CONCORDIA UNIVERSITY
Department of Economics
ECON 437 SECTION A/ ECON 537 SECTION A
ECONOMICS OF PUBLIC EXPENDITURE
Professor Georgi Boichev
Summer 2020 – ASSIGNMENT 1
Due: Thursday, July 9th at 16:00 in my Moodle mailbox
TOTAL POINTS: 75
TEAM SCORE: ___________________________________________________
SUBMISSION INSTRUCTIONS: Submit the responses of your team, of either 2 or 3 members, in a single
PDF file. Late submissions, submissions in files other than a PDF, individual submissions and/or
submissions of teams of more than 3 members will be disqualified and receive a score of 0 pts.
The submission of identical responses across teams is a form of academic dishonesty.
Question 1 (34 pts.)
There are two individuals, A and B, with quasi-linear preferences over a private good, 𝑥𝑖 , and over a
public good, 𝐺, represented by the following utility function, 𝑈(𝑥𝑖 , 𝐺) = 𝑥𝑖 + ln 𝐺. Each of them has
income, 𝑚𝑖 , which can be either spent on the private good, 𝑥𝑖 , or contributed to the public good, 𝐺. 1
unit of individual contributions, 𝑔𝑖 , from either individual is used to produce 1 unit of the public good, 𝐺.
The social planner has a Rawlsian social welfare function given by 𝑊(𝑈𝐴 , 𝑈𝐵 ) = min{5𝑈𝐴 , 5𝑈𝐵 }.
a) (3 pts.) Derive the Lagrangean corresponding to the optimization problem faced by individual B.
The objective function of the Individual B’s optimization problem is max 𝑈(𝐺, 𝑥𝐵 ) = 𝑥𝑖 + ln 𝐺 subject
𝑔𝐵 ,𝑥𝐵
to her budget constraint 𝑥𝐵 + 𝑔𝐵 ≤ 𝑚𝐵 and the production function of the public good 𝐺 = (𝑔𝐴 + 𝑔𝐵 ).
Substituting the production function directly into the objective function and keeping the budget
constraint as a constraint yields the Lagrangean:
max 𝐿 = 𝑥𝐵 + ln(𝑔𝐴 + 𝑔𝐵 ) − 𝜆[𝑥𝐵 + 𝑔𝐵 − 𝑚𝐵 ]
𝑔𝐵 ,𝑥𝐵 ,𝜆
b) (3 pts.) Calculate the first-order conditions.
𝜕𝐿
1
=
−𝜆 =0
𝜕𝑔𝐵 𝑔𝐴 + 𝑔𝐵
𝜕𝐿
= 1−𝜆 = 0
𝜕𝑥𝐵
𝜕𝐿
= 𝑚𝐵 − 𝑥𝐵 − 𝑔𝐵 = 0
𝜕𝜆
1
c) (2 pts.) Derive the best response function of individual B, 𝑔𝐵 .
From
𝜕𝐿
𝜕𝑥𝐵
𝜕𝐿
1
+𝑔
𝐴
𝐵
= 1 − 𝜆 = 0, 𝜆∗ = 1. Substituting 𝜆∗ = 1 into 𝜕𝑔 = 𝑔
𝐵
− 𝜆 = 0 yields 𝑔𝐴 + 𝑔𝐵 = 1.
Individual B’s best response function is 𝑔𝐵 = 1 − 𝑔𝐴 .
d) (4 pts.) Calculate the demand functions for 𝑥𝑖∗ and 𝐺 ∗ .
To solve for 𝐺 ∗, first we need to determine the individual contributions, 𝑔𝐴 and 𝑔𝐵 using the best
response functions 𝑔𝐵 = 1 − 𝑔𝐴 and 𝑔𝐴 = 1 − 𝑔𝐵 , where we exploit the symmetry in the optimization
problem they are facing. Solving this system of two equations with two unknowns yields a continuum of
Nash equilibria (𝑔𝐴∗ , 𝑔𝐵∗ ) such that 𝑔𝐴 + 𝑔𝐵 = 1 and 𝑔𝑖 ∈ [0, 1]. Only one of them is a symmetric Nash
equilibrium, 𝑔𝐴∗ = 𝑔𝐵∗ = 0.5.
The rest of the answer key uses the numerical values corresponding to the symmetric Nash equilibrium.
Substituting the individually optimal contributions into the production function yields 𝐺 ∗ =
(𝑔𝐴∗ + 𝑔𝐵∗ ) = (0.5 + 0.5) = 1
Finally, substituting 𝑔𝐵∗ = 0.5 into the budget constraint yields 𝑥𝐵∗ = 𝑚𝐵 − 0.5.
e) (2 pts.) State the conceptual difference between the best response function, 𝑔𝐵 , and the
function, 𝑔𝐵∗ .
For an arbitrary value of the individual contribution of person A, the best response function, 𝑔𝐵 ,
specifies a best response contribution by person B in the sense that it generates the highest utility for
person B among all affordable values of 𝑔𝐵 . In contrast, the function, 𝑔𝐵∗ , is the Nash equilibrium
contribution to the public good chosen by person B that depends only on numerical and/or parameter
values.
f)
(2 pts.) Briefly explain whether the bundle (𝑔𝐴∗ − 0.5, 𝑔𝐵∗ ) constitutes a Nash equilibrium.
The bundle (𝑔𝐴∗ − 0.5, 𝑔𝐵∗ ) does not constitute a Nash equilibrium since person A has a unilateral
incentive to deviate from 𝑔𝐴 = 𝑔𝐴∗ − 0.5 and choose 𝑔𝐴 = 𝑔𝐴∗ .
g) (2 pts.) Assuming that 𝑚𝐴 = 𝑚𝐵 , calculate for what income level 𝐺 ∗ = 1.
For 𝑚𝑖 < 0.5, there is a corner solution and all income is being spent on the public good, i.e. 𝑥𝑖∗ = 0 and
𝑔𝑖∗ = 𝑚𝑖 . By exploiting the symmetry in preferences and income levels, the production function for the
public good becomes 𝐺 ∗ = 2𝑚𝑖 . 𝐺 ∗ = 1 is supplied whenever 𝑚𝑖 ≥ 0.5 for both individuals.
2
h) (2 pts.) State whether the individual contributions, 𝑔𝑖 , would differ across two high-income
individuals who earn different income levels, 𝑚𝑖 > 2. Justify your answer using your findings
from parts d) and e).
For 𝑚𝑖 < 0.5, there is a corner solution and all income is being spent on the public good, i.e. 𝑥𝑖∗ = 0 and
𝑔𝑖∗ = 𝑚𝑖 . In contrast, for 𝑚𝑖 > 0.5, 𝑔𝑖∗ = 0.5, i.e. individual contributions are independent of income.
Since both individuals earn 𝑚𝑖 > 0.5, the individual contributions are the same even when different
income levels.
i)
(4 pts.) If 𝑚𝐴 = 2 and 𝑚𝐵 = 1, calculate the optimal bundle for each individual and their utility
level.
Since 𝑚𝑖 > 0.5, 𝐺 ∗ = 2. Using the demand functions of the private good for an interior solution, 𝑥𝐵∗ =
𝑚𝐵 − 0.5 = 1 − 0.5 = 0.5 and 𝑥𝐴∗ = 𝑚𝐴 − 0.5 = 2 − 0.5 = 1.5.
The utility levels are 𝑈(𝑥𝐴 , 𝐺) = 𝑥𝐴 + ln 𝐺 = 1.5 + ln 2 = 1.5 + 0.7 = 2.2 and 𝑈(𝑥𝐵 , 𝐺) = 𝑥𝐵 + ln 𝐺 =
1.5 + ln 2 = 0.5 + 0.7 = 1.2
j)
(2 pts.) Write down the social planner’s problem.
The social planner maximizes the social welfare function max 𝑊(𝐺, 𝑥𝐴 , 𝑥𝐵 ) = min{5𝑈𝐴 , 5𝑈𝐵 }
𝐺,𝑥𝐴 ,𝑥𝐵
subject to the resource constraint: 𝑥𝐴 + 𝑥𝐵 + 𝐺 ≤ 𝑚𝐴 + 𝑚𝐵 and the reservation utility of one
individual, e.g. person B. 𝑈(𝑥𝐵 , 𝐺) = 𝑥𝐵 + ln 𝐺 ≥ ̅𝑢̅̅𝐵̅ , where ̅𝑢̅̅𝐵̅ = 1.2.
k) (2 pts.) Show that the social welfare function implies that 𝑥𝐴 = 𝑥𝐵 .
The Rawlsian social welfare function is maximized whenever its two argument are equalized, i.e. 5𝑈𝐴 =
5𝑈𝐵 , which implies that 𝑈𝐴 = 𝑈𝐵 and, in turn, 𝑥𝐴 + ln 𝐺 = 𝑥𝐵 + ln 𝐺. Cancelling out ln 𝐺 from both
sides yields 𝑥𝐴 = 𝑥𝐵 .
l)
(2 pts.) Use the resource constraint and your result from part i) to find an expression of
individual A’s participation constraint where 𝐺 is the only endogenous variable contained in it.
Substituting 𝑥𝐴 = 𝑥𝐵 into the resource constraint yields, 2𝑥𝐴 + 𝐺 = 3, or equivalently, 𝑥𝐴 = 1.5 − 0.5𝐺.
Substituting 𝑥𝐴 = 1.5 − 0.5𝐺 into A’s utility function yields 𝑈 𝐴 (𝐺) = 1.5 − 0.5𝐺 + ln 𝐺.
m) (2 pts.) Briefly explain why the Rawlsian welfare function captures only equity considerations.
The Rawlsian social welfare function captures only equity considerations in the sense that the utility
levels of the two individuals are equalized. For this numerical problem, this implies that the
consumption allocation of the private good will be equalized irrespective of individual income levels.
3
Due to the non-excludability property of a public good, both individuals also consume the same amount
of the public good.
n) (2 pts.) Use your result from part j) to calculate the socially efficient quantity of the public good.
The FOC
𝑑𝑃𝐶
𝑑𝐺
= −0.5 +
1
𝐺
= 0 implies that 𝐺 ∗ = 2.
o) (2 pts.) Verify that the Samuelson condition yields the same 𝐺 ∗ as in part k).
𝐴
𝐵
𝑀𝑅𝑆𝐺,𝑥
+ 𝑀𝑅𝑆𝐺,𝑥
= 𝑀𝑅𝑇𝐺,𝑥
𝐴
𝐵
𝑀𝑈𝐺𝐴
𝑀𝑈𝐺𝐵
+
= 𝑀𝑅𝑇𝐺,𝑥
𝑀𝑈𝑥𝐴𝐴 𝑀𝑈𝑥𝐵𝐵
1
1
1
Substitute 𝑀𝑈𝐺 = 𝐺 , 𝑀𝑈𝑥𝐴 = 1 and 𝑀𝑈𝑥𝐵 = 1 into the Samuelson condition to calculate 𝐺 + 𝐺 = 1,
which yields 𝐺 ∗ = 2.
p) (2 pts.) Briefly explain why the decentralized provision of the public good is (in)efficient for this
economy.
The decentralized problem of the amount of the public good is underprovided relative to the socially
efficient level. This is due to the free-riding problem captures by the discrepancy that each person
consumes 𝐺 but contributes only the fraction 𝑔𝑖 as an input for its production.
Question 2 (41 pts.)
You are tasked to determine which one of two policy options yields better results in reducing corruption
at the village level. Option 1: top-down monitoring by an upper-level government; Option 2: grassroots
participation of citizens in village council accountability meeting. Corruption is measured as a reduction
in the percentage of missing materials in infrastructure projects, e.g. road construction.
As part of an experiment, some villages are randomly assigned to be audited by the federal government
(group A), while the remainder are not (group N).
A public economist uses data on monitoring type 𝐷𝑖 and the amount of missing materials, 𝑌𝑖 , to estimate
the population regression function (PRF) 𝑌𝑖 = 𝛽1 + 𝛽2 𝐷𝑖 + 𝑢𝑖 by using the following sample:
𝑖
𝐷 (observed)
𝑌 (observed)
𝐷 (hypothetical)
𝑌 (hypothetical)
1
1
5
0
3
2
0
6
1
2
4
3
1
7
0
1
4
0
8
1
4
The benchmark group of villages are subject to top-down monitoring.
a) (2 pts.) Write down the optimization problem of the method of least squares.
̂1 , 𝛽
̂2 ) = Σi 𝑢̂𝑖 2
min 𝑆(𝛽
̂1 ,𝛽
̂2
𝛽
b) (3 pts.) Briefly describe the nature of the optimization problem by specifying the dependent,
explanatory and exogenous variables from a statistical standpoint.
To have the regression best fit the data, minimize the sum of squared of squared residuals, Σi 𝑢̂𝑖 2 ,
(dependent variable) by choosing the optimal quantities of the sample estimators of the intercept and
̂1∗, 𝛽
̂2∗ (explanatory variables), as a function of the numerical values of the
the slope coefficients, i.e. 𝛽
exogenous variables available in the data, 𝑋𝑖 and 𝑌𝑖 .
c) (4 pts.) Show that
̂1 ,𝛽
̂2 )
𝑑𝑆(𝛽
̂1
𝑑𝛽
̂1 − 𝛽
̂2 𝐷𝑖 ) = 0 by using the chain rule and the
= −2Σ𝑖 (𝑌𝑖 − 𝛽
summation rules where appropriate. Do NOT skip steps.
̂1 , 𝛽
̂2 ) = min Σi (𝑌𝑖 − 𝛽
̂1 − 𝛽
̂2 𝐷𝑖 )2
𝑆(𝛽
̂1 𝛽
̂2
𝛽
̂1 − 𝛽
̂2 𝐷1 )2 + (𝑌2 − 𝛽
̂1 − 𝛽
̂2 𝐷2 )2 + (𝑌3 − 𝛽
̂1 − 𝛽
̂2 𝐷3 )2 + (𝑌4 − 𝛽
̂1 − 𝛽
̂2 𝐷4 )2
= (𝑌1 − 𝛽
The chain rule:
̂1 ,𝛽
̂2 )
𝜕𝑆(𝛽
̂
𝜕𝛽1
=
̂1 ,𝛽
̂2 ) 𝜕𝑢
𝜕𝑆(𝛽
̂1
̂1
̂1
𝜕𝑢
𝜕𝛽
+
̂1 ,𝛽
̂2 ) 𝜕𝑢
𝜕𝑆(𝛽
̂2
̂1
̂2
𝜕𝑢
𝜕𝛽
+
̂1 ,𝛽
̂2 ) 𝜕𝑢
𝜕𝑆(𝛽
̂3
̂1
̂3
𝜕𝑢
𝜕𝛽
+
̂1 ,𝛽
̂2 ) 𝜕𝑢
𝜕𝑆(𝛽
̂4
̂1 ,
̂4
𝜕𝑢
𝜕𝛽
where 𝑢̂𝑖 = 𝑌𝑖 −
̂1 − 𝛽
̂2 𝐷𝑖 .
𝛽
̂1 , 𝛽
̂2 )
𝜕𝑆(𝛽
̂1 − 𝛽
̂2 𝐷1 )(−1) + 2(𝑌2 − 𝛽
̂1 − 𝛽
̂2 𝐷2 )(−1) + 2(𝑌3 − 𝛽
̂1 − 𝛽
̂2 𝐷3 )(−1) = 0
= 2(𝑌1 − 𝛽
̂1
𝜕𝛽
Collecting terms yields:
̂1 ,𝛽
̂2 )
𝜕𝑆(𝛽
̂
𝜕𝛽1
̂1 − 𝛽
̂2 𝐷𝑖 ) = 0.
= −2Σ𝑖 (𝑌𝑖 − 𝛽
Distributing the summation operator yields:
̂1 ,𝛽
̂2 )
𝜕𝑆(𝛽
̂
𝜕𝛽1
̂1 − 𝛽
̂2 Σ𝑖 𝐷𝑖 ) = 0, where
= −2(Σ𝑖 𝑌𝑖 − 𝑛𝛽
̂1 is an additive constant and 𝛽
̂2 is a multiplicative constant.
𝛽
n
n
n
n
n
d) (2 pts.) Calculate Σ𝑖=1
𝑌𝑖 , Σ𝑖=1
𝑌𝑖2, Σ𝑖=1
𝐷𝑖 , Σ𝑖=1
𝐷𝑖2 , Σ𝑖=1
𝐷𝑖 𝑌𝑖 .
𝑖
1
2
3
4
Σ𝑖
𝑌𝑖
5
6
7
8
26
𝐷𝑖
1
0
1
0
2
𝑌𝑖2
25
36
49
64
174
𝐷𝑖2
1
0
1
0
2
5
𝐷𝑖 𝑌𝑖
5
0
7
0
12
e) (2 pts.) Use your results from part c) as well as
̂1 ,𝛽
̂2 )
𝑑𝑆(𝛽
̂
𝑑𝛽2
̂1 − 𝛽
̂2 𝐷𝑖 )𝐷𝑖 = 0 to
= −2Σ𝑖 (𝑌𝑖 − 𝛽
̂1 and 𝛽
̂2 .
calculate the estimates of 𝛽
FOCs become:
̂1 ,𝛽
̂2 )
𝜕𝑆(𝛽
̂1
𝜕𝛽
̂1 − 26 + 2𝛽
̂2 = 0 (1) and
= 4𝛽
̂1 ,𝛽
̂2 )
𝜕𝑆(𝛽
̂2
𝜕𝛽
̂2 − 12 + 2𝛽
̂1 = 0 (2).
= 2𝛽
Solve the FOCs as a system of two equations with two unknowns.
̂1 − 14 = 0. Solve this resulting equation for 𝛽
̂1 : 𝛽
̂1 = 7.
Subtract equation (2) from (1) to obtain: 2𝛽
̂1 = 7 into equation 2: 2𝛽
̂2 − 26 + 4(7) = 0. Solve this resulting equation for 𝛽
̂2 : 𝛽
̂2 =
Substitute 𝛽
−1.
̂1 is always equal to the conditional
To check your work: For a regression with a single binary regressor, 𝛽
̂1 + 𝛽
̂2 is always equal to the conditional
mean of the dependent variable for the 𝐷𝑖 = 0 group and 𝛽
mean of the dependent variable for the 𝐷𝑖 = 1 group.
f)
(2 pts.) Calculate the true parameter, 𝛽1 .
𝛽1 = 𝐸(𝑌0𝑖 |𝐷𝑖 = 0) =
𝑌01 + 𝑌03 3 + 1
=
=2
2
2
g) (4 pts.) Calculate the causal effect measured by the true parameter 𝛽2 . Use appropriate
mathematical notation before substituting any numerical values. Do NOT skip steps.
𝛽2 = 𝐸(𝑌1𝑖 |𝐷𝑖 = 1) − 𝐸(𝑌0𝑖 |𝐷𝑖 = 1) =
𝑌11 + 𝑌13 𝑌01 + 𝑌03 5 + 7 3 + 1
−
=
−
=6−2=4
2
2
2
2
h) (4 pts.) State and briefly interpret each of the two terms of the causal effect by specifying the
relevant group(s), the state(s) of nature and the outcome of interest. Be pedantic about the
clarity and precision of your response.
𝐸(𝑌1𝑖 |𝐷𝑖 = 1) – expected percentage reduction of missing materials for the villages who (chose to) be
enrolled in the grassroots participation mechanism observed in the actual state of nature.
𝐸(𝑌0𝑖 |𝐷𝑖 = 1) – expected percentage reduction of missing materials for the villages who (chose to) be
enrolled in the grassroots participation mechanism observed in the counterfactual top-down
mechanism.
6
i)
(2 pts.) Calculate the selection bias. Do NOT skip steps.
𝐸(𝑌0𝑖 |𝐷𝑖 = 1) − 𝐸(𝑌0𝑖 |𝐷𝑖 = 0) =
j)
𝑌01 + 𝑌03 𝑌02 + 𝑌04 3 + 1 6 + 8
−
=
−
= 2 − 7 = −5
2
2
2
2
(2 pts.) Briefly explain why Olken (2007) is concerned that the geographic terrain might lead to
making an erroneous conclusion about its effectiveness in reducing corruption.
The villages in audit group tend to be located disproportionately in the mountains, where roads are
more likely to be washed away during heavy rainfall. The observed effect captures both the influence of
monitoring type and all other influences including geographic terrain om missing materials. If the impact
of the geographic is larger than that of monitoring type, the sign of the observed effect that we estimate
would be the opposite of the true impact (the causal effect).
k) (2 pts.) Briefly explain why Olken (2007) conducts an experiment to find the causal effect
instead of relying on your method in part g).
Relying on non-experimental data means that we cannot distinguish between the causal effect and the
selection bias since hypothetical data does not exist.
The experimental data uses randomization, e.g. tossing a coin, to assign villages to a monitoring
mechanism, i.e. stripping them of the ability to choose the monitoring mechanism. Randomization
ensures that in a reasonably large sample all factors other than the monitoring mechanism will have no
impact on the amount of missing materials, e.g. approximately equal share of villages located in the
mountains for each assigned group.
l)
(2 pts.) For the data point (𝑋2 , 𝐷2 ), calculate the deterministic component 𝐸(𝑌2 |𝐷2 ), the error
term 𝑢2 , the predicted value 𝑌̂2 and the residual 𝑢
̂.
2
𝐸(𝑌2 |𝐷2 ) = 𝛽1 + 𝛽2 𝐷2 = 2 + 2(0) = 2
𝑢2 = 𝑌2 − 𝐸(𝑌2 |𝐷2) = 6 − 2 = 4
̂1 + 𝛽
̂2 𝐷2 = 7 + 1(0) = 7
𝑌̂2 = 𝛽
𝑢
̂2 = 𝑌2 − 𝑌̂2 = 6 − 7 = −1
m) (3 pts.) In a single graph:
- Draw the sample regression function, the population regression function and all data points,
i.e. (𝐷𝑖 , 𝑌𝑖 ).
- For the data point (𝐷2 , 𝑌2 ), illustrate the decomposition of 𝑌2 into its deterministic
component 𝐸(𝑌2 |𝐷2 ) and its error term 𝑢2 ;
7
-
For the data point (𝐷2 , 𝑌2 ), illustrate the decomposition of 𝑌2 into its predicted value 𝑌̂2 and
its residual 𝑢
̂.
2
The graph appears on the last page of this file.
̂1 ) =
n) (5 pts.) Suppose that the sample variances of the intercept and slope coefficients are 𝑣𝑎𝑟(𝛽
̂2 ) = 4 respectively. Test the hypothesis at the 90 % level of significance that the
2 and 𝑣𝑎𝑟(𝛽
centralized provision of public goods is at least twice as effective in reducing the amount of
missing materials relative to the corresponding decentralized provision. In your answer,
- (2 pt.) Specify the appropriate hypotheses and statistical test.
- (2 pt.) Demonstrate your calculations, including the appropriate expansion of the statistical
test’s numerator and denominator. Do not skip steps.
- (1 pt.) Provide a conclusion.
To specify the hypothesis, use the fact that 𝛽1 measures the average of 𝑌 corresponding to the topdown mechanism, while 𝛽1 + 𝛽2 measures the average of 𝑌 corresponding to the grassroots
participation mechanism. What we would like to find is that 2𝛽1 > 𝛽1 + 𝛽2 , which is equivalent to
: 𝛽1 > 𝛽2 .
𝐻0 : 𝛽1 ≤ 𝛽2
𝐻𝐴 : 𝛽1 > 𝛽2
𝑡=
̂2 − 𝛽
̂1 − 𝛽2 − 𝛽1
̂2 − 𝛽
̂1
𝛽
𝛽
=
=
̂2 − 𝛽
̂1 − 𝛽2 − 𝛽1 )
̂2 − 𝛽
̂1 )
𝑠𝑒(𝛽
𝑠𝑒(𝛽
̂2
𝛽
̂
̂
√𝑣𝑎𝑟(𝛽1 ) + 𝑣𝑎𝑟(𝛽2 )
𝑁−𝐾
=
−1 − 0
√ 6
4−2
=
−1
= −0.58
1.73
There is insufficient evidence to reject 𝐻0 since 𝑡 = −0.58 < 2.92 = 𝑡𝛼=0.1,𝑁−𝐾=2 .
o) (2 pts.) Briefly explain why there is a discrepancy between your answers in parts f) and g) and
those in part n).
In part n), we use only sample information that turns out to be from a biased sample. In contrast, we
use the true population parameters in parts f) and g).
8