Chapter 4 A Model of Production 4.1 Introduction In this chapter, we learn • how to set up and solve a macroeconomic model. • the purpose of a production function and its use for understanding differences in GDP per capita across countries. • the role of capital per person and technology in explaining differences in economic growth. • the relevance of “returns to scale” and “diminishing marginal products.” • how to look at economic data through the lens of a macroeconomic model. Introduction A model • is a mathematical representation of a hypothetical world that we use to study economic phenomena. • consists of equations and unknowns with real-world interpretations. Macroeconomists • document facts. • build a model to understand the facts. • examine the model to see how effective it is. 4.2 A Model of Production Vast oversimplifications of the real world in a model can still allow it to provide important insights. Consider the following model: • Single, closed economy • One consumption good Inputs in the production process: • Labor • Capital Production function: • Shows how much output (Y) can be produced given any number of inputs Setting Up the Model – Production Function Model Output growth corresponds to changes in ππ. There are three ways that ππ can change: • Capital stock (K) changes. • Labor force (L) changes. • Ability to produce goods with given resources (K, L) changes. • Technological advances occur (changes in A). • TFP is assumed to be exogenous in the Solow model. Cobb-Douglas Production Function The Cobb-Douglas production function is a particular production function that takes the form of where α is assumed to be 1/3. F(K,L) is increasing in both K and L. • More inputs yield more output. • A production function exhibits constant returns to scale if doubling each input exactly doubles output. Constant Returns to Scale (CRS) If πΎπΎ and πΏπΏ increase by x%, • ο¨ ππ also increases by x% Mathematically, • • Homogeneous function (of degree 1) Standard replication argument Output per Person (Intensive Form) Divide output by the number of workers • Per capita = per person = per worker • Lowercase letters denote per capita. We can rewrite output per person as where and Typical Production Function Graph of Note: If ππ = 0 then π¦π¦ = ππ(ππ) = 0. 10 8 y 6 4 k 2 0 0 200 400 600 800 1000 k Returns to Scale Comparison • Sum of exponents • Sum to 1 • Sum to > 1 • Sum to < 1 • Result • Constant returns to scale • Increasing returns to scale • Decreasing returns to scale Allocating Resources • π: profits • r: rental rate of capital • w: wage rate The rental rate and wage rate are taken as given under perfect competition. • Hire capital until MPK = r. • Hire labor until MPL = w. For simplicity, the price of the output is normalized to one. Marginal Products The marginal product of labor (MPL) is the additional output that is produced when one unit of labor is added, holding all other inputs constant: The marginal product of capital (MPK) is the additional output that is produced when one unit of capital is added, holding all other inputs constant: Diminishing Marginal Product of Capital in Production Diminishing Returns Formally: Suppose we have one unit of K and one unit of L. Assume this results in one unit of Y. • Add a second unit of L. • Previously, each unit of L had one unit of K to work with. • Now, each unit of L has ½ unit of K to work with. • We are assuming that this will make each unit of L less productive, and output will increase, but not double. Solving the Model: General Equilibrium—1 • Five endogenous variables • Output (Y) • The amount of capital (K) • The amount of labor (L) • The wage (w) • The rental price of capital (r) • Five equations • The production function • The rule for hiring capital • The rule for hiring labor • Supply equals the demand for labor. • Supply equals the demand for capital. Solving the Model: General Equilibrium—2 Solution—1 A solution to the model • A new set of equations that express the five unknowns in terms of the parameters and exogenous variables General equilibrium • Solution to the model when more than a single market clears Supply and Demand in the Capital and Labor Markets Solution—2 In This Model… The solution implies that • firms employ all the supplied capital and labor in the economy. • the production function is evaluated with the given supply of inputs: • π€π€ = πππππΏπΏ evaluated at the equilibrium values of Y, K, and L. • ππ = πππππΎπΎevaluated at the equilibrium values of Y, K, and L. Interpreting the Solution The equilibrium wage is proportional to output per worker. • Output per worker = (Y/L) The equilibrium rental rate is proportional to output per capital. • Output per capital = (Y/K) In the United States, empirical evidence shows: • of production is paid to labor • of production is paid to capital • The factor shares of the payments are equal to the exponents on the inputs in the Cobb-Douglas function. Labor’s Share of Income Equilibrium All income is paid to capital or labor. • Results in zero profit in the economy • Verifies the assumption of perfect competition • Also verifies that production equals spending equals income • Income = production 4.3 Analyzing the Production Model Development accounting: • The use of a model to explain differences in incomes across countries • Setting the productivity parameter = 1 Comparing Models with Data Our example uses only one good, but countries produce far more than one good. • How can our model help to understand per capita GDP across widely different countries? Models are simplified versions of reality. The best models are those that tend to be very close to true, despite the simplicity. • If the gap between the model and the world is too large, we will see this failure when we use data. The Empirical Fit of the Production Function If the productivity parameter is 1, the model overpredicts GDP per capita. Diminishing returns to capital implies that • countries with low K will have a high MPK. • countries with a lot of K will have a low MPK, and cannot raise GDP per capita by much through more capital accumulation. The Model’s Prediction for Per Capita GDP (United States = 1) Predicted Per Capita GDP in the Production Model The Model’s Prediction for Per Capita GDP Case Study: Why Doesn't Capital Flow from Rich to Poor Countries? If MPK is higher in poor countries with low K, why doesn't capital flow to those countries? • Short Answer: A simple production model with no difference in productivity across countries is misguided. • We must also consider the productivity parameter. • Various elements of risk that impact TFP may also be embedded • Caselli subsequently questions why MPK is poor countries is not much higher given that they have so little. Productivity Differences: Improving the Fit of the Model The productivity parameter measures how efficiently countries are using their factor inputs. • Total factor productivity (TFP) • If TFP is ≠ to 1 ο¨ better model Total Factor Productivity Data on TFP is not collected. • It can be calculated because we have data on output and capital per person. • See national growth accounting literature. • TFP is referred to as the “residual.” A lower level of TFP implies that workers produce less output for any given level of capital per person. Measuring TFP so the Model Fits Exactly—1 United States and Chinese Production Functions Measuring TFP so the Model Fits Exactly—2 4.4 Understanding TFP Differences Output differences between the richest and poorest countries? • Differences in capital per person explain about onethird of the difference. • TFP explains the remaining two-thirds. Thus, rich countries are rich because • they have more capital per person. • more importantly, they use labor and capital more efficiently. Why are some countries more efficient at using capital and labor? Understanding TFP Differences • • • • • Institutions, Institutions, Institutions Human capital Technology Misallocation Management Institutions Even if human capital and technologies are better in rich countries, why do they have these advantages? Institutions are in place to foster human capital and technological growth. • Private property rights • The rule of law • Contract enforcement • Freedom of the press • Government systems Institutions What are institutions exactly? Douglass North offers the following definition: “Institutions are the rules of the game in a society or, more formally, are the humanly devised constraints that shape human interaction. In consequence they structure incentives in human exchange, whether political, social, or economic”. Of primary importance to economic outcomes are the economic institutions in society such as the structure of property rights and the presence and perfection of markets. Without property rights, individuals may not have the incentive to invest in physical or human capital or adopt more efficient technologies. Economic institutions also help to allocate resources to their most efficient uses. When markets are missing or ignored, gains from trade go unexploited and resources are misallocated. Societies with economic institutions that facilitate and encourage factor accumulation, innovation and the efficient allocation of resources will prosper. Global Competitiveness Index Human Capital Human capital • Stock of skills that individuals accumulate to make them more productive • Education and training Returns to education • Value of the increase in wages from additional schooling Accounting for human capital reduces the residual from a factor of 11 to a factor of 6. Technology Richer countries may use more modern and efficient technologies than poor countries. • Increases productivity parameter Misallocation Misallocation • Resources not being put to their best use Examples • Inefficiency of state-run resources • Political interference 4.5 Evaluating the Production Model Per capita GDP is higher if capital per person is higher and if factors are used more efficiently. Constant returns to scale imply that output per person can be written as a function of capital per person. Capital per person is subject to strong diminishing returns because the exponent is much less than one. Weaknesses of the Model In the absence of TFP, the production model incorrectly predicts differences in income. The model does not provide an answer as to why countries have different TFP levels. Clicker Question 1 a. b. c. d. constant returns to scale decreasing returns to scale diminishing returns to scale increasing returns to scale Clicker Question 1 – Answer a. b. c. d. constant returns to scale decreasing returns to scale diminishing returns to scale increasing returns to scale Clicker Question 2 A firm uses capital and labor to produce a good. Which of the following is greatest? a. accounting profit b. economic profit c. It depends on the situation. d. All of these choices are equal. Clicker Question 2 – Answer A firm uses capital and labor to produce a good. Which of the following is greatest? a. accounting profit b. economic profit c. It depends on the situation. d. All of these choices are equal. Clicker Question 3 In the production model from the text, which of the following is NOT an exogenous variable or a parameter? a. the supply of capital b. the supply of labor c. the amount of capital d. the productivity of parameter Clicker Question 3 – Answer In the production model from the text, which of the following is NOT an exogenous variable or a parameter? a. the supply of capital b. the supply of labor c. the amount of capital d. the productivity of parameter Clicker Question 4 Output per person is higher when a. a country is more efficient in adopting a technology. b. a country has a higher capital-to-population ratio. c. a country has stronger property rights and contract enforcement. d. All of these choices are correct. Clicker Question 4 – Answer Output per person is higher when a. a country is more efficient in adopting a technology. b. a country has a higher capital-to-population ratio. c. a country has stronger property rights and contract enforcement. d. All of these choices are correct. Clicker Question 5 Output per capita and output per worker are a. always equal. b. equal in the production model, but output per capita is smaller in general. c. equal in general, but output per capita is smaller in the production model. d. not equal, because output per capita is always greater. Clicker Question 5 – Answer Output per capita and output per worker are a. always equal. b. equal in the production model, but output per capita is smaller in general. c. equal in general, but output per capita is smaller in the production model. d. not equal, because output per capita is always greater. Clicker Question 6 Which of the following is included in TFP? a. the amount of labor b. the amount of capital c. the quality of labor d. All of these choices are correct. Clicker Question 6 – Answer Which of the following is included in TFP? a. the amount of labor b. the amount of capital c. the quality of labor d. All of these choices are correct. Clicker Question 7 If the productivity parameter is assumed to equal 1, the production model a. correctly identifies that countries are richer if they have more capital. b. incorrectly predicts that poor countries are substantially richer than they are. c. incorrectly predicts that some countries are richer than the United States. d. All of these choices are correct. Clicker Question 7 – Answer If the productivity parameter is assumed to equal 1, the production model a. correctly identifies that countries are richer if they have more capital. b. incorrectly predicts that poor countries are substantially richer than they are. c. incorrectly predicts that some countries are richer than the United States. d. All of these choices are correct. Clicker Question 8 Suppose India has a per capita GDP that is 0.074 times the United States GDP. It has a capital-per-person ratio that is 0.035 times that of the United States. Compared to the United States, the implied value of total factor productivity for India is approximately a. 0.07. b. 0.12. c. 0.23. d. 0.30. Clicker Question 8 – Answer Suppose India has a per capita GDP that is 0.074 times the United States GDP. It has a capital-per-person ratio that is 0.035 times that of the United States. Compared to the United States, the implied value of total factor productivity for India is approximately a. 0.07. b. 0.12. c. 0.23. d. 0.30. Clicker Question 9 a. true b. false Clicker Question 9 – Answer a. true b. false Clicker Question 10 Capital per person explains about one-half of the difference in per capita income between the richest and poorest countries. a. true b. false Clicker Question 10 – Answer Capital per person explains about one-half of the difference in per capita income between the richest and poorest countries. a. true b. false Clicker Question 11 In a Cobb-Douglas production function, the factor share of income going to each input is equal to the exponent on the input in the production function. a. true b. false Clicker Question 11 – Answer In a Cobb-Douglas production function, the factor share of income going to each input is equal to the exponent on the input in the production function. a. true b. false Clicker Question 12 The standard replication argument implies that Italy can raise its per capita GDP by doubling the amount of capital per person. a. true b. false Clicker Question 12 – Answer The standard replication argument implies that Italy can raise its per capita GDP by doubling the amount of capital per person. a. true b. false Clicker Question 13 If the marginal product of capital is less than the rental rate of capital, the firm should rent more capital. a. true b. false Clicker Question 13 – Answer If the marginal product of capital is less than the rental rate of capital, the firm should rent more capital. a. true b. false Clicker Question 14 a. b. c. d. 10 24 48 1,600 Clicker Question 14 – Answer a. b. c. d. 10 24 48 1,600 Clicker Question 15 You plot the production function for the United States on a graph with output per person on the vertical axis and capital per person on the horizontal axis. If a shock occurs causing the productivity parameter to increase, the production function would shift upward. a. true b. false Clicker Question 15 – Answer You plot the production function for the United States on a graph with output per person on the vertical axis and capital per person on the horizontal axis. If a shock occurs causing the productivity parameter to increase, the production function would shift upward. a. true b. false Clicker Question 16 What is the marginal product of capital (MPK) for the 1 1 Μ 2 πΏπΏ2 production function ππ = π΄π΄πΎπΎ a. b. c. d. 1 Μ 2 1 π΄π΄πΎπΎπΏπΏ 2 πΎπΎ 1 Μ 2 1 π΄π΄πΎπΎπΏπΏ 2 πΏπΏ 1 1 Μ 2 πΏπΏ2 1 π΄π΄πΎπΎ 3 πΏπΏ 1 1 Μ 2 πΏπΏ2 1 π΄π΄πΎπΎ 2 πΏπΏ Clicker Question 16 – Answer What is the marginal product of capital (MPK) for the 1 1 Μ 2 πΏπΏ2 production function ππ = π΄π΄πΎπΎ a. b. c. d. 1 Μ 2 1 π΄π΄πΎπΎπΏπΏ 2 πΎπΎ 1 Μ 2 1 π΄π΄πΎπΎπΏπΏ 2 πΏπΏ 1 1 Μ 2 πΏπΏ2 1 π΄π΄πΎπΎ 3 πΏπΏ ππ ππ οΏ½ π²π²ππ π³π³ππ ππ π¨π¨ ππ π³π³ Clicker Question 17 a. b. c. d. 5 25 125 625 Clicker Question 17 – Answer a. b. c. d. 5 25 125 625 Clicker Question 18 According to Figure 4.5 (on the next slide), does the production model accurately predict the level of per capita GDP for Singapore? a. Yes, because the predicted value of per capita GDP for Singapore is close to the U.S. level (which is equal to 1). b. Yes, because the predicted value of per capita GDP for Singapore is close to the actual value of its per capita GDP. c. No, because the predicted value of per capita GDP for Singapore is different than the U.S. level (which is equal to 1). d. No, because the predicted value of per capita GDP for Singapore is different than the actual value of its per capita GDP. Clicker Question 18 (Figure 4.5) Clicker Question 18 – Answer According to Figure 4.5, does the production model accurately predict the level of per capita GDP for Singapore? a. Yes, because the predicted value of per capita GDP for Singapore is close to the U.S. level (which is equal to 1). b. Yes, because the predicted value of per capita GDP for Singapore is close to the actual value of its per capita GDP. c. No, because the predicted value of per capita GDP for Singapore is different than the U.S. level (which is equal to 1). d. No, because the predicted value of per capita GDP for Singapore is different than the actual value of its per capita GDP. Clicker Question 19 a. It cannot be determined because we don’t know the wage rate and the rental rate of capital. b. 152 tons of cream cheese c. 40 tons of cream cheese d. $40 Clicker Question 19 – Answer a. It cannot be determined because we don’t know the wage rate and the rental rate of capital. b. 152 tons of cream cheese c. 40 tons of cream cheese d. $40 Clicker Question 20 If Country A has the Cobb-Douglas production 1 2 Μ 3 πΏπΏ3 , function: ππ = π΄π΄πΎπΎ and Country B had the Cobb-Douglas production 2 1 Μ 3 πΏπΏ3 , function: ππ = π΄π΄πΎπΎ which country pays a greater production share to capital? a. neither Country A nor B b. Country A c. both Country A and B d. Country B Clicker Question 20 – Answer If Country A has the Cobb-Douglas production 1 2 Μ 3 πΏπΏ3 , function: ππ = π΄π΄πΎπΎ and Country B had the Cobb-Douglas production 2 1 Μ 3 πΏπΏ3 , function: ππ = π΄π΄πΎπΎ which country pays a greater production share to capital? a. neither Country A nor B b. Country A c. both Country A and B d. Country B Clicker Question 21 Country A and Country B are allies. Country A shares its production function with Country B. The production function for both countries is: 1 2 π΄π΄πΎπΎ 3 πΏπΏ3 ππ = Both countries have πΏπΏ = 100 and πΎπΎ = 100, but Country A has ππ = 400 and Country B has ππ = 300. Given the information in this chapter, why is this happening? a. differences in human capital b. differences in technology c. all of the above d. none of the above Clicker Question 21 – Answer Country A and Country B are allies. Country A shares its production function with Country B. The production function for both countries is: 1 2 π΄π΄πΎπΎ 3 πΏπΏ3 ππ = Both countries have πΏπΏ = 100 and πΎπΎ = 100, but Country A has ππ = 400 and Country B has ππ = 300. Given the information in this chapter, why is this happening? a. differences in human capital b. differences in technology c. all of the above d. none of the above