as follows：
log p   0    k log pk 
k
1
kl log pk log pl
2 k l
(2)
Since (1) (2) are non-linear equations, the parameter estimations are too troublesome. Thus
Deaton and Mulbauer (1980) proposed to make use of the Stone indexes:

k
log pk to
replace (2), which give rise to (3):
n
n
j 1
j 1
i  i  ij log p j  i log M   j log p j
(3)
where  j   i j .
However, empirical research found that the parameters in the AIDS often failed to satisfy the
coefficient constraints required by demand functions (see Thomas J. Lareau and Joel Darmstadter,
1983). To deal with this problem, a lot of scholars began to introduce the linear expenditure
utility function proposed by Klein and H. Rubin:
n
n
i 1
i 1
U   U i ( X i )   bi ln( X i  ri )
104
(4)
Research of Mathematical Economics No. 1 2011
This function assumed that the utility of commodity i could be represented by the logarithm
of the difference between actual demand X i and basic demand ri , thus total utility function
could be calculated by adding up utility functions of all commodities; bi represented the
marginal budget share of the commodity i.
Among all the theories derived from this utility function, a rather famous one is the Linear
Expenditure System (LES) proposed by English economist R. Stone:
n
Pi X i  Pr
, n)
i i  bi (C   Pj rj ) (i  1, 2,
(5)
j 1
where Pi X i is the consumption expenditure of the ith good or service; C is the total
consumption expenditure of all goods and services, and C 
n
PX
i 1
i
i
; Pi is the market price
of the ith good or service; ri is the basic demand of the ith good or service; bi is the marginal
budget share of the ith good or service, namely, the ratio of additional expenditure leaving basic
n
expenditure outside used to consume the ith good or service. Obviously,
b
i 1
i
 1.
Again, because the parameters of LES model are nonlinear, which makes it difficult to
estimate them, later economist Lluch revised the LES model with income Y replacing total
budget expenditure C, and marginal propensity to consume bi
*
taking the place of marginal
n
budget share bi . As total expenditure is less than total income, we have
b
i 1
i
*
 1 . Thus the
Extended Linear Expenditure System (ELES) follows:
n
*
Pi X i  Pr
i i  bi (Y   Pj rj ) (i  1, 2,
, n)
(6)
j 1
The Extended Linear Expenditure System not only explains the impact income and prices
bring on residential consumption pattern, but also considers varies residential consumption
expenditure as interrelated and mutually constraining behaviors. In addition, it divides people’s
demand into basic needs and beyond-basic needs, in which basic needs have nothing to do with
the level of total expenditure, and residents would not arrange beyond-basic needs according to
105
A Study of Chinese Residential Electricity Demand Elasticities
corresponding marginal propensity to consume unless basic needs were satisfied. Theoretically,
this model makes it possible to analyze residential basic needs, marginal propensity to consume
and different kinds of demand elasticity.
2.2 Application of the Model
Due to the fact that the Extended Linear Expenditure System itself is a function of demand,
elasticity can be obtained by closing into corresponding elasticity formula.
For income elasticity, based on the formula:  
X Y
. , we have
Y X
n


*
b
(
Y

Pj rj ) 

i

*
*
X Y
 
j 1
 . Y  bi Y  bi Y
i  i . 
ri 
Y X i Y 
Pi
Vi
 X i Pi X i




(7)
where Vi  Pi X i .
For own price elasticity of demand, according to the formula:  
X P
. , there is
P X
n


*
b
Y

Pj rj 

i

X P
 
j 1
 . Pi
 ii  i . i 
ri 
Pi X i Pi 
Pi
 Xi




*
n
b
* Pr
i i
  i ( Pr
1
i i  Y   Pj rj )  (1  bi )
Vi
Vi
j 1
(8)
Adapting the ELES model to calculate residential electricity usage, we are able to establish
the regression equation of electric power consumption:
n
n
j 1
j 1
Pe X e  Pe re  be* (Y   Pj rj )  ( Pe re  be*  Pj rj )  be*Y
where the parameters need estimation are:
(1) The basic demand for electricity Pe re ;
106
(9)
Research of Mathematical Economics No. 1 2011
n
(2) The total basic demand
Pr ;
j 1
j j
*
(3) The propensity to consume be .
Among the parameters above,
be* can be obtained from regression equation
Ve  ae  be*Y , but the estimation of Pe re and
n
Pr
j 1
requires us to establish econometric
j j
equations of all residential consumptions. The classification of statistical yearbooks gives us a
wonderful guidance: foods, clothing, household equipment and service, medical care, transport
and communications, education and recreation, housing, and miscellaneous items. We mark each
of them successively with a number ranging from 1 to 8. And because electricity is included in
housing, it is possible to remove it from the residence. In the final analysis, we are able to
establish 9 econometric equations:
Vi  ai  bi*Y (i  1, 2, ,8)
(10)
Ve  ae  be*Y
Obtaining a and b through regression, and then according to
n
n
 Pr 
i 1
i i
a
i 1
n
i
1   bi
(11)
*
i 1
and
 n

  aj 

Pe re  ae  be*  j 1n

*
1

b

j


j 1


(12)
we find the way to estimate all the necessary parameters.
Based on the data between 1999 and 2008 gained from Beijing Statistical Yearbook, after
calculation, this paper gets the following OLS regression equations:
107
A Study of Chinese Residential Electricity Demand Elasticities
Table 1
OLS Regression Equations in ELES Application
Item
Regression Equation
R2
Foods
Clothing
Household equipment and
service
Medical care
Transport and communications
Education and recreation
Housing(without electricity fee)
Miscellaneous items
Electricity fee
C1 = 0.1502Y + 1295.1
C2 = 0.0562Y + 93.386
0.9857
0.9763
C3 = 0.0206Y + 490.8
0.8259
C4 = 0.0571Y + 29.543
C5 = 0.1125Y - 476.71
C6 = 0.079Y + 536.02
C7 = 0.0368Y + 76.823
C8 = 0.017Y + 221.4
Ce = 0.0069Y + 107.45
0.8456
0.8286
0.7461
0.8218
0.6998
0.8180
The smallest R 2 in this table is 0.6998, moreover, items except for the miscellaneous or
education and recreation have R 2 larger than 0.81, which means that the data fit relatively well.
Bringing the regression coefficients into (11) (12) (7) (8), we have the following results:
(1) The residential marginal propensity to consume power is around 0.0069;
(2) The basic power need is about 137.16 yuan per person per year;
(3) The income elasticity of demand is about 0.533 ;
(4) The self-price elasticity is -0.38 approximately.
3.
Terminal-Energy-Need Method
3.1 Brief Introduction of the Method and Model
Terminal-Energy-Need method aims at calculating energy consumption through terminal
equipment. Its principle is to determine the energy consumption from the number of devices, the
utilization rate of appliances and the efficiency of energy usage, rather than the quantity of
energy consumption itself. Generally speaking, residential electricity terminal is considered to
include lighting, appliances, heating, air conditioning and so on. The elasticity of residential
power demand can be estimated on the basis of knowledge about the power usage of typical
devices. This method is directly based on the user's demand, thus avoiding the more complex
theoretical analysis and enigmatic problems such as the failure of solving mathematical models,
so that the accuracy of estimation is improved.
Building on Terminal-Energy-Need method, and making use of double-log model, this
108
Research of Mathematical Economics No. 1 2011
paper will go on measuring the elasticity of residential electricity usage. The reason why a
double-log model is used here is that it is not only easier to explain, but also more logical. The
specific model is set as follows:
ln Q  a1  a2 ln Y  a3 ln W  a4 ln Pe  a5 ln Pf  u
(13)
where Q is the annual electric power consumption per capita (in degrees); Y is the annual income
per head (in yuan); W is the total annual electricity consumption of three representative
residential power terminals (lighting, TV, air conditioning) per year per capita (in degrees); Pe
is the consumer electricity price index; Pf is the fuel retailing price index; and u represents the
random error.
The income elasticity of power demand can be measured by a2 , and the self-price elasticity
can be got by estimating a4 .
3.2 Numerical Calculation
In order to determine the relation between ln Q and ln Y , lnW , ln Pe as well as
ln Pf , I begin with adopting SPSS18.0 to make multiple regression and at the same time
diagnose multi-collinearity. The result is displayed in Table 2.
Table 2
Multiple Regression and Multicollinearity Diagnosisa
Non-standardized
coefficient
Standard
B
deviation
-2.985
0.491
0.933
0.086
-0.044
0.044
(Constant)
Income
Appliance
Electricity
-0.181
price
Fuel price
0.270
a. Dependent variable: electricity
Standardized
coefficient
t
Sig.
Multicollinearity
statistics
Tolerance
VIF
0.912
-0.041
-6.077
10.794
-1.006
0.002
0.000
0.361
0.064
0.276
15.613
3.625
0.139
-0.068
-1.307
0.248
0.170
5.870
0.181
0.142
1.492
0.196
0.050
19.811
Observation finds that the regression results are inconsistent with common sense: the
coefficient of three typical power consumption terminals was negative. Further observation
reveals that the variance inflation factors are relatively great, especially in terms of income and
109
A Study of Chinese Residential Electricity Demand Elasticities
fuel prices, each of which has a VIF of 15 or more. These results show that there exists
significant multicollinearity in the regression equation.
In view of this, I apply the commonly used method in eliminating multicollinearity--Ridge
regression to fix the results of multiple regression. Table 3 displays the standardized regression
coefficients of independent variables under different values of k.
From Table 3, we can see that the regression coefficients change quickly when k varies from
0 to 0.25, which is the abnormal change caused by multicollinearity. However, when k reaches
0.25, the ridge regression coefficients become stable, indicating that multicollinearity has been
almost eliminated. Reference to the value of R 2 , one can find that R 2 =0.988 when k = 0.25,
which is still a great value. Thus I select the ridge parameter k = 0.25.
110
Research of Mathematical Economics No. 1 2011
Table 3
Standardized regression coefficients under different ridge parameters
k
R2
Income
Appliance
Electricity price
Fuel price
0.00
0.999
0.928
0.067
-0.050
0.051
0.05
0.997
0.666
0.261
-0.077
0.095
0.10
0.995
0.521
0.325
-0.078
0.137
0.15
0.992
0.445
0.341
-0.075
0.144
0.20
0.991
0.402
0.340
-0.074
0.196
0.25
0.988
0.344
0.338
-0.084
0.186
0.30
0.985
0.337
0.334
-0.083
0.181
0.35
0.982
0.331
0.330
-0.083
0.177
0.40
0.978
0.324
0.326
-0.082
0.173
0.45
0.974
0.318
0.323
-0.082
0.170
0.50
0.970
0.313
0.319
-0.081
0.166
0.55
0.965
0.307
0.316
-0.081
0.162
0.60
0.961
0.302
0.312
-0.081
0.157
0.65
0.956
0.297
0.309
-0.080
0.154
0.70
0.952
0.292
0.306
-0.080
0.149
0.75
0.947
0.287
0.302
-0.079
0.145
0.80
0.942
0.283
0.300
-0.079
0.141
0.85
0.937
0.279
0.297
-0.078
0.137
0.90
0.932
0.275
0.294
-0.078
0.133
0.95
0.927
0.271
0.291
-0.077
0.129
1.00
0.922
0.267
0.288
-0.077
0.126
When k = 0.25, redo the ridge regression, and non-standardized coefficients are obtained, as
below:
Table 4
Non-standardized regression coefficients of independent variables
a2
a3
a4
a5
0.501
0.417
-0.297
0.207
111
A Study of Chinese Residential Electricity Demand Elasticities
Results displayed in Table 4 means that the income elasticity of power demand is nearly
0.501, while the self-price elasticity is -0.297 approximately.
4.
Conclusions and Analysis
A comparison with the elasticities calculated from ELES model (0.533 and -0.38) shows
that the results derived from terminal energy needs (0.501 and -0.297) are smaller; This may be
due to the reason that terminal-energy-need method focuses on three representative appliances,
and some power consumption projects (such as dryer) whose usage is more flexible are not taken
into account for the purpose of building a more simplified model. But because the differences
between these two methods are relatively small, and since no perfect method is proposed, this
paper considers the differences between results as understandable. In other words, we can
conclude that the income elasticity of Chinese residential electricity demand is about 0.5, and the
self-price elasticity is in the -0.3 or so.
Compared with overseas studies, China's short-term price elasticity is at an intermediate
level, while the income elasticity is slightly larger than that of abroad (James A. Espey and Molly
Esper, 2004). The possible explanation is that since most families in China are rural households
whose economic conditions are not very good, combined with the traditional concept of
regarding thrift as a virtue, most families consume electricity frugally in everyday life, and the
usage rate of household appliances is not very high; that is, appliances are not fully utilized.
When income increased in a short term, residents are able to make use of appliances more
frequently, thus the potential increase in electricity consumption is relatively large.
In addition, a comparison of the income elasticity and price elasticity reveals that the
previous is clearly greater than the latter, indicating that the major reason for a significant
increase in residential electricity consumption is the rise of income. Although electricity price
and income have been increasing simultaneously in recent years, the impact revenue brings on
power consumption is larger than the influence brought about by price.
A relatively small price elasticity also shows that the policies cannot achieve significant
control effect in the short term which try to reduce the power consumption by raising the price of
electricity so that energy can be saved and environment can be protected, although it is true that
to judge the effectiveness of such policies we need to further measure the long-term elasticity.
112
Research of Mathematical Economics No. 1 2011
Furthermore, such a small price elasticity of demand also shows that the burden adds to residents
which is caused by a rise in electricity price cannot be ignored.
5.
Possible Directions for Future Development
This paper studies only short-term elasticity of residential electricity demand, but in fact
long-term elasticity is also very important. In particular, to policy makers who make effort to
determine a electricity price which can encourage residents to save power and reduce energy
consumption, whether the policies they make will receive the intended effect is largely depends
on long-term elasticity. Moreover, since long-term elasticity is more significant than short-term
one in general case, the effect it brings out will be more apparent. Therefore, I will provide some
possible directions in this section for further research on long-term elasticity.
For the ELES model, theoretically, it is possible to dynamize it so that it can be used in
estimate long-term elasticity. But the model derived in this way will be extremely complicated in
parameter estimation, and there may be no satisfactory solution under all parameter constraints.
Due to the additivity of utility functions, we can separate utilities; that is, isolating energy
consumption from total consumption system and build a dynamic model considering only the
energy consumption system. We may establish the power demand function following the
dynamic form of Stone-Geary model proposed by Pollak in 1970:
ue    qit    i  i qi (t 1)  
n
i
(14)
i 1
where
ue
is the total utility function;
qit
is the consumption of the ith energy in period t;
qi ( t 1) is the consumption of the ith energy in period
the ith energy;
i
t 1;
 i is the basic need coefficient of
is the dynamic adjustment factor of the ith energy; and
i is the marginal
budget share of the ith energy.
Supposing that the energy expenditure constraint on utility maximization is:
n
met   pit qit
(15)
i 1
113
A Study of Chinese Residential Electricity Demand Elasticities
where
met
is the total energy expenditure in period t; pit is the price of the ith energy in
period t; qit is the consumption of the ith energy in period t.
Maximizing the utility function (14) under the constraint (15), we get the following demand
function:
qit   i  i qi (t 1) 
i 
n

m

p



q



et
it
i
i
i
(
t

1)

pit 
i 1

(16)
whose corresponding expenditure function is:
n


mit  pit   i  i qi (t 1)   i  met   pit  i  i qi (t 1)  
i 1


(17)
where mit is the expenditure on consuming the i th energy during period t.
From formula (17), one can see that the expenditure on the ith energy is still divided into
two parts: one is the basic need part, namely, the first part on the right-hand side of formula (17),
pit   i  i qi (t 1)  ; the other is the additional need expenditure, as the second part on the right

side, i  met 

 p 
n
i 1
it
i

 i qi (t 1)  .

For the terminal-need-method, because residents are able to change the stock of household
appliances in the long run, there is no need to suppose a fixed stock in interpretation of the
change in power consumption. Thus model (13) is modified to exclude W.
The long-term model can be set as follows:
ln Q  a1  a2 ln Y  a3 ln Pe  a4 ln Pf  u
(18)
Similarly, due to the existence of multicollinearity, we can apply ridge regression to fit the
model in actual operation.
Only when we have a comprehensive understanding of residential electricity demand
elasticity of not only the short term but also the long run, are we able to get a full range of
knowledge about the characteristics of China’s residential power consumption, which is the
foundation of developing effective policies and researching more in-depth.
114
Research of Mathematical Economics No. 1 2011
Appendix
The Derivation of Formula (16)
Setting up the Lagrangian expression


n
n
i


L =   qit   i  i qit 1     met   pit qit 


i 1


i 1
Then setting the partial derivatives of L (with respect to qit and
(19)
 ) equal to zero,
which gives rise to:
 q  
j i
jt
  j q j (t 1)  i  qit   i  i qi (t 1) 
j
j
(i  1, 2,
i 1
  pit  0
(20)
, n)
and
n
met   pit qit
(21)
i 1
From (20), we have
qi t 
n


i B
  i  qi ( i 1 t)
 pit
(22)
i
where B    qit   i  i qi  t 1  .


i 1
Therefore
n
 B

p
q

pit  i   i  i qi (t 1) 


it it
i 1
i 1
  pit

n
n
B

   i    pit   i  i qi (t 1) 
  i 1  i 1
 met
n
115
A Study of Chinese Residential Electricity Demand Elasticities
Thus
B

 met   pit   i  i qi (t 1) 
n
(23)
i 1
Finally,
qi t 
i 

m et  p it  i q ( i 1 )i  t    qi

pit 
i 1

n
(
i1 ) i t
(16)
References
http://www.un.org/millenniumgoals/.
http://www.cpnn.com.cn/.
Beijing Statistical Yearbook, China Statistics Press, 2000-2009.
United Nations Millennium Development Goals Report of 2009.
Abdallah Al-Shenri, “Artificial Neural Network for Forecasting Residential Electrical Energy”, International
Journal of Energy Research, June 1999, 23(8), pp. 649-661.
Angus Deaton and John Muellbauer, “An Almost Ideal Demand System”, The American Economic Review,
June 1980, 70(3), pp. 312-326.
Constantino Lluch, “The Extended Linear Expenditure System”, European Economic Review, April 1973,
4(1), pp. 21-32.
Ferda Halicioglu, “Residential electricity demand dynamics in Turkey”, Energy Economics, March 2007,
29(2), pp. 199-210.
George Hondroyiannis, “Estimating residential demand for electricity in Greece”, Energy Economics, May
2004, 26(3), pp. 319-334.
James A. Espey and Molly Esper, “Turning On the Lights: A Meta-Analysis of Residential Electricity
Demand Elasticities”, Journal of Agricultural and Applied Economics, April 2004, 36(1), pp. 65-81.
L. R. Klein and H. Rubin. “A Constant-Utility Index of the Cost of Living”, The Review of Economic Studies,
1947-1948, 15(2), pp. 84-87.
Richard Stone, “Linear Expenditure Systems and Demand Analysis: An Application to the Pattern of British
Demand”, The Economic Journal, September 1954, 64(255), pp. 511-527.
Robert A. Pollak, “Habit Formation and Dynamic Demand Functions”, The Journal of Political Economy,
Part I July-August 1970, 78(4), pp. 745-763.
Thomas J. Lareau and Joel Darmstadter, Energy and Household Expenditure Patterns, Washington, D.C.:
Resources for the Future, Inc., 1983.
T. R. Lakshmanan and William Anderson, “Residential energy demand in the United States: A regional
116
Research of Mathematical Economics No. 1 2011
econometric analysis”, Regional Science and Urban Economics, August 1980, 10(3), pp. 371-386.
Tsuyoshi Ueno, Ryo Inada, Osamu Saeki and Kiichiro Tsuji, “Effectiveness of Displaying Consumption
Data in Residential Houses. Analysis on How the Residents Respond”, ECEEE 2005 Summer Study Proceedings,
pp. 1289-1299.
Xavier Labandeira, José M. Labeaga and Xiral López-Otero. “Estimation of Elasticity Price of
Electricitywith Incomplete Information”, FEDEA Working Paper No. 09-18, 2009.
(Responsible Editor: Zhang Lingyun) (Proofreader: Xu Lingjue)
117
The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and
Manufacturing
The Factors that Determine a Firm’s Optimal Allocation of Capital
between Research and Manufacturing
Xu Lingjue, Wang Danna
(School of Economics, Renmin University of China)
Abstract: This paper is based upon the endogenous growth theory model written by Philippe Aghion and Peter
Howitt (1992). Assuming drastic innovation, the model explains how a firm with limited endowment chooses the
optimal amount of resources used in research. We apply the model in analyzing the exogenous factors affecting
the choice made by producer, who is also innovation motivator. Firstly, given capital endowment, rental rate,
market competitiveness, time preference, innovation arrival rate, productivity increase level and monopoly power,
we construct a model from which the optimal amount of capital used in research derives. Then we conduct
comparative-statics analysis of these exogenous factors and give both mathematical and economical explanations
of the result. Furthermore, we apply the conclusions to explain the innovation inadequacy in China. Finally, we
give appropriate policy recommendations.
Keywords: Innovation, Creative Destruction, Comparative-Statics Analysis
1. Background
After China joined the WTO, the government’s preferential policies would be gradually
cancelled. Firms are confronted with a competitive market featured in variability and uncertainty.
With intense competitions, international environment and ever-decreasing product lifecycle,
innovation plays an increasingly important role in a firm’s existence and development (Fujita,
1997; Nakaha ra, 1997). In the process of modern production which is distinct in resource
scarcity, new products just spring out on end. However, the crucial creative contents often come
from transnational corporations of developed countries rather than domestic companies. Even if
in some departments, manufacturing of these innovative products is going on domestically, we
have to buy patents or technologies which are very costly. From the late 19th century, China’s
high value-added products in manufacturing have been climbing continuously, but still it is only a
118
Research of Mathematical Economics No. 1 2011
quarter of America’s. Actually, America manufactures the most highly value-added products
owing to its ever-increasing investment in innovation in the field of IT and biochemical industries.
Considering the circumstance we have been in, it is high time that the firms of China emphasized
more on the innovation in order to promote their core competencies.
There is no doubt that innovation is crucial to both firms and the whole nation, what we care
now turns out to be how firms’ innovation decision can be affected. However, the variables that
are faced by firms have been very complex. If we could find out the comparatively decisive
parameters among these variables, we can be able to explain the firms’ strategy in innovation
from these dimensions. Furthermore, if social planner can change some of these parameters, there
will be a possibility that the governor indirectly influence firms’ decision through policy-making.
2. Literature Review
The factors that affect the input of resources in research have been explained in
multi-dimension.
In the field of management, the effort in research chosen is determined by the entrepreneur’s
innovative spirit, ability in creating new inventions, corporate culture and intra-incentive
mechanism. Joseph Alois Schumpeter (1950) defined entrepreneur as an organization’s leader
with innovation ability and positively evaluated the effect of entrepreneur. The innovation ability
refers to human capital involving the strategy, innovative spirit, confidence, determination and
specific skills. In another word, the human capital is the core of innovation in a corporate. Yu Jin
(2008) considers the absence of innovative spirit of entrepreneur as the reason for the failure in
research. Cai Bing (2006) views the lack of self-conducted innovation capability in a corporate
culture crucial, whereas Wang Bingjie and Wang Jingyu (2010) relate the quality of staff, the
ability in creating new invention and the goal of the firm to the process of innovation.
From the point of Industrial Organization, the succession of new product research will lead
to a certain level of production differentiation. In the new product market, the research promoter
will be the only supplier and thus attains complete monopoly rent. The chase of profit finally
determines the input in research. So, to some extent, a higher degree of monopoly power will
create more monopoly rent which gives the firm more incentive to innovate (Wang Junfeng,
Zhou Shaodong, Zhu Quanzhen, 2010).
119
The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and
Manufacturing
Considering the policy making, the research and development (R&D), especially
fundamental research, owns the property of positive externality, which the firm does not take into
account. Meanwhile, the non-competitiveness feature also lures the firm to be a free-rider instead
of striving in innovation. These two effects tend to reduce the firm’s incentive to invest in R&D
of new products. Consequently, public sectors ought to take the responsibility of supporting
research, through making reasonable and sound policies like guaranteeing patent and franchise
right. Zhang Jingyi (2004) explores the cause of inadequate innovation in small and
medium-sized enterprise and found that the indistinctive definition of property plays the key role.
Cao Yan (2005) focuses on the property rights involving human capital and owes the lack of
innovation to the absence of it. Anthony Bartzokas (2003) emphasizes the role of venture capital
as a core element of the institutional framework that supports innovation dynamics in developing
countries.
There is also a group of research explaining the reason of innovation insufficiency with
Institutional Economics. Research from Jia Liqun and Wang Yingluo (1994) perceive the pursue
of innovation is determined by the “potential” of its initial state. Potential will be distributed to
multi degrees of freedom, and variant path will lead to the final goal of a firm, in which the path
without innovation will be preferred by the firm because of less cost and risk.
3.
Inadequacies of the Existing Literature and Our Point of View
Most existing research starts from the practical side in interpreting the factors that influence
firms’ innovation decisions. Rarely are there economical models explaining the phenomenon
theoretically. In this article, we utilize part of the work by Aghion and Howitt, and get a standard
formula with a series of exogenous parameter. Then we discuss how the resource will be
allocated between researching and manufacturing when these parameters change. Finally,
empirical analysis is made on the basis of data of China and policy recommendations are given.
4.
The Model
4.1 Assumptions
We assume a random sequence of quality improving innovations that result from research
120
Research of Mathematical Economics No. 1 2011
activities. For simplicity, in the model, the new inventions make old technologies or products
obsolete. In another word, we take the innovations drastic. This assumption is partly true with
reality. For example, we can imagine the flow of reform in the field of software. Windows 7 has
been the latest invention among Windows series, and hardly can we find Windows 2000
nowadays.
The economy is populated by a continuous mass of capital K, and  represents the time
preference.
The economy only consists of two sectors, the intermediate good sector and the final
consumption good sector whose production depends on the intermediate good. We denote the
intermediate good x and the final consumption good y. The market of y is perfectly competitive,
whereas the producer of x is a monopolist. The relation between x and y is defined as:
y  AF (x)
(1)
where F ' ( x)  0, F '' ( x)  0 , the parameter A represents the productivity of x.
Innovations consist of the invention of a new variety of intermediate good that replaces the
old one. It is the obsolescence process that raises the productivity parameter A, by the constant
factor,   1 . That is, denoting At as the productivity level after the tth innovation takes
place,
At 1  At   t A0
(2)
where A0 is the initial value given by history.
The capital supply is divided into two parts. It can produce the intermediate good x, one unit
of K for one unit of x, so the x t equals the capital used in manufacturing during the tth
innovation instant. And it can also be used in research. So K  x  n , where x is the amount of
capital in manufacturing and n is the amount of capital in research.
Research produces a random sequence of innovations. The poisson arrival rate of
innovations in the economy at any instant is  (n) , where
 >0 is a parameter indicating the
productivity of the research technology.  (n ) is a constant-returns, concave production
function. It satisfies  ' (n)  0 and  (0)  0 . Both
 and  (n ) are given by the
technology of research. The arrival rate  (n) means that given the number of capital used in
121
The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and
Manufacturing
research at some instant, the  (n) amount of innovations arrive naturally. Here the
assumption of Poisson process well ensures the no memory feature between the two adjacent
research instants.
Given the assumption of the model above, we can construct the model next.
4.2 Production Decision Made by the Producer of the Intermediate Sector
There is a sequence of discrete period, t  1,2,  , representing the tth innovation, where
the variable t does not refer to real time, but rather to the sequence of innovations.
During the tth innovation interval, we denote the production quantity of the intermediate
good x t , the rental rate of the capital used in manufacturing v t , the profit of the monopolist
 t , the price of xt pt ( xt ) , the productivity parameter At , the capital used in research nt
and the value of the tth innovation Vt .
In this part, our discussion only concerns the tth innovation, i.e. t is given.
Recalling that the market of y is perfectly competitive, the price of x t is determined by
p t ( xt ) 
yt
xt
This is the equation that decides the amount of x t used in producing yt . The left hand
side (LHS) is the marginal cost of x t and the right hand side (RHS) is the marginal revenue of
xt (we may wish to set the price of y equals 1). According to the production function in
equation (1), the price of x t is determined by
p t ( xt ) 
yt
 At F ' ( xt )
xt
(3)
Accordingly, the monopolist producing x t chooses the output targeting
max  t  { pt ( xt )  vt }xt  { At F ' ( xt )  vt }xt
xt
Take first order condition of the equation (4), we get
122
(4)
Research of Mathematical Economics No. 1 2011
 t
 At F " ( xt ) xt  At F ' ( xt )  vt  0
xt
We denote rt 
vt
, which has the economical meaning of effective rental rate.
At
Rearranging the above equation, we get
rt  rt ( xt )  F " ( xt ) xt  F ' ( xt )
or
xt  xt (rt )
(5)
(6)
As a result, the monopoly rent generated from the drastic innovation is
 t   At F " ( xt ) xt 2  At~(rt ( xt ))
(7)
2
where ~(rt ( xt ))   F " ( xt ) xt . Noting that we have assumed that F '' ( x)  0 , so here the
profit must be positive.
4.3 Research Decision Made by the Producer of the Intermediate Sector
As was stated in the last part, if tth innovation takes place, there will be a positive monopoly
rent, therefore impelling the producer to allocate part of its capital in research, which means
nt  0 . This part will explain how the producer of xt chooses the optimal amount of nt .
We should be careful that the value of the (t+1)th innovation Vt is not the complete
monopoly profit. Allowing for the possibility of another innovation occurs after the (t+1)th
innovation succeeds, there should be Vt 1   t 1 . Consequently, Vt must satisfy the following
equation,
Vt 1   t 1   (nt 1 )Vt 1
(8)
The LHS of equation (8) is the net expected present value of the (t+1)th innovation, which
equals the RHS, that is, the profit  t 1 attainable by the (t+1) intermediate good monopolist
minus the expected loss that will occur when the (t+1)th innovation is replaced by a new
innovator, which it does at the expected rate  (nt 1 ) .
Rearranging the equation (8), we get
123
The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and
Manufacturing
Vt 1 
 t 1
   (nt 1 )
(9)
With more usage of capital in research, given fixed amount of capital endowment, less
capital will be available for manufacturing. This trade-off will balance the distribution of limited
capital by the arbitrage condition
vt   ' (nt )Vt 1
(10)
The LHS of equation (10) is the value of using one more unit of capital in manufacturing
whereas the RHS is the expected value of investing one more unit of capital in research — the
'
flow probability  ( nt ) of an innovation times the value Vt 1 . That is, the equation (10)
represents the condition that the marginal value of manufacturing must equal to the marginal
value of research.
Substituting the Vt 1 in (10) with (9) and combining equation (2), we can get
rt   ' (nt )
~ (rt ( xt ))
   (nt 1 )
(11)
4.4 Steady-State equilibrium
The steady-state equilibrium is simply defined as a stationary solution with
rt  r , nt  n, t  1,2,3. Note that K  x(r )  n and x(r )  K  n , we can get
r ( K  nˆ ) ~ (r ( K  nˆ ))

 ' (nˆ )
   (nˆ )
(12)
This is the equation that determines the optimal amount of research that the firm chooses.
Specifically, consider the Cobb-Douglas example: F ( x)  x  ,0    1 with a linear
research technology  (n)  n .
Correspondingly, the equation (3)(6)(7)(12) will be transferred to
pt ( xt )  At xt
124
 1

vt

(3)’
Research of Mathematical Economics No. 1 2011
xt (rt )  (
2
1
t 

1
1
 (
1
rt
)
 1
(6)’
vt xt
(7)’
)( K  nˆ )

  nˆ
1
The condition for n̂ to be positive is
5.
1
 (


(13)’
)K
.
Comparative-Statics Analysis
We will go further to discover the economic meaning of equation (12).
r ( K  nˆ ) ~ (r ( K  nˆ ))

   (nˆ )
 ' (nˆ )
(12)
The RHS of equation (12) stands for the marginal benefit of research — the expected
present value of the intermediate goods’ profit. The LHS of equation (12) stands for the marginal
cost of research — the expected amount of rental rate that has to be paid to increase one unit of
innovation arrival rate.
We denote marginal benefit and marginal cost of research as follows,
b(nˆ ) 
~(r ( K  nˆ ))
   (nˆ )
(13)
r ( K  nˆ )
 ' (nˆ )
(14)
c(nˆ ) 
d~
dr
dx
 0,
0 ,
 0 ,  ' (nˆ )  0 and  '' (nˆ )  0 , we get
Noting that
dr
dx
dnˆ
d~ dr dx

[    (nˆ )]   ' (nˆ )~
d
ˆ
dr
dx
d
n
b(nˆ ) 
0
dnˆ
[    (nˆ )] 2
125
The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and
Manufacturing
dr dx
 ' (nˆ )   '' (nˆ )r
d
c(nˆ )  dx dnˆ
0
dnˆ
[ ' (nˆ )] 2
Assume
lim r(x) = 
x 0
,
lim r(x) = 0
x 
and
c(0)  b(0) (we do not consider
c(0)  b(0) , since a firm will not innovate under this situation).
Given the assumption, we can obtain the marginal benefit curve and marginal cost curve as
illustrated in Graph 1.
Graph1
Marginal Benefit Curve and Marginal Cost Curve
From equation (13), we find a lower discount rate
   (nˆ) or a higher profit
~(r ( K  nˆ )) or a higher level of productivity advance  will shift b(nˆ ) upward and thus
increase n̂ (Graph 2).
126
Research of Mathematical Economics No. 1 2011
Graph2
Holding
c (nˆ )
constant, any change that can shift
b(nˆ )
upward will increase
n̂
From equation (14), we find a higher effective rental rate r ( K  nˆ ) or a lower innovation
arrival rate
 ' (nˆ ) will shift c (nˆ ) upward and thus decrease n̂ (Graph 3).
127
The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and
Manufacturing
Graph 3
Holding
b(nˆ )
constant, any change that can shift
c (nˆ )
upward will decrease
n̂
5. Empirical Analysis on the Basis of Reality in China
In the rest of this article, we are going to give some possible explanations of the innovation
inadequacy in China by applying what we have concluded before.
5.1 Firms in China has a High Discount Rate    (nˆ ) .
A reasonable explanation is related to State owned enterprises (SOE). As is well known, in
China, these enterprises make up a large part of all firms. Consequently, their decisions of
innovation will greatly influence the total amount of innovations. Most managers in state-owned
enterprises are not concerned about the consequence of their policies after they leave. They just
focus on their firms’ performances when they are still in charge. As a result, they have little
incentive to invest in innovation whose value is accomplished in the future. In other words, they
are not patient enough to wait for the return of innovation. This impatience can be characterized
128
Research of Mathematical Economics No. 1 2011
by a high discount rate.
Another reasonable explanation is the variability of government’s policy. There isn’t a sound
system to guarantee the returns of innovation. If a relevant policy can be altered at will, how dare
firms innovate? How can firms attach high value to something uncertain in the future? As a
consequence, variability of government’s policy results in a high discount rate.
5.2 Firms in China has a Lower Level of Productivity Advance  .
Compared to other countries, we find that invention patents only account for a small
percentage of all patent applications (the other two kinds of patent is design patent and utility
model patent). It means the level of productivity advance  in China is comparatively low
(Table1).
Table 1
Three Kinds of Patent Applications Accepted and the Percentage of Inventions
Item
Country
Percentage of Inventions
Inventions
Utility Models
Designs
Total
China
194,579
223,945
298,620
717,144
27.13
USA
231,588
**
15,463
247,051
93.74
Japan
330,110
7,717
29,621
367,448
89.84
France
14,743
121(2007)
4,093
18,836
78.27
Australia
2,821
1,024
2,727
6,572
42.92
Italy
8,588
1,981
1,184
11,753
73.07
Switzerland
1,594
**
1,123
2,717
58.67
Austria
2,298
682
805
3,785
60.71
Czech Republic
712
1,107
288
2,107
33.79
Denmark
1,634
218
183
2,035
80.29
Korea
61,115
16,971
52,786
130,872
46.70
Russian Federation
27,712
10,483
2,356
40,551
68.34
(%)
Source: World Intellectual Property Indicators (2010)
5.3 Firms in China has a Lower Innovation Arrival Rate  ' (nˆ ) .
129
The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and
Manufacturing
A fundamental breakthrough will lead to a wave of innovations and thus increase innovation
arrival rate. Fundamental breakthrough often derives from basic research. As a result, to some
degree, the more the basic research the higher the innovation arrival rate.
Data in Table 1 evidently shows that compared to other countries, China’s investment in
basic research is extremely low. The inadequacy of basic research can be the cause of a low
innovation arrival rate.
Table 2
International Comparison of R&D Activities
Item
Country
By types of research (%)
year
Basic Research
Applied Research
Experimental
Development
China
(2007)
4.7
13.28
82.01
USA
(2006)
18.56
23.12
58.31
Japan
(2005)
12.65
22.18
65.17
France
(2005)
23.70
38.99
37.31
Australia
(2004)
23.17
38.14
38.69
Italy
(2005)
27.71
44.40
27.89
Switzerland
(2004)
28.70
33.32
37.98
Austria
(2002)
17.80
37.6
44.60
Czech
(2006)
29.32
24.06
46.62
Denmark
(2005)
18.71
28.08
53.22
Korea
(2006)
15.15
19.86
64.99
Russian Federation
(2003)
15.10
15.60
69.40
Source: China Statistical Yearbook on Science and Technology
5.4 Patent Protection System in China is Imperfect
To some extent, having a patent means enjoying a certain degree of monopoly power.
Since a patent prevents others from utilizing the innovation without payment, other
competitors will lose competitiveness or at least be in a disadvantageous position in the market.
In this way, innovator can be regarded as having some monopoly effect.
The monopoly power can be measured by the value of price minus marginal cost divided by
price. We can examine the monopoly effect in the linear Cobb-Douglas example. Recalling
130
Research of Mathematical Economics No. 1 2011
equation (3)’ and (13)’
pt ( xt )  At xt
1
1
 (
 1

vt
(3)’

)( K  nˆ )

  nˆ
from (3)’, we can easily acquire the monopoly power, that is
(13)’
p t  vt
p  pt
 t
 1   . The
pt
pt
degree of monopoly power is measured inversely by the parameter
 . From equation (13)’, we
can easily find that, the higher the monopolistic degree is, the higher the marginal benefit is.
As we have mentioned before, here a firm’s monopoly power should be interpreted as the
ability to gain revenue from its own innovation. A perfect patent protection system can exactly
grant the utmost to gain revenue, that is, the strongest “monopoly power”, which motivate the
firms to innovate.
However, piracy is rampant in China, or in other words, patent protection is so weak in
China. Since innovation may be plagiarized or copied by other firms, the innovator cannot enjoy
the complete outcome of its innovation. As a result, firms would have less incentive to innovate
and the innovation tends to be insufficient.
6. Policy Recommendations
(1) Construct a reasonable and sound system to evaluate the performances of SOEs’ managers.
This evaluation system should include assessment of innovation performances.
(2) Try to keep relevant policy stable. Policy makers should let firms operate in a less uncertain
environment and ensure their investments’ safety.
(3) Pay more attention to the development of basic science and the education of basic science
personnel. Attach more importance to basic research and invest more resources in this field.
(4) Encourage more invention-oriented innovations by preferential policy, rewards and so forth.
(5) Establish a more effective policy to protect patent. Make sure that firms can enjoy their
achievements and be free from plagiarism.
131
The Factors that Determine a Firm’s Optimal Allocation of Capital between Research and
Manufacturing
(6) Give preferential interest rate to R&D loans and at the same time set up a supervision system
to guarantee that the loans are used appropriately.
7. Conclusions
This paper studies the factors that determine a firm’s optimal allocation of resources
between research and manufacturing. We discover that the amount of research input moves in the
opposite direction of discount rate and effective rental rate whereas it moves in the same
direction with profit of innovation product, productivity advance level and innovation arrival
rate.
Through empirical analysis, we find that there are mainly four reasons for the innovation
inadequacy in China. Firstly, absence of innovation performance evaluation of SOE’s managers
and the variability of government’s policy result in a high discount rate. Secondly, level of
productivity advance is small since firms focus more on design and utility improvement rather
than invention. Thirdly, innovation arrival rate is low due to the inadequacy of basic research.
Fourthly, patent protection system is imperfect.
At last, we have to point out that this model could be extended in several ways. For example,
introducing labor markets and a variety of intermediate goods into this model will make it closer
to the reality. The assumption of drastic innovation can also be relaxed. After learning
Econometrics systematically in next year, we can also use this tool to do more empirical analysis.
References
Jin Xiangrong, Yu Dongyun, “Industrial Characteristics, Innovative Efficiency and Regional Economic
Growth”, Journal of Zhejiang University, 2010 (5), pp. 73-81.
Li yuan, Wang Yingluo, “Exploration on the Components of Enterprises’ Technology Innovation
Motivation”, Scientific Management Research, 1994 12(4), pp. 44-45.
M. Ishaq Nadiri, “Innovation and Technological Spillovers”, NBER Working Paper No. 4423, 1993.
N. L. Stokey, “Learning by Doing and the Introduction of New Goods”, Journal of Political Economy, 1988
96(4), pp. 701-717.
Philippe Aghion, Peter Howitt, “A Model of Growth Through Creative Destruction”, Econometrica, 1992
60(2), pp. 323-351.
132
Research of Mathematical Economics No. 1 2011
Wang Junfen, Zhou Shaodong, Zhu Quanzhen, “Research on the Relationship between Market
Concentration and Innovation Efficiency of Non-incumbent Enterprises”, Science &Technology Progress and
Policy, 2010 27(14), pp. 81-85.
Wu Hanhong, “The Theory of Industrial Organization”, Beijing: Renmin University of China Publishing
House, 2007.
Yu Ji, “Research on Enterprises’ Technology Innovation Motivation”, Study & Exploration, 2007(6),
pp. 113-175.
(Responsible Editor: Xu Lingjue) (Proofreader: Zhang Lingyun)
133
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
A Quantitative Analysis of the Influences of China’s Inflation to
Different Asset-Holding Classes
Chen Jun
(School of Economics, Renmin University of China)
Abstract: Based on the knowledge of the China’s economic development and the characters of China’s inflation
welfare cost, the paper constructed a Bewley type model with market incompleteness and individual
heterogeneity, by which the paper has an analysis of the China’s inflation welfare cost and the differences of the
inflation welfare cost to different groups. The calibration results of the model show us that there are great costs of
the inflation to the society as a whole; however, different groups are related to different inflation welfare costs.
Generally speaking, if the inflation is very low, the welfare loss is smaller than the wealth loss, but from the
wealth distribution viewpoints, the inflation worsen the China’s wealth distribution. The wealth share of the poor
declines with the increase of the inflation while the one of the wealthy increases with the increase of the inflation.
Therefore, the wealth Gini index increases with the increase of the inflation.
Keywords: Inflation, Incomplete Market, Bewley Type Model, Welfare Cost, Wealth Distribution, Individual
Heterogeneity
1. Introduction
Inflation is one of the core problems concerned by the government all over the world. It is
the common dream for each government to achieve a high growth rate with a low inflation rate.
In order to achieve the goal, the central banks all over the world emphasize the role of
stabilization of price; therefore, the average inflation rate in the developed government falls from
9% in 1980s to 2% in 2005 and that of China has fallen from more than 10% in 1995 to 1.8% in
2005 (Chen and Ma, 2007). However, with the economic development, there is a trend to
increase for the inflation rate in China. Although the subprime crisis has suppressed the increase
of price, the stimulus policy in China has brought about the hidden danger of the inflation in the
future. Under these circumferences, what’s the rate of inflation should be in China? How the
134
Research of Mathematical Economics No. 1 2011
Chinese government should do to keep a relatively sound inflation rate? The government are
faced with there problems. However, in order to solve the problems, the government should
know the mechanism of inflation formation in China; what’s more, the quantitative analysis of
inflation to the macro economy is needed.
From the traditional research view, the quantitative analysis on inflation in China is usually
based on the Taylor’s Rule and the Phillips Curve. However, the accurate estimation of Taylor’s
Rule should be based on the accurate estimation of the output gap, but it is impossible to have an
accurate estimation on the output gap in China.1 At the same time, when the macro economy is
faced with uncertainty, the Phillips curve cannot give us estimation of optimal inflation rate and
relevant monetary policy (Zhao, 2006). Comparing with the two traditional methods, the
quantitative analysis of inflation welfare cost can give us the estimation of optimal inflation rate,
hence more rational policy suggestion can be acquired through the method.
The analysis of inflation welfare cost is to know about the inflation through estimating the
loss of family and the whole society caused by the inflation. Since the increase of price would
change the behavior pattern of individual and family, inflation would cause deadweight loss.
Through estimating the inflation welfare cost, the government can acquire the optimal inflation
rate of the economy and make more reasonable measures; however, the welfare cost is usually
covered and very difficult to estimate (Chen and Ma, 2007). With the development of
macroeconomics, economists have offered several methods to estimate the inflation welfare cost.
The consumer surplus method developed by Bailey (1956) is the first step for economists to
estimate the inflation welfare cost. According to the method, inflation welfare cost should be
represented by the area under the inverse money demand curve. With the definition, Bailey
estimates that the inflation welfare cost under the rate of 6% in the USA is about 0.3%-1% of
national income. Although the method keeps in accordance with the definition of consumer
surplus in the microeconomics and hence provides a convenient method to estimate the inflation
welfare cost, there is a fatal disadvantage: when the nominal interest rate is changed, the money
1
Specifically, there are several reasons that make the accurate estimation of China’s output gap impossible: (1)
there is no united definition for the output gap (Guo and Jia, 2004), therefore, different standards would be
adopted for different researches; (2) there is no united method to estimate the output gap and each method has its
own disadvantages. On one hand, time series methods such as detrended method consider relatively few factors,
hence ignoring supply side (Xie and Luo, 2002); on the other hand, production function method require higher
quality of data, though it can avoid the disadvantage of time series methods, but we cannot get there data in
current China (Guo and Jia, 2004).
135
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
demand curve should be required to keep unchanged when calculating the welfare cost; this
would further require that the marginal utility of real balance should be independent from the
demand of other commodities, therefore, the method cannot give us an accurate estimation of
inflation welfare cost (Laidler, 1990).2 In order to surmount the disadvantage, lots of general
equilibrium methods are developed. Specifically, Cooley and Hansen (1989) first constructed a
one-sector stochastic growth model with CIA (Cash-In-Advance). With the model, they estimated
that the rate of 10% would cause a 0.38% of gross product. Later on, with the development of
MIU (Money-In-Utility) model, lots of MIU models are adopted to estimate the inflation welfare
cost. For example, Lucas (2000) raised a general equilibrium model with MIU and showed us
that the inflation rate of 10% would cause a loss of about 1.3% of gross consumption to the
macro economy. In order to investigate the inflation in a further step, lots of economists try to
portray the micro mechanism of keeping money. For example, McCallum and Goodfriend (1987)
assume that both time and money can not only provide transaction service and also substitute for
each other, hence constructing the money demand function from the microcosmic point; while
Logos and Wright (2005) raised a general equilibrium model from the view of money search and
estimated that the rate of 10% would cause a loss of about 3%-5% of gross consumption.
Although there have been lots of researches based on these methods, but most of them are
based on the representative-agent model. In the representative-agent model, all agents have the
same preferences and inflation expectation, therefore, the influences of inflation to all the agents
would be the same; however, the influences in reality would be obviously different. Doepke and
Schneider (2006a) and Meh and Terajima (2008), with the help of micro data, investigate the
redistribution effect of inflation in U.S. and Canada, respectively. The results show us that even
the moderate inflation would cause great wealth transfer from the olds to the youth. Based on
these facts, more and more scholars try to investigate the influences of inflation under the
heterogeneous-agent frame. For example, Erosa and Ventura (2002) introduces two kinds of
families such as the poor families and wealthy families and two kinds of transaction methods
such as cash transaction and credit transaction; the estimation results show us that the wealthy
families would be faced with less loss due to greater frequency to adopt the credit transaction.
Cysne (2006) introduces the families with different production efficiency and transaction
efficiency into the shopping time model and the results show us that the welfare cost would be
2
Numerical explanation about the disadvantage of the method can be seen from İmrohoroğlu (1992).
136
Research of Mathematical Economics No. 1 2011
different from the traditional shopping time model. Doepke and Schneider (2006b) and Meh et al.
(2010) take the different portfolios of the individuals with different ages and different
asset-holding classes into consideration and investigate the influences of unexpected inflation
shock to different asset-holding classes. Algan et al. (2009) investigates the influences of
inflation to macro economy and different asset-holding classes from the labor supply view; the
results show us that inflation would cause that the wealth transfers from the wealthy to the poor,
hence increasing the labor supply of the wealthy and the increasing of the gross labor supply.
In current China, wealth distribution has been worsened and has become the bottleneck of
the economic development (Li et al., 2000; Li et al., 2005; Chen et al., 2009). With the change of
wealth distribution, the portfolios of different asset-holding classes would be different, too. (Li et
al., 2005; Chen, 2008) Since different assets would have different yields in the inflation, the
inflation would influent the wealth distribution and the behavior pattern. Therefore, there would
be great distinctions of inflation welfare cost for different asset-holding classes. Therefore, only
with the help of heterogeneous-agent model can we know about the influences of inflation in
China.
At the same time, for the developing countries such as China, there would be fewer financial
products due to the limit of laws (Li and Han, 2010); therefore, the market incompleteness would
be more serious in developing countries (Li and Li, 2004; Zhu and Liu, 2009; Huang et al., 2009).
The incomplete market would influence the inflation welfare cost (İmrohoroğlu, 1992；Chiu and
Molico, 2008；Algan et al., 2009); therefore, we should take the market incompleteness into
consideration when we investigate the influences of inflation in China.
In order to investigate the influences of China’s inflation with a model with incomplete
market and heterogeneous agents, the paper raised a Bewley-type model. This kind of model is
developed first by Bewley (1983), and then developed in a further step by Huggett (1993),
Aiyagari (1994) and Krusell and Smith (1998). In the model, the agents are faced with
idiosyncratic income shock and the incomplete market make the agents cannot fully insure
against the income risk. In order to smooth the consumption, the agent would self-insure through
keeping the money, capital or adjusting the labor supply. Since the shock history of each agent
would be different, the asset holding and the portfolio would be different. In a further step, we
can conclude that the influences of inflation to different people would be different, too. Therefore,
the model would endogeneize the inflation heterogeneity. What’s more, since asset holding and
137
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
labor supply would change with the inflation rate, the gross capital and gross labor supply would
be different, too. This means that gross production would be influenced by the inflation; therefore,
the paper links the inflation and gross product. With the dynamic general equilibrium model, the
paper has an investigation into influence heterogeneity of inflation to different asset-holding
classes.
The remainder of this paper is organized as follows: in section 2, the model would be
introduced; the calibration procedure would be given in section 3; in section 4, the results of the
model are given; with the analysis of the model, the paper gives some suggestions about the
monetary policy, and this can be seen in section 5. In the appendix, some technical details are
introduced.
2. The Model
In order to have an investigation into the China’s inflation welfare cost, the paper raises an
Bewley-type model with idiosyncratic risk and borrowing constraints according to the work of
İmrohoroğlu and Prescott (1991), İmrohoroğlu (1992), Algan and Ragot (2010). However, there
are great distinctions with the model in the paper and the models in İmrohoroğlu and Prescott
(1991) and İmrohoroğlu (1992): (1) the model in İmrohoroğlu and Prescott (1991) and
İmrohoroğlu (1992) is in fact an endowment economy where there is no production sector,
therefore, the link between the inflation and gross product are ignored. The paper, however,
introduces the production sector into the model; therefore the inflation would influence the gross
product and the consumption decision in the model.3 (2) The model in İmrohoroğlu and Prescott
(1991) and İmrohoroğlu (1992) only takes the saving function of money into consideration;
however, the paper raises a MIU model according to Sidrauski (1967), hence introducing the
trade medium function and the consumption smoothing function of money into the model. (3)
The paper introduces the leisure into the utility function; hence, the choice of leisure would be a
method to smooth the consumption (Heathcote et al., 2009). (4) There is only one kind of asset in
İmrohoroğlu and Prescott (1991) and İmrohoroğlu (1992) while in the model economy in the
paper, there is productive capital except the money. From the whole model setting, the model in
3
During the investigation of inflation welfare cost, the link between the gross product and the inflation should be
taken into consideration. This is, in fact, very important. More details can be seen in Gomme (1993). Temple
(2000) gave a good literature survey about the relationships of inflation and the gross product.
138
Research of Mathematical Economics No. 1 2011
the economy is very similar to the Algan and Raogt (2010), except for the differences of
government; however, the aim of the paper is different from that of Algan and Ragot (2010).4
2.1 Problems for the Family
The model is composed by continuous families standardized to be 1 and one-period utility
function form of each family is the same; however, the production efficiency, the real balance and
asset holding of each family are different. Assume further that the one-period utility of each
family mainly comes from the consumption, leisure and real balance. Therefore, the objective
function that each family is faced with is as follows:

max E0   tU  ct , mt pt , lt 
(1)
t 0
where 0    1 is the subjective discount factor, E0 means the expectation based on the
information in Period 0. ct , lt and mt pt are the consumption, labor supply and real
balance in period t . The function U () is the one-period utility function and the function is
differentiable and continuous for each variable.
In order to make the utility function coincide with the balanced growth and satisfy the nature
of utility function listed above as well as to reduce the calculation cost, the paper, referring to
Castañeda et al. (2003) and Lucas (2000), assumes that the one-period utility function is as
follows:


1 1
 
  1 1      1  l 1 1      1
c
m
p


t
t


U  ct , mt pt , lt   
(2)
 ln c  1    ln  m p    1  l 1 1    ,  1
t
t

where
 is the constant relative risk aversion coefficient,  and  are weight parameter of
consumption and leisure, respectively.
   1 is the curvature of leisure in the utility
function.
From the one-period utility function in (2), we can see that the function satisfies the
4
The aim of Algan and Ragot (2009) is to investigate the influences of inflation to the gross saving in the
economy while the aim of the paper is to investigate the inflation heterogeneity, hence taking the influences of
inflation to the gross capital and gross product into consideration.
139
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
conditions that the first-order derivative is larger than 0 and the second-order derivative is smaller
than 0, this means that the utility function satisfies the law of diminishing marginal utility.
2.2 Problems for the Firm
Assume that the market is competitive and the production technology is represented by
Cobb-Douglas production function; therefore, the whole market can be seen as a factory. Denote
the gross capital in period t as K t and gross labor as Lt , then the gross production in period
t can be expressed as:
Yt  A  Kt 

 Lt 
1
(3)
When the market is competitive, the price of capital and labor are given by the marginal
production; therefore, the wage rate and interest rate are given, respectively, as follows:
wt  1    A  K t 

 Lt 

(4)
and
rt   A  K t 
 1
 Lt 
1
(5)
2.3 Exogenous Driving Processes
There is no aggregate uncertainty in the model, but the agents are faced with idiosyncratic
efficiency shock. Assume that the logarithm of the efficiency shock follows the AR(1) process. If
the production efficiency in period t is  t , then the motion function can be expressed as:
ln  t   ln  t 1  t , t ~ N  0,  2 
(6)
where  t , a normal distribution with a mean of 0 and a variance of   , represents the stochastic
2
shock of production efficiency in period t .
Once we know the motion function, we can adopt the Tauchen (1986) to discretize the
stochastic process and acquire the transitional probability matrix P and stable distribution.
2.4 Problems for the Government
According to the literatures, the fiscal policy and monetary policy would play a key role in
the inflation welfare cost, however, we do not pay much attention on the influences of fiscal
140
Research of Mathematical Economics No. 1 2011
policy and monetary policy to the inflation welfare cost; therefore, the paper simplifies the role of
the government in the model economy.5
Referring to the method in Freeman et al. (2010) and Henriksen and Kydland (2010), the
paper assumes that the government only controls the currency issue. The motion function of
money stock in the model economy can be expressed as:
M t  1    M t 1
(7)
where M t is the gross nominal money stock in period t , and  is the increase rate of
nominal money stock.
Generally speaking, the government would have some income when issuing the currency. In
order to avoid the government policy to the model results, the paper assumes that government
would spread out the income evenly to each family. Since the gross number is standardized to be
1, the transfer acquired by each family is the gross income of the government. Assume that the
transfer of each family acquired in period t is  t , and then the balanced budget constraint
equation can be expressed as:
t   M t 1
(8)
2.5 Insurance Market
The insurance market in the model is incomplete. Because of the incompleteness of the
insurance market, the family cannot fully insure against the risks through the insurance market;
however, the family can hold money or productive capital or adjust the labor supply to smooth
the consumption. What’s more, the paper assumes that agents can only lend or borrow productive
capital and the nominal money holding must be non-negative.
2.6 Bellman Equation
From the assumptions above, we can see that the family income in the model economy can
be divided into three parts: wage income, capital income and the transfer from the government;
the expenditure can be divided into two parts: consumption and saving, where the saving can be
5
For more information about the government monetary policy and the inflation welfare cost, see Meh et al.
(2008); for more information about the government fiscal policy and the inflation welfare cost, see Gomme
(2008).
141
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
both money and productive capital.
Assume that the productive capital holding of one family in period t is at , the nominal
money stock is mt 1 , the depreciation rate is
 and the price of commodities in period t . Then
the budget constraint equation can be expressed as:
pt ct  pt at 1  mt  mt 1  t  pt wt lt  t  pt 1  r    at
(9)
Replace the government transfer  t by Equation (8). At the same time, denote the real
balance holding and the gross real balance in period t as mt and M t , respectively. What’s
more, assume the inflation rate in period t is  t , then the budget constraint can be expressed
as:
ct  at 1  mt   mt 1   M t 1  1   t   wt lt  t  1  rt    at
(10)
From Equation (10), we can see that the saving of the family can be both money and
productive capital; however, once entering into the next period, the forms of the asset holding
have no influences on the consumption and saving decision in the next period. Therefore, in order
to reduce the number of state variable, the wealth in the beginning of period t can be expressed
as:
qt  mt 1 1   t   1  rt    at
(11)
Since there is no aggregate uncertainty, the joint distribution of the individual states should
be stable under the equilibrium; hence, the joint distribution would not change with time.
Therefore, we can denote that the joint distribution of asset-holding and the productive efficiency
as
  q,   . At the same time, when the joint distribution is acquired, the gross labor supply,
gross capital and the gross real balance would be acquired. Considering the factors, the state
variable of the family should be asset holding and productive efficiency, the families make the
optimal consumption, optimal asset holding, optimal real balance and the optimal labor supply.
Denote the value function to be V , then the Bellmen function can be expressed as:


V  q,    max U  c, m ', l    E V  q ',  ' |  q,  
c , a ', m ',l
s. t.   c a' m'
142
q  M
 1  
w l
(12)
(13)
Research of Mathematical Economics No. 1 2011
q '  m ' 1     1  r    a '
(14)
m'  0
(15)
a '   a, a 
(16)
where a and a denote the lower limit and the upper limit of asset holding, a ' and m '
represent the real productive capital and real balance in the next period. The family solves the
dynamic optimum problem and acquires a set of optimal policy function based on the state
variables
c  q,   , a '  q,   , m '  q,   , l  q,   .
2.7 Recursive Competitive Equilibrium
The recursive competitive equilibrium is composed by the optimal value function V  q,   ,
the optimal policy function
state variables
variables
c  q,   , a '  q,   , m '  q,   , l  q,   , the joint distribution of
  q,   , the price of capital and labor r, w and the macroeconomic
K , L, M ,   . All the parts should satisfy the conditions as follows:
(1) The gross labor, gross capital and gross real balance should be acquired through aggregating
the decisions of all the families, that is to say:
K   ad 
(17)
L   l d 
(18)
M   md 
(19)
and
and
It should be noted that the equation (18) is also the clearing condition for the labor market
and the equation (19) is the clearing condition for the money market.
(2) Given the stable joint distribution
aggregate
real
balance
M
,
c  q,   , a '  q,   , m '  q,   , l  q,  
  q,   , aggregate capital K , aggregate labor L ,
wage
rate
w
and
interest
rate
r
,
should be the solutions of the equations (12)-(16).
143
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
(3) The product market clears, that is to say:

 c  q,   d   q,    K  AL 
1
 1    K
(20)
(4) The balanced budget constraint is satisfied, that is to say, Equation (7) is satisfied.
(5) The joint distribution of state variables
  q0 ,  0  
  q,   should be stable and the motion function is:



1
a
'

a
'
q
,


1
m
'

m
'
q
,


P

'
|

d


dq ' d  ' (21)










 

q0 , 0  Q , 



where the function 1  is an indicator function, when the variable is true, the value of the
function is 1, otherwise, it is 0. P
productive efficiency

 ' |  
represents the probability of transfer from
 in the current period to the efficiency  ' in the next period. Q is the
asset-holding space and  is the efficiency space. At the same time, in order to guarantee that
the wealth space and the efficiency space are static, the equation (21) should be always satisfied
q ,   Q  .
0
for any point in the space
0
Since the recursive equilibrium is stable, the joint distribution would not change with time.
While from equation (6), we can see that the gross nominal money is growing at a constant rate;
therefore, under the recursive equilibrium, the growth rate of the model economy should coincide
with the inflation rate, that is to say, the equation    should be satisfied.
2.8 First-order Condition
With dynamic program method, the first-order condition can be expressed as:
1 1 
c 1 1  m '
 c 1 1  m '
 1 1
  E 1  r    c '

1 1 
 1    c 1   m ' 
 m ''
1 1 


1 1  1
 1  1
1 1
 1
  E 
  c '
 m ''   
1  

 1  l    c 1 1  m '

1 1 
w
c  a ' m '  q   M 1     wl
144
(22.a)
(22.b)
(22.c)
(22.d)
Research of Mathematical Economics No. 1 2011
From the first-order condition, we can see that the equation (22.d) is in fact the budget
constraint of the family. (22.a) is the Euler function of productive capital, where the right hand
represents the marginal cost of holding productive capital while the left hand represents the
marginal return. Equation (22.b) is the Euler function of money holding where the right hand is
the marginal cost of holding money while the left hand of the equation is the marginal returns of
holding money. Equation (22.c) is the first-order condition for labor supply. Only if l  0,1
can the equation be satisfied. Otherwise, agent will choose the corner solution and the optimal
policy will be given by the equations (22.a), (22.b) and (22.d).
3.
Calibration
In order to solve the model, there are several parameters to be set at first. Specifically, these
parameters include: subjective discount rate  , relative risk aversion coefficient
parameter  and  , the curvature of leisure  , depreciation rate
capital
 , weight
 , output elasticity of
 , technical progress A and the upper limit as well as the lower limit of the
asset-holding space. What’s more, the autocorrelation coefficient  and the standard deviation
  should be given to estimate the motion function of productive efficiency. During the
calibration, the paper assume the technical progress is 1 at first; for other parameters, the paper
refers to some research literatures and adopts the data sets in reality to make the results of the
model fit the reality including wealth distribution, income distribution, the ratio of capital to
output and the ratio of money to capital as well as the average labor supply. Considering there is
few data about the China’s wealth distribution, it become inappropriate to adopt the monthly or
quarterly data to calibrate the parameter. Therefore, the model period is set to be 1 year. At the
same time, the inflation rate is assumed to be 3%, which is the average China’s inflation rate
during the year 1995 and the year 2007. All the calibration results are shown in Table 1.
145
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
146
Research of Mathematical Economics No. 1 2011
Table 1
Preference Parameters:
Relative Risk Aversion
Weight of Consumption
Weight of Leisure
Curvature of Leisure
Depreciation Rate
Motion Function of Efficiency:
Autocorrelation Coefficient
Standard Deviation
Technical Parameters:
Technical Progress
Output Elasticity of Capital
Depreciation Rate
Space of Asset Holding
Lower Limit of Productive Capital
Upper Limit of Productive Capital
Lower Limit of Money
Upper Limit of Money
Calibration Results
Parameters
Calibration


1.5





0.98
1.35
1.5
0.86
0.98
0.254
A
1
0.45
0.052
a
a
0
25
0.001
5


3.1 The Preference Parameters
The relative risk aversion coefficient
 is taken from other literatures directly while the
other preference parameters are calibrated to make the results of the model fit certain model
targets. Specifically, referring to Chen et al. (2009), the paper set the relative risk aversion
coefficient to be 1.5, which is in accordance with the value in the relative literatures.
For the subjective discount rate  , the paper calibrates it with the help of China’s ratio of
capital to output. Because there is no capital stock data in the dataset, we have no choice but to
estimate the gross capital stock to acquire the ratio of capital to output. With the method
developed by Wang and Yao (2003) and Liu et al. (2009), the paper estimates the gross capital
stock in China from the year 1978 to the year 2007; at the same time, the gross product is
represented by the real GDP. With these data, the ratio of capital to output is about 2.45. In order
to generate a relatively higher value of capital to output ratio, the subjective discount rate should
be very small. Finally, the subjective discount rate is determined to be 0.86.
147
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
We choose the weight parameter of consumption  so as to match China’s ratio of money
stock to gross output. According to the data from the website of Chinese Economic Information,
the paper estimates that the average share of M0 to the gross output from the year 1990 to the
year 2009 is about 13.7%. In order to match the value, the weight parameter of consumption 
is determined to be 0.98 at last.
There are still two preference parameters to calibrate. Since there are few literatures on the
labor supply and leisure choice in China, it is quite difficult to calibrate the other two parameters
with China’s datasets. Considering the form of one-period utility function and the calibration in
Castañeda et al. (2003), the paper choose the weight parameter of leisure and the curvature of
leisure so as to make the share of average working time to the gross labor endowment is about
1/3. After trying several times, the paper finally sets the weight parameter  to be 1.35 and the
curvature of the leisure  to be 1.5. We can see that the calibration results are different from
Castañeda et al. (2003), but all the parameters are greater than the value 1.
3.2 The Motion Function of Productive Efficiency
There are few empirical literatures on the China’s productive efficiency, let alone the motion
function of the individual productive efficiency. In order to overcome the problem, the paper
adopts the calibration method to estimate the motion function of individual productive efficiency.
The autocorrelation coefficient  and the standard deviation   are set so as to match the
wealth distribution and the wage distribution in reality. According to the estimation from survey
data conducted by Ao Er Duo Investment Center in the year 2005 and the year 2007, the wealth
Gini index lies in the interval between 0.56 and 0.58, therefore, the paper sets the Gini index to
be 0.57 at last. At the same time, according to the survey data, the wage Gini index lies in the
interval between the value 0.42 and the value 0.45; hence, the paper sets the wage Gini index to
be 0.435 and takes it as one target to calibrate. With the help of two calibration targets, the paper
finally sets the autocorrelation coefficient  to be 0.98 and the relevant standard deviation  
is 0.254. The results coincide with the calibration results of Chen et al. (2010).
After estimating the motion function of productive efficiency, the continuous process can be
discretized with the method developed by Tauchen (1986). Specifically, the paper sets the
interval width to be 1 and the number of points to be 5.
148
Research of Mathematical Economics No. 1 2011
3.3 Technical Parameters
Since the technical progress is standardized to be 1, there are still depreciation rate
output elasticity
 and
 to calibrate. According to Chen et al. (2009), the estimations of output
elasticity of capital in different literatures are quite different due the limit of the availability of
data in China. Referring to Chen et al. (2009), the paper sets
  0.45 . Considering that the
estimation is quite different, the paper would investigate the influences of the output elasticity of
capital to the inflation welfare cost in the robust analysis.
For the depreciation rate
 , the paper adopts the method in Chen et al. (2009) and average
the depreciation rate during all the years with the help of the data in Chow and Li (2002). Finally,
the depreciation rate is set to be 0.052.
3.4 Space of Asset Holding
The space of productive capital should satisfy two conditions: first of all, the lower limit of
the productive capital should be larger than the natural borrowing constraint so as to guarantee
that the model economy is with incomplete insurance market; secondly, the upper limit of the
space of productive capital should be set so as to make the productive capital choice of all the
agents lie in the interval. In order to improve the accuracy of the inflation welfare cost, the paper
should keep the space of the productive capital unchanged; therefore, the upper limit of the space
should be larger. After trying several times, the paper finally set the upper limit of productive
capital to be 25. At the same time, in order to simplify the computation procedure, the paper
assumes that each agent cannot borrow, this means that the lower limit of the space is 0.
For the space of money holding, on one hand, the form of one-period utility function means
that the money holding should be larger than 0; on the other hand, the low value of ratio of
money stock to the output means that the money stock should be very few. Therefore, the paper
sets the lower limit of the money stock space to be 0.001 and the upper limit is set to be 5.
What’s more, since the gross asset holding is taken as the state variable, the space of gross asset
holding is very important to the computation. Because the gross asset holding is determined by
the money holding and the productive capital holding, the choice of the gross asset holding
should reply on the space of money capital and the productive capital; however, if the space of
gross asset holding is determined directly by the equation (14), the upper and lower limit of the
149
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
gross asset holding would change with the change of interest rate, this would increase the
computation cost. Considering all the factors above, the paper sets the lower limit of the space to
be the sum of the lower limit of money holding and productive capital holding, while the upper
limit of the gross asset holding is set to be the sum of upper limit of the money holding and upper
limit of productive capital multiplied by 0.9.
4. Results
With the help of the solution methods given in Appendix 1, the paper solves the dynamic
stochastic general equilibrium model with Matlab Programming. We can acquire the optimal
policy function, the macroeconomic indexes and the inflation welfare cost. The section is divided
into two parts: in the part 4.1, the paper introduces the main results of the model when the
inflation rate is 3% so as to compare the calibration results and the data in reality; in part 4.2, the
influences of inflation to different asset-holding classes are introduced.
4.1 The Numerical Nature of the Benchmark Model
As the starting point of the analysis, the paper first consider the reasonability of the
calibration in the benchmark model and see whether the model results in the benchmark model fit
the macro economy in reality. Table 2 shows us the macroeconomic index and relevant targets.
Table 2
The Comparison of Model Results and the Macroeconomic Index in Reality
Macroeconomic Index
Model Results
Model Targets
Ratio of Capital to Output
2.5343
2.45
Ratio of Money Stock to Output
13.51%
13.7%
Ratio of Working Time to Labor Endowment
32.51%
33%
Wage Gini Index
0.4394
0.435
Note: The capital stock is estimated with the help of the original capital stock in the year 1978 and the method
developed by Wang and Yao (2003); the ratio of working time to labor endowment is in accordance with the
eight-hour working day; the estimation of wage Gini index relies on the survey data from Ao Er Duo Investment
Center.
From the results in Table 2, we can see that the macroeconomic index in the model results fit
the model target very well. Specifically, the ratio of capital to output in the model economy is
2.53, while the ratio in reality is 2.45. This means that the ratio in the model economy is higher
150
Research of Mathematical Economics No. 1 2011
than the model target; however, the gap is very small, it is only 0.08. The ratio of money stock to
output and the share of working time to the labor endowment are also quite close to the model
target. The share of working time in the model economy is 0.49% lower than the share in reality
while the ratio of money stock to output is 0.19% lower than the model target. We can also see
that the model economy fits the model target quite well in the wage Gini index.
At the same time, for a heterogeneous-agent model, it should be guaranteed that the model
results should fit the wealth distribution in reality with some degree (Krusell and Smith, 1997);
therefore, before we can investigate the heterogeneity of inflation, we should see that whether the
wealth distribution generated by the model fits the reality very well. The wealth distributions in
the model economy and in reality are shown in Table 3. However, only the wealth distribution in
reality is chosen as the model target while the shares of asset holding for different asset holding
classes are not chosen as the model targets.
Table 3
Comparison of Wealth Distribution in the Model Economy and in Reality
Gini
1%
5%
10%
20%-30%
30%-40%
40%
Index
wealthiest
wealthiest
wealthiest
wealthiest
wealthiest
poorest
Data in
2005
0.56
8.62%
25.88%
39.21%
30.84%
18.25%
7.62%
Reality
2007
0.58
8.74%
23.32%
36.79%
34.39%
16.94%
4.67%
0.5759
3.87%
17.52%
32.42%
40.86%
19.22%
3.16%
Model Results
Note: Data in reality is from Ao Er Duo Investment Center, the results can be seen in Chen et al. (2009).
From the results in Table 3, we can see that the wealth distribution fits the data in reality
quite well. Specifically, according to the calibration procedure listed in Section III, the model
target of the wealth Gini index is set to be 0.57, while the Gini index in the model economy is
0.5759, which is very close to the model target and lies in Gini indexes in the year 2005 and the
year 2007. From the share of asset holding for different asset-holding classes view, we can see
that although the share of the 1% wealthiest and the 5% wealthiest in the model economy is
lower than the data in reality, the model economy fits the reality for the share of 10% wealthiest,
20%-30% wealthiest and 40% wealthiest. Considering that there is no particular mechanism to
generate a higher share of the wealthy and we do not take the share of different asset-holding
classes as the model targets, we can say that the model economy fits the wealth distribution in
reality quite well. This means that we can use the benchmark model to study the inflation
151
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
heterogeneity.
4.2 The Influences of Inflation
With the help of method in Appendix 2, we can acquire the overall inflation welfare cost and
the welfare cost heterogeneity for different asset-holding classes.
First of all, we can see the overall inflation welfare cost to the society. The results are shown
in Table 4.
Table 4
Inflation Rate
Inflation Welfare Cost
0
0
Overall Inflation Welfare Cost（%）
1%
-0.46
2%
0.83
3%
4.19
5%
5.2
7%
5.44
10%
6.15
From Table 4, we can see that the overall inflation welfare cost is relatively high.
Specifically, when the inflation rate is 2%, the overall inflation welfare cost is about 0.83% of the
gross consumption, while when the inflation rate is 4%, the inflation welfare cost rises in a
further step to 2.37%. When the inflation rate is 10%, the inflation welfare cost is as much as
6.15% of the gross consumption. This means that the loss to the society due to the inflation in
China is quite high. At the same time, Table 4 further shows us that we should allow a relatively
low inflation rate when we have inflation control measures. The results show us that when the
inflation rate is 1%, there is no welfare cost but welfare gain. The welfare gain is about 0.46% of
the gross consumption. Therefore, the optimal inflation rate in China should lie in the interval
between the values of 1% and 2%, which coincides with the conclusions of Bewley (1983). That
is to say, the optimal inflation rate according to the Friedman Rule is not optimal any more.
Taking the market incompleteness in reality into consideration, and considering that the market
incompleteness would cause the difference of marginal substitution for different individuals and
the complete market would assume that the marginal substitution is same for each agent in the
model economy (Li and Han, 2010), it is reasonable for us to believe that the inflation welfare
cost calculated through the Bewley type model with incomplete market and the heterogeneous
agents would be more accurate for the Chinese economy.
In order to have a deeper understanding on the inflation welfare cost, the paper adopts the
calculated inflation welfare cost in Table 4 and the consumption and production data in reality to
acquire the inflation welfare cost in terms of money. The results can be seen in Table 5.
152
Research of Mathematical Economics No. 1 2011
Table 5
China’s Inflation Welfare Cost in the Years 2000-2009
Real Consumption
Inflation Rate
Welfare Cost
Welfare Cost
Welfare Cost
(100 Million RMB)
(%)
(Consumption %)
(GDP, %)
(100 Million, RMB)
2000
39105.70
0.4
-0.2780
-0.1096
-108.7091
2001
42756.11
0.7
-0.4233
-0.1662
-180.9749
2002
48524.09
-0.8
0.4099
0.1640
198.8875
2003
51893.58
1.2
-0.4000
-0.1547
-207.5774
2004
57267.56
3.9
5.4656
2.0341
3129.9906
2005
65988.80
1.8
0.3450
0.1253
227.6554
2006
75280.79
1.5
-0.1470
-0.0519
-110.6951
2007
85124.05
4.8
5.2902
1.7755
4502.2287
2008
102443.53
5.9
5.0053
1.7291
5127.5767
2009
133613.70
-0.7
0.3926
0.1553
524.5081
Note: Data is from the website of Chinese Economic Information. The gross consumption is represented by the
Year
total volume of retail sales of consumption. Both GDP and consumption have been adjusted by the CPI and the
base year is the year 2000. The inflation welfare cost in each year is acquired through the cubic spline
interpolation. Column 4 represents the welfare cost in terms of the gross consumption, Column 5 represents the
welfare cost in terms of GDP and the Column 5 represents the inflation welfare cost in terms of money.
From Table 5, we can see that the inflation has been very low before the year 2006 since
entering the new century. However, the inflation began to rise since 2006 and the trend did not
stop until the influences of the subprime crisis. A glance at the 10 years, we can see that there is
no hyperinflation. Even in the year 2008 when the inflation rises to its peak, the inflation rate is
only 5.9%. However, the inflation welfare cost is very large. In 2007, the inflation welfare cost is
as much as 5.3% of the gross consumption and the absolute amount is as much as 450 billion
RMB. The inflation welfare cost in 2008 is also as much as 5% of the gross consumption and
1.73% of GDP; what’s more, the absolute amount is as much as 513 billion RMB, higher than the
absolute amount in 2007. Therefore, it is quite necessary to take effective measures to have the
inflation under control. What’s more, we can see from Table 5 that the results are different from
the results generated by the model with complete markets: the calibration results show us that
when the inflation is quite low, the inflation is good for the economic development and the
Friedman Rule is not always optimal. Therefore, the market incompleteness would influence the
relationship between the inflation and the macro economy; hence we should try our best to
incorporate the market incompleteness into the model.
However, it is not enough to investigate the overall influences of the inflation to make
153
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
rational measures. What’s more important is to have an investigation into the inflation
heterogeneity to the different asset-holding classes. Table 6 shows us the inflation welfare cost to
different asset-holding classes.
Table 6
Inflation Rate
Inflation Welfare Cost Heterogeneity
1%
2%
3%
5%
7%
10%
1%
0.12
-0.72
-0.45
-0.62
-0.25
1.08
5%
0.09
-0.56
-0.30
-0.45
-0.10
1.29
10%
-1.19
0.43
1.85
1.72
2.11
3.47
20%
-1.50
0.73
0.68
2.87
4.50
2.17
30%
-2.40
0.40
1.28
4.32
5.10
4.56
1%
0.77
-0.66
0.07
-0.70
1.48
2.90
5%
3.14
1.23
3.32
1.81
2.67
1.71
10%
2.96
0.43
2.45
0.88
2.84
1.96
20%
1.16
0.04
2.65
0.74
1.85
2.37
30%
1.61
0.52
4.61
2.21
3.01
3.60
The Poor:
The Wealthy:
From Table 6, we can see that there are great distinctions of inflation welfare cost to
different asset-holding classes. For example, when the inflation welfare is quite low, the inflation
would usually benefit the 1% poorest and the 5% poorest while the most agents in the economy
would suffer great welfare loss. Therefore, even the policy would cause a moderate inflation to
the economy; the government should also evaluate the different influences to the poor and the
wealthy.
When evaluating the inflation heterogeneity to the agents, there is one important index to
consider: the influences of inflation to wealth distribution. Table 7 and Figure 1 give us the
wealth distribution under different inflation rates.
154
Research of Mathematical Economics No. 1 2011
Table 7
Inflation Rate
Inflation and Wealth Distribution
0
1%
2%
3%
5%
7%
10%
0.563
0.558
0.567
0.576
0.582
0.581
0.585
0.448
0.437
0.447
0.439
0.444
0.442
0.440
1%
0.007
0.007
0.005
0.006
0.005
0.005
0.005
5%
0.036
0.036
0.027
0.028
0.027
0.026
0.025
1%
3.602
3.656
3.678
3.866
3.866
3.785
3.757
5%
16.966
16.792
16.941
17.520
17.469
17.393
17.614
1
0.077
0.082
0.061
0.062
0.057
0.053
0.050
2
0.240
0.253
0.229
0.191
0.157
0.133
0.138
3
0.897
1.018
0.797
0.781
0.623
0.623
0.592
4
2.276
2.443
2.183
2.126
1.896
1.810
1.731
5
4.536
4.671
4.492
4.328
4.241
4.260
4.129
6
8.081
7.924
7.888
7.322
7.213
7.301
7.340
7
12.322
12.283
12.357
11.904
11.970
12.292
11.954
8
17.342
17.344
17.414
17.089
17.269
17.547
17.365
9
23.040
23.189
23.318
23.774
23.996
23.778
23.850
10
31.188
30.792
31.262
32.424
32.576
32.202
32.850
Wealth
Distribution
Wage
Distribution
The Poor:
The Wealthy:
In terms of
Tenths
Note: Line 2 and Line 3 represent the wealth Gini index and wage Gini index, respectively. Others represent the
share of wealth for each asset-holding class, the unit is 1%.
155
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
1
0
5%
10%
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
Figure 1
0.4
0.5
0.6
0.7
0.8
0.9
1
Inflation and Wealth Distribution
From Table 7, we can see that inflation would worsen the wealth distribution. Specifically,
with the increase of inflation, the wealth Gini index would increase; the wealth share of the poor
would decrease while the share of the wealthy would increase.6 On the Lorenz curve, we can see
that with the increase of the inflation rate, there is a trend to convex to the lower right. This can
be seen from Figure 1. From Figure 1, we can see that the Lorenz curve under the inflation rate of
5% would convex more to the lower right than the Lorenz curve without inflation while convex
less to the lower right than the Lorenz curve under the inflation rate of 10%. For China, the
worsening of the wealth distribution has limited the development of the economy and
harmonious of the society and it is very important to control the wealth distribution. Since
inflation would influence the wealth distribution, we should take the effects into consideration
when we have some measures to control the inflation.
5. Conclusions
Although there have been lots of economic models to investigate the China’s inflation
welfare cost, most of them are based on representative-agent models and the models with
6
For more details, see Appendix 3.
156
Research of Mathematical Economics No. 1 2011
complete market (Chen, 2003; Chen and Ma, 2007; Zhao, 2008; Wu and Cui, 2006). The
complete market usually assumes that the marginal substitutions for different individuals are
same, which contradicts with the reality. In China, the insurance market is not quite fully
developed; therefore, agents cannot fully insure against all kinds of risks through the insurance
market. This means that it is quite difficult to smooth the consumption, which would lead to the
differences of marginal substitution for different agents and different periods; hence, the
individual decision would change and thereafter the inflation welfare cost would change.
Therefore, we should try to incorporate the market incompleteness into the model when studying
the China’s inflation welfare cost. What’s more, under the circumferences that the wealth
distribution has been worsened, it is not enough to investigate the overall influences of inflation;
in fact, it would be helpful for making rational measures to taking the heterogeneity into
consideration. Based on the consideration, the paper constructs a Bewley model with production
sector and with the help of the model, the paper has an investigation into the inflation welfare
cost and the influences of inflation to different asset-holding classes.
After computation, we can find that inflation would cause great influences to the macro
economy from the overall viewpoint and the overall inflation welfare cost is quite large. At the
same time, the paper investigates the influences of inflation to the wealth shares and welfare
costs of different asset-holding classes. Numerical results show us that there are great distinctions
of inflation to different asset-holding classes. When the inflation is quite moderate, inflation
would benefit the poor while causing great welfare cost to the wealthy; however, this does not
mean that the inflation would improve the wealth distribution condition. Contrarily, the inflation,
in fact, has played a role of transferring the wealth from the poor to the wealthy: with the increase
of the inflation rate, the wealth share of the poor would decrease while the wealth share of the
wealthy would increase. This means that inflation does not improve China’s wealth distribution
but worsen the distribution.
Therefore, with the Bewley model, the paper combines the heterogeneity and the market
incompleteness into one model to study the inflation and hence provides new insights of the
China’s welfare inflation. However, there are still lots of work to do to improve the work. First of
all, it is very important for policy to do transition dynamic analysis on the inflation welfare cost
(Gomme, 2008; Burdick, 1997). It would be surely quite helpful to investigate the heterogeneous
influences of transitional dynamics of inflation. Secondly, it is quite important to investigate the
157
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
influence heterogeneity of unexpected inflation shock (Doepke and Schneider, 2006a; Meh et al.,
2010). I am sure it would be helpful to study the problem under the framework of Bewley type
model with incomplete market and heterogeneous agents. Thirdly, Comparing with other
developed countries, there is dualistic structure in urban and rural areas. Not only the income and
wealth structures in rural and urban areas are different, but also the consumption and behavior
habits of rural and urban residents are different. What’s more, there is dualistic structure for the
inflation in China (Chen and Chen, 2009). How to combine all there heterogeneity into one
model? This would be one project full of difficulty. However, I think it would be quite helpful to
know more about the dualistic structure in rural and urban areas and the influences of China’s
inflation to the macro economy.
References
Aiyagari, S. R., 1994, “Uninsured Idiosyncratic Risk and Aggregate Saving”, The Quarterly Journal of
Economics, Vol. 109, No. 3, pp. 659-684.
Algan, Y., E. Challe and X. Ragot, 2009, “Incomplete Markets and the Inflation-Output Tradeoff”,
Forthcoming of Economic Theory.
Algan, Y., and X. Ragot, 2010, “Monetary Policy with Heterogeneous Agents and Borrowing Constraints”,
Review of Economic Dynamics, Vol. 13, No. 2, pp. 295-316.
Bailey, M. J., 1956, “The Welfare Cost of Inflationary Finance”, Journal of Political Economy, Vol. 64, No.
2, pp. 93-110.
Bewley, T. F., 1983, “A Difficulty with the Optimum Quantity of Money”, Econometrica, Vol. 51, No. 5, pp.
1485-1504.
Burdick, C. A., 1997, “A Transitional Analysis of the Welfare Cost of Inflation”, Federal Reserve Bank of
Atlanta Working Paper, No. 97-15.
Castañeda, A., J. Díaz-Giménez and J. Ríos-Rull, 2003, “Accounting for the U.S. Earnings and Wealth
Inequality”, Journal of Political Economy, Vol. 111, No. 4, pp. 818-857.
Chen Liping, 2003, “’Catchting Up with the Joneses’ and the Welfare Cost of Inflation”, China Economic
Quarterly, Vol. 2, No. 3, pp. 573-590.
Chen Yanbin and Lili Ma, 2007, “The Study of the Cost of Inflation in China”, Economic Research Journal,
Vol. 53, No.4, pp. 30-42.
Chen Yanbin and Jun Chen, 2009, “Analysis on China’s Insufficient Consumption: Based on the Perspective
of Household Wealth”, Journal of Renmin Unversity of China, Vol. 23, No.6, pp. 80-86.
Chen Yanbin, Zhen Huo and Jun Chen, 2009, “Disaster Risk and Wealth Distribution of Chinese Urban
158
Research of Mathematical Economics No. 1 2011
Residents”, Economic Research Journal, Vol. 55, No. 11, pp. 144-158.
Chen Yanbin, Zhesheng Qiu and Fangxing Li, 2010, “Advance in Macroeconomics: Bewley Model”,
Economic Research Journal, Vol. 56, No. 7, pp. 141-151.
Chen Y., F. Li and Z. Qiu, 2010, “Accounting for the Household Saving Rates in China”, Renmin University
of
China Working Paper.
Chow, G. C., and K. Li, 2002, “China’s Economic Growth: 1952-2010”, Economic Development and
Cultural Change, Vol. 51, No. 1, pp. 247-256.
Chiu, J., and M. Molico, 2008, “Uncertainty, Inflation, and Welfare”, Bank of Canada Working Paper No.
2008-13.
Cooley, T. F., and G. D. Hansen, 1989, “The Inflation Tax in a Real Business Cycle Model”, The American
Economic Review, Vol. 79, No. 4, pp. 733-748.
Cysne, R. P., 2006, “An Intra-Household Approach to the Welfare Costs of Inflation”, Getulio Vargas
Foundation (Brazil) Working Paper No. 612.
Doepke, M., and M. Schneider, 2006a, “Inflation and the Redistribution of Nominal Wealth”, Journal of
Political Economy, Vol. 114, N0. 6, pp. 1069-1097.
Doepke, M., and M. Schneider, 2006b, “Inflation as a Redistribution Shock: Effects on Aggregate and
Welfare”, NBER Working Paper No. 12319.
Erosa, A., and G. Ventura, 2002, “On Inflation as a Regressive Consumption Tax”, Journal of Monetary
Economics, Vol. 49, No. 4, pp. 761-795.
Freeman, S., E. R. Henriksen and F. E. Kydland, 2010, “The Welfare Cost of Inflation in the Presence of
Inside Money”, Chapter 1 in Monetary Policy in Low-Inflation Economies, pp. 1-20.
Guo Qingwang and Junxue Jia, 2004, “Estimating Potential Output and the Output Gap in China”, Economic
Research Journal, Vol. 50, No. 5, pp. 31-39.
Gomme, P., 1993, “Money and Growth Revisited: Measuring the Costs of Inflation in an Endogenous
Growth Model”, Journal of Monetary Economics, Vol. 32, No. 1, pp. 51-77.
Gomme, P., 2008, “Measuring the Welfare Costs of Inflation in a Life-cycle Model”, Concordia University
Working Paper, No. 08-001.
Huang Zuhui, Xichuan Liu and Enjiang Chen, 2009, “Explanations for the Low Participation Rate in the
Formal Credit Market by Rural Households in the Poor Areas”, Economic Research Journal, Vol. 55, No. 4, pp.
116-128.
Huggett, M., 1993, “The Risk-Free Rate in Heterogeneous-agent Incomplete-insurance Economies”, Journal
of Economic Dynamics and Control, Vol. 17, No. 5-6, pp. 953-969.
Krusell, P. and A. Smith, 1997, “Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset
Returns”, Macroeconomic Dynamics, Vol. 1, No. 2, pp. 387-422.
Krusell, P. and A. Smith, 1998, “Income and Wealth Heterogeneity in the Macroeconomy”, Journal of
Political Economy, Vol. 106, No. 5, pp. 867-896.
159
A Quantitative Analysis of the Influences of China’s Inflation to Different Asset-Holding Classes
Laidler, D., 1990, “Taking Money Seriously”, Cambridge: The MIT Press.
Lagos, R., and R. Wright, 2005, “A Unified Framework for Monetary Theory and Policy Analysis”, Journal
of Political Economy, Vol. 113, No. 3, pp. 463-484.
Lucas, Jr. R. E., 2000, “Inflation and Welfare”, Econometrica, Vol. 68, N0.2, pp. 247-274.
Li Junqing and Qiheng Han, 2010, “Incomplete Financial Market, Precautionary Saving and Welfare Cost of
Inflation”, China Economic Quarterly, Vol. 9, No. 1, pp. 191-212.
Li Rui and Ninghui Li, 2004, “Studies on the Borrowing Behavior of Rural Households and Welfare Effects”,
Economic Research Journal, Vol. 50, No. 12, pp. 96-104.
Liu Danhe, Shilei Tang and Du Li, 2009, “Technical Progress and Quality of China’s Economic Growth:
1978-2007”, On Economic Problems, Vol. 31, No. 3, pp. 30-33.
McCallum, B. T., and M. S. Goodfriend, 1987, “Money: Theoretical Analysis of the Demand for Money”,
NBER Working Paper, No. 2157.
Meh, C. A., and Y. Terajima, 2008, “Inflation, Nominal Portfolios and Wealth Redistribution in Canada”,
Bank of Canada Working Paper, No. 2008-19.
Meh, C., J. Rĺos-Rull and Y. Terajima, 2010, “Aggregate and Welfare Effects of Redistribution of Wealth
Under Inflation and Price-Level Targeting”, Journal of Monetary Economics, Vol. 57, No. 6, pp. 637-652.
Heathcote, J., K. Storesletten and G. L. Violante, 2009, “Quantitative Macroeconomics with Heterogeneous
Households”, Annual Review of Economics, Vol.1, No.1, pp. 319-354.
Henriksen, E., and F. E. Kydland, 2010, “Endogenous Money, Inflation and Welfare”, Review of Economic
Dynamics, Vol. 13, No. 2, pp. 470-486.
İmrohoroğlu, A., and E. Prescott, 1991, “Seigniorage as a Tax: A Quantitative Evaluation”, Journal of Money,
Credit and Banking, Vol.23, No. 3, pp. 462-475.
İmrohoroğlu, A., 1992, “The Welfare Cost of Inflation Under Imperfect Insurance”, Journal of Economic
Dynamics and Control, Vol. 16, No. 1, pp. 79-91.
Sidrauski, M., 1967, “Inflation and Economic Growth”, Journal of Political Economy, Vol. 75, No. 6, pp.
796-810.
Tauchen, G., 1986, “Finite State Markov-Chain Approximations to Univariate and Vector Autogressions”,
Economic Letters, Vol. 20, No. 2, pp. 177-181.
Temple, J., 2000, “Inflation and Growth: Stories Short and Tall”, Journal of Economic Surveys, Vol. 14, No.
4, pp. 395-426.
Wang, Y., and Y. Yao, 2003, “Sources of China’s Economic Growth 1952-1999: Incorporating Human
Capital Accumulation”, China Economic Review, Vol. 14, No. 1, pp. 32-52.
Wu Hanhong and Yong Cui, 2006, “Seigniorage-maximizing and Inflation in China: 1952-2004”, Economic
Research Journal, Vol. 52, No. 9, pp. 27-38.
Zhao Liuyan, 2008, “Inflation, Government Revenue and Welfare Costs”, China Economic Journal, Vol. 8,
No. 1, pp. 179-196.
160
Research of Mathematical Economics No. 1 2011
Zhao Zhenyu, 2006, “An Exploration of the Inflation Path in Two Theoretic Frameworks”, China Economic
Journal, Vol. 5, No. 2, pp. 497-510.
Zhu Xinkai and Gang Liu, 2009, “Dual Financial Institution and Farmers’ Choice on Consumer Credit
Behavior: The Explanation and Analysis of ROSCA”, Economic Research Journal, Vol. 55, No. 2, pp. 43-55.
(Responsible Editor: Ren Junqiushi) (Proofreader: Wang Jue)
161
Analysis of the Mechanism of “Dutch Disease”
Analysis of the Mechanism of “Dutch Disease”
Wang Qizhi
(School of Economics, Renmin University of China)
Abstract: This article describes the mechanism of “Dutch Disease” by setting natural resources into the
production function and creating an independent sector, the resource exploiting sector. With the help of the
Ramsey Model and the general equilibrium analysis framework, I will try to explain the influence of the “Dutch
Disease” to the manufacturing sector of the economy and consumption-investment decision making. At last, I’ll
give out some advice on how to avoid it.
Keywords: Dutch Disease, Ramsey Model
1. Introduction
In recent years, the relation between natural resources and economic growth draws
increasing attention among the public. Natural resources are the material basis and prerequisites
of economic growth, while its abundance will affect the country’s gross economic growth to
some extent. In short, there is a positive correlation between the two，but there also exists such
phenomenon of so-called “resources curse” in the relationship between the two. For a long time,
some economists believe that natural resources are the foundation of economic growth and have
played a positive role. While other economists argue that natural resources may hinder economic
growth because of the facts that those countries with abundant resources generally can’t get rid of
“resources curse”. Resources occupy an extraordinary position in sustainable development.
When it comes to the “Resources curse”, the Dutch disease is a typical one. During the
1960s, the Netherlands, which was already one of the main exporting countries of manufactured
goods, found a great amount of gas in the North Sea. Dutch government strongly support that
area, thus its exporting leapt, and so did its international balance of payment, the whole economy
showed all signs of prosperity. Robbert van Eerd (2010) describes the situation at that time like
this:
162
Research of Mathematical Economics No. 1 2011
The year 1959 saw the first exploratory gas drillings in the Dutch province of Groningen,
which would turn out to be one of the largest gas fields in the world. Initially, the recoverable
deposits were estimated to be 50 billion cubic meters. The estimates were repeatedly revised
upwards, and by 1963, when the first concessions were granted, it was believed that the total
reserve could be as large as 1,100 billion cubic meters. By 1967, estimates stood at 2,200 billion
cubic meters (to illustrate how large this volume is, last year’s natural gas consumption in the
whole of the Netherlands was about 50 billion cubic meters).1
But the booming gas sector hurt the other part of the Dutch economy badly, such as the
agricultural sector and some other industrial sectors which also reduce the competitive ability of
them. In the 1970s, Netherlands suffered a severe inflation thus reducing exports of manufactured
goods and citizen’s income. Here is a description of the situation at that time:
The wage pressure in the primary industry leads to wage pressure in other sectors of the
economy. As workers seek out the highest wages, wage demands in the secondary sector need to
mirror those in the primary sector (extraction and related industries). The apparel industry, for
example, declined from 60,000 to 20,000 jobs between 1968 and 1978.2
When resources are abundant, the resources are often "curse" instead of "bless" the economy.
The "Dutch Disease", which means that a country digging out considerable amount of
resources would firstly boom the economy and then drag the economy into recession, this
phenomenon was first brought to light by Corden (1984). Corden divided a country's economy
into three sectors----tradable manufacturing sector, tradable resources sector and non-tradable
goods sector, the resources sector exert influence on other sectors (in the model, the
manufacturing sector and service sector), in which mechanism works through wages and
exchange rate, thus affects the country's long-term sustainable development. But the analysis is
more focused on the manufacturing sector in its long-term changes, while ignoring the whole
economy as an organic whole.
In the analysis of economic growth factors, neoclassical growth theory ignored the elements
of natural capital, but focusing only on capital and labor (including human capital), neglecting
the restriction of resource constraints. Hence, Solow-Swan (1956) established neoclassical
1
Robbert van Eerd, 2010, Of Dutch Disease and Other Ailments, The Bologna Center Journal of International
Affairs.
2 The same as above.
163
Analysis of the Mechanism of “Dutch Disease”
growth model which laid the cornerstone for economic growth theory, and the model maintains
as the orthodoxy for a very long time. Solow Model uses the "new classical" production function
in which capital and labor can be smoothly replaced, which means given a certain amount of
labor, with the increase in capital stock, the law of diminishing marginal returns of capital
ensures an equilibrium economic growth rate . Variance between the particular equilibrium value
and the actual growth rate disappear due to lack of investment return expectations. All the above
guarantee the conclusion of steady economic growth.
However, as we know, the savings rate in the Solow model is exogenous, which doesn’t
quite fit the fact, so Cass-Koopmans (1965) pulled the analysis of the consumer optimization into
the new classical economic growth model, which makes the savings rate endogenous, thus
establishing a more general economic growth model. Cass-Koopmans referred to the
mathematical form used in Ramsey’s paper (1928), in which Ramsey constructed a dynamic
optimization model, as well as using the classical calculus of variation to get the famous
‘Keynes-Ramsey rule’ on deciding the savings rate. Anyway, Ramsey’s paper laid the foundation
of research on accumulation and growth issues.
Although the savings rate achieved became endogenous, Ramsey-Cass-Koopmans model
consists with Solow-Swan growth model in the long-term occasion, namely, long-term economic
growth rate depends only on the exogenous factors of population growth and technological
progress. To fix this situation, in the twentieth century, mid-eighties, new economic growth
theory of the neoclassical model came into being. Theoretically, Romer (1990) firstly put forward
a model which makes the economic growth depending on endogenous technological progress rate.
Around the same time, Lucas (1988) proposed a model on human capital in which the
endogenous growth of human capital explains the economic growth.
Although the growth theory is kept in the process of being modified and improved, basically
speaking, the input factors has always been only two, capital and labor, without any factor of
natural resources into consideration. Let’s think, when resources exhausted, no state, even those
with the best equipment and personnel, can put into production,
thus, resources is certainly a
significant restriction to a country's economic growth. Stiglitz (1974) investigated in the
economic growth pattern under exhaustible resource constraints, the model shows that the
scarcity of natural resources can be offset by technological progress, in other words, as long as
keeping a positive rate of technological progress, aggregate output will not fall.
164
Research of Mathematical Economics No. 1 2011
In this paper, I put the Ramsey growth model into Corden’s analytical framework, equate
resources with labor and capital as input factors, and separate resource-exploiting department
from the economy, thus lead to a complete study concerning what would happen to small
countries’ economy once abundant resources are exploited, meanwhile my study also shows that
the price variance of resources in international market can produce the same effect as to the small
countries’ economy.
The second part of my paper describes the basic model, the third part is my study on the
mechanism of “Dutch disease” with the use of my model, and the last part ( the fourth one)
shows my policy suggestions as well as the deficiency of this paper.
2.
Basic Model
The economy is divided into 3 parts: household, manufacturing sector and resource
exploiting sector. Household hopes to maximize its inter-temporal utility and makes consumption
decisions; the manufacturing and resource exploiting sectors both seek maximum profits. We
assume that labor and capital markets are perfectly competitive. Assume that the product of the
manufacturing sector has no access to export while the resource market is open and any amount
of resources can be sold at the world price P .
2.1 Manufacturing Sector (MS)
Assume that the MS has the following production function:
where Q represents quantity of production, A1 represents technique, K stands for the total
capital stock, (1  v) K is the capital stocked in the MS, L stands for the labor in the whole
society, (1  u ) L is the labor working in MS, Rd is the resources used in MS.
The manufacturing sector seeks maximum profit:
165
Analysis of the Mechanism of “Dutch Disease”
where wQ is the wage in the MS, rQ is the interest rate，and
 is the depreciation, the
first-order condition is:
2.2 Resource Exploiting Sector (RES)
Assume that the RES is labor intensive and its production function is:
0  v  1, 0  u  1
where R represents the total amount of resources that are exploited, A2 can be used to
measure the quality of one country’s resources: when A2 is large, little inputs can exploit large
amounts of resources; when A2 is small, exploiting will cost more. uL represents the labor in
the RES.
RES is facing an open resource market with a world price P . And MS also seeks
maximum profit:
The first-order condition is:
2.3 Household
166
Research of Mathematical Economics No. 1 2011
Household has the utility function:
and could make the investment decision:
(1)
where
rate，
represents the household’s total assets,
represents wage，
represents interest
is consumption.
When the control variable is consumption, household’s utility-maximizing decision making
can be expressed by the current-value Hamiltonian:
The first-order condition is:
and
’s movement function is:
So we get the Ramsey’s rule：
(2)
and the transversality condition:
lim[a (t ) (t )]  0
t 
3. Analysis and Applications
3.1 Equilibrium
When this economy reaches the equilibrium, families and firms are faced with the same
interest rate and wage level expressed here,
167
Analysis of the Mechanism of “Dutch Disease”
(3)
(4)
(5)
Because of our assumption of closed capital market, families’ assets
society’s capital stock
is equal to the
. Besides, the market is perfectly competitive, which means all firms
get zero profit, so the yield of the economy equals the sum of wage and interest. Combining (1)
(4) and (5), we get:
(6)
If we transform both sides of (3) (4) (5) to their Logarithm and then differentiate them with
respect to time, we find that3
(7)
(8)
(9)
3.2 Steady State
In the Ramsey model, it was proved that this economy has a steady state. When the
population is not growing, the growth of consumption and capital stock is zero. On the condition
of TVC and making the growth of
equal to 0, from (2) we can calculate:
(10)
which corresponds to the vertical line in Figure 1.
Then after we make the growth of
equal to zero, from (6) we can understand:
(11)
This equation illustrates a situation that is similar to the Solow-Swan Model. We can
.
3
168
g c represents c c , and the same way goes for other variables.
Research of Mathematical Economics No. 1 2011
to get the “golden rule” level of
differentiate the right side of this equation with respect to
capital stock. To clarify this, we can assume that the “production function” of the economy is F ,
which satisfies F ( K , L)  Q( K , L)  R( K , L) , then
(11)
which corresponds with the solid curve in Figure 1.

c=0
c

k=0
MPK*
Kgold
Figure
k
1
Figure 1 illustrates that, under the condition of different capital stock, households should
select different consumption strategies in order to move to the steady point through stable arm.
Now I’ll show how this works:
When the economy starts with small capital stock, if households select the consumption
lower than the amount steady arm points out, which actually is a phenomena of higher saving rate,
.
it’s trajectory eventually crosses c  0 locus. After crossing,
keeps on rising while
starts to decline. At last, the path will hit the horizontal axis. Similarly, if households select the
.
higher consumption rate, the trajectory will cross the k  0 locus. After intersecting,
169
Analysis of the Mechanism of “Dutch Disease”
continues to rise while
starts to decline. So, in finite time it will hit the vertical axis.
3.3 Exogenous Shocks
Because the finding of new resources is unpredictable, this article assumes that it’s a
exogenous shock to the economy. When new resources with high quality are found, it means the
average quality of domestic resources rises which indicates a rise of
in the production
function of RS. So we can get more resources with the same amount of input in RES. Consider
the influence of this shock to the steady state.
Proposition 1: The finding of new resources will lead to a change in domestic industrial
pattern, which means RES will be relatively booming while MS will be relatively shrinking.
We’ll try to prove this by observing the changes in the labor market4
From (8) we can reason that:
Then we use
and get:
When R has an exogenous growth
, the wage in RES must rise. And because the labor
market is competitive, it has a tendency to flow from MS to RES. So
must be positive, and
that indicates
(12)
That means RS grows faster than MS. So undoubtedly, that effect will induce the decline of
the proportion of MS and increase the proportion of RES. This is evidence for what was
explained earlier that “RES will be relatively booming while MS will be relatively shrinking”.
Hence, we’ve proved proposition 1, and we could call this effect a pull-out effect.
4We
170
can get (11) as well by using the same way to analyze the capital market through the change of interest rate.
Research of Mathematical Economics No. 1 2011
c


c0 = 0
c1 = 0
B
C1*

C0*
k1 = 0
A

k0 = 0
K0*
K1*
k
Figure
2
Proposition 2: The finding of new resources will pull the economy to a new steady state
where c and K are higher, thus the utility is higher than the formal one.
As we assume the production function is linear homogeneous, and the markets are perfectly
competitive, households’ revenue Y equals the sum of wages and interests, hence
(13)
Combining (13) with (11),
.
From the equation above we can see that the locus of k  0 also moves upward in the
phase diagram.
.
From (10) we can know, when r rises, c  0 will move right, shown in Figure 2. Variables
with subscript 0 show the initial situation of steady state, while 1 shows the situation after shock.
We can find the equilibrium point moves from A to B, and the equilibrium consumption and
*
*
capital stock also move from C0 to C1 and K 0
*
*
to K1 . So compared with the initial
situation, we can now enjoy more consumption and capital stock. Because utility is an increasing
171
Analysis of the Mechanism of “Dutch Disease”
function of c, social utility also increases. We can call this effect the growth effect.
Proposition 3: When world price P increases, pull-out effect and growth effect will also
occur.
Essentially, no matter if the increase is in P or A2 , the economy will react the same way.
Both shocks lead to the same result, that being more outputs with same inputs. So the procedure
has no actual difference from each other and can be omitted here.
3.4 Empirical Analysis
Proposition 1 tells us that when the quality of domestic resources increases, RES will
occupy a bigger part of this economy by pull out some part of MS. Meanwhile, proposition 2
shows us consumption and capital stock also increase. So, if we assume that a country suddenly
finds a lot of resources with high quality, then the pull-out effect and growth effect will both
happen and the economy will enjoy a booming time.
But because the exploited resources are always from the high-quality or low-cost resources,
after exploiting for a while the average quality of domestic resources will decline instead of
continuing to increase. When no other high-quality resources are found, the reversal effects of
both effects will happen. At that time, the economy will suffer a declining time, and the speed of
decline is related to that of resource quality. One thing needs to be noticed, the MS may still
shrink compared with the initial condition (before new resources’ finding). In real life, due to the
imperfect substitution in the labor force between two sectors, when labor flows into the RES, the
efficiency of the new workers must be lower. Besides, the cost of management may augment.
Those will all lead to a decrease in the production efficiency (which is the opposite effect of the
shock’s), which will reinforce the reversal effects. Besides, the works may not find new jobs at
once, so the unemployment undoubtedly will occur. All in all, the imperfection of market leads to
the decay of the economy.
So we can see that, the fact of the Netherlands between 1960s and 1970s mainly matches the
result we conclude from this model. But, there are still some points that I should mention: (1)
Because the manufacturing sector of the Netherlands is not in a closed market, selling gas must
have an impact on its exchange rate to force it to appreciate which will hurt the MS more. This
model can explain this by a little modification that opens the market of manufactured products.
But it will not fundamentally change the conclusion of this paper, so I didn’t do this work. (2)
172
Research of Mathematical Economics No. 1 2011
Because the market in the real world cannot adjust as swiftly as it’s imagined in the model, such
as the change in wages, labor’s transfer, and the economy’s moving to the steady point will all
take some time, so the real economy shows some lagging and mixed characters. That is, some
phenomena are not showing immediately or showing some combined results.
3.5 Policy Advices
The “Dutch Disease” makes the manufacturing sector decay in the country with rich
resources. The problem is, when the manufacturing sector goes down, the country’s economy is
essentially done in the long run and the manufacturing sector has more externalities like
technological innovation and nurturing great managers. Moreover, the resource exploiting sector
mostly needs lower human ability than the manufacturing sector, so if the manufacturing sector
fades, the outflow of human talents is inevitable.
More importantly, the manufacturing sector can develop and sustain in the long run while
the resource exploiting sector has an inherent limit, which is its reserve. Once all resources are
exploited, the resources sector will disappear. So it could be really dangerous for one country’s
sustainable development when the manufacturing sector goes down. Due to the huge uncertainty
of the resource sector, the economy will be quite unstable when it takes a great part in it. When
the production of the resource sector is undulating, it will influence the economy so badly that
may cause some social problems.
So, an effective method to control the consequence of the “Dutch Disease” means a lot to
both economic growth and society stability.
Firstly, the government can tax on the exporting resources. If we do so, the domestic price of
resources will stay low which will reduce the cost of the manufacturing sector. At the same time,
tax makes it possible for the economy to stick to the optimal exploiting path5, not too much or too
quickly.
Secondly, the government can invest in new resource developing activities. When the
current resources’ exploiting cost is high enough, the economy can turn to new resources. That
action can also reduce the instability of the economy induced by resource production.
At last, the advice I mentioned above is coping with the pull-out effect. So these kinds of actions
reach their goal by sacrificing current consumption and investment. So if the government is
5
More details in Stiglitz (1974), etc.
173
Analysis of the Mechanism of “Dutch Disease”
going to do so, take that cost into consideration.
4.
Concluding Remarks
This paper uses the general equilibrium to illustrate that the finding of new high-quality
resources may cause two effects to its economy; pull-out effect and growth effect, which will
influence the growth of this economy in steady state. And then provides some advice of coping
with the negative effects and keeping the economy sustainable.
Generally speaking, this paper shows three things that are thought to benefit the economy
ordinarily:
(1) Rich in resources may “bless” and “curse” one economy at the same time.
(2) Opening is not always a good way to benefit one economy. Sometimes, other sectors may
sacrifice to the open sector. Meanwhile, it will make the economy less stable.
(3) The growth of one economy doesn’t always show the prosperity. Sometimes it shows the loss
of potential development.
This paper also has some shortcomings.
First of all, in reality the capital market is not open. Because the including of this factor will
bring much more variables and more changes, thus more difficulties. So I’m not going to work it
in this paper.
Secondly, the analysis of this model is not thorough enough. Because of the complexity of
the mathematical procedure, this paper mainly focuses on the difference between the two steady
states (before and after the shock), instead of the procedure of moving towards the new steady
state(including speeds of convergence , path of convergence, etc).
Besides, this model omitted the human capital and knowledge capital. Because the inclusion
of these two factors would not change the conclusion of this paper, I didn’t take these into
consideration. But when it comes to the research of the real world, speeds of convergence or path
of convergence, they are essential.
Finally, the development of new resources can be added in this model. When the world price
is higher than a certain level, high price may stimulate the development of substituent resources.
That may influence the investment decision and the steady state.
174
Research of Mathematical Economics No. 1 2011
References
R.M., Solow, “A contribution to the theory of economic growth”, The Quarterly Journal of Economics,
Vol.70, No.1 (Feb., 1956), pp, 65-94.
F.P., Ramsey, “A Mathematical Theory of Saving”, The Economic Journal, Vol. 38, No. 152 (Dec., 1928), pp.
543-559.
P.M., Romer, “Endogenous Technological Change”, The Journal of Political Economy, Vol. 98, No. 5, Part 2:
The Problem of Development: A Conference of the Institute for the Study of Free Enterprise Systems (Oct., 1990),
pp. S71-S102.
R.E., Lucas, Jr. “On the mechanics of economic development”, Journal of Monetary Economics, Volume 22,
Issue 1 (July, 1988), pp. 3-42.
W.M., Corden, “Booming Sector and Dutch Disease Economics: Survey and Consolidation”, Oxford
Economic Papers, New Series, Vol. 36, No. 3 (Nov., 1984), pp. 359-380.
J., Stiglitz, “Growth with Exhaustible Nature Resources: Efficient and Optimal Grows Paths”, The Review of
Economic Studies, Vol. 41, 1974.
(Responsible Editor: Ren Junqiushi) (Proofreader: Zhang Lingyun)
175