# 4.2 A Model of Production. 4.1 Introduction. A Model of Production. Setting Up the Model. Chapter 4

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1 Chapter 4 A Model of Production By Charles I. Jones Media Slides Created By Dave Brown 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 Penn State University 4.1 Introduction In this chapter, we learn: How to set up and solve a macroeconomic model. How a production function can help us understand differences in per capita GDP across countries. The relative importance of capital per person versus total factor productivity in accounting for these differences. The relevance of returns to scale and diminishing marginal products. How to look at economic data through the lens of a macroeconomic model. Setting Up the Model A certain number of inputs are used in the production of the good Inputs Labor (L) Capital (K) Production function Shows how much output (Y) can be produced given any number of inputs 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. Others variables with a bar are parameters. Production function: Output Productivity parameter Inputs 1

2 A production function This function takes as inputs the number of workers, L, and the number of machines, K And returns as output the amount of ice cream, that is produced by those workers using those machines Standard replication argument A firm can build an identical factory, hire identical workers, double production stocks, and can exactly double production. Implies constant returns to scale. is a positive constant term, which we will call productivity because when it is higher, output is higher even with the same K and L The Cobb-Douglas production function is the particular production function that takes the form of Usually assumed to be 1/3. Explained later. Returns to Scale (RS) In general for any production function. Show what happens to the output if we double the inputs: If the output is exactly double constant RS If the output is less than double decreasing RS If the output is more than double increasing RS Allocating Resources It is a Cobb-Douglas production function, and it is the most commonly used model in all of economics Why? The first key property is constant returns to scale: If you increase all inputs by some proportion, you will increase output by the very same proportion In other words, if K doubles and L doubles, then Y doubles too Firm chooses inputs to maximize profit Rental rate of capital Wage rate The rental rate and wage rate are taken as given under perfect competition. For simplicity, the price of the output is normalized to one. 2

3 The marginal product of labor (MPL) The additional output that is produced when one unit of labor is added, holding all other inputs constant. The marginal product of capital (MPK) The additional output that is produced when one unit of capital is added, holding all other inputs constant. Solving the Model: General Equilibrium The solution is to use the following hiring rules: Hire capital until the MPK = r Hire labor until MPL = w The model has five endogenous variables: Output (Y) the amount of capital (K) the amount of labor (L) the wage (w) the rental price of capital (r) If the production function has constant returns to scale in capital and labor, it will exhibit decreasing returns to scale in capital alone. The model has five equations: The production function The rule for hiring capital The rule for hiring labor Supply equals the demand for capital Supply equals the demand for labor The parameters in the model: The productivity parameter The exogenous supplies of capital and labor 3

4 A solution to the model A new set of equations that express the five unknowns in terms of the parameters and exogenous variables Called an equilibrium General equilibrium Solution to the model when more than a single market clears 4

5 In this model The solution implies firms employ all the supplied capital and labor in the economy. The production function is evaluated with the given supply of inputs. The wage rate is the MPL evaluated at the equilibrium values of Y, K, and L. The rental rate is the MPK evaluated at the equilibrium values of Y, K, and L. All income is paid to capital or labor. Results in zero profit in the economy This verifies the assumption of perfect competition. Also verifies that production equals spending equals income. Interpreting the Solution If an economy is endowed with more machines or people, it will produce more. 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) Case Study: What Is the Stock Market? Economic profit Total payments from total revenues Accounting profit Total revenues minus payments to all inputs other than capital. The stock market value of a firm Total value of its future and current accounting profits The stock market as a whole is the value of the economy s capital stock. In the United States, empirical evidence shows: Two-thirds of production is paid to labor. One-third 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. 5

6 4.3 Analyzing the Production Model Per capita = per person Per worker = per member of the labor force. In this model, the two are equal. We can perform a change of variables to define output per capita (y) and capital per person (k). Part 2 Output per person equals the productivity parameter times capital per person raised to the one-third power. A recap of last class: Our model takes factors capital, K, and labor, L, as inputs and returns output, Y The special functional form that describes output so well, the Cobb-Douglas production function, tells us that Output per person Productivity parameter Capital per person In words: output per person, y, equals a productivity constant, A, times capital per person, k, to the one-third y will be higher if A or k is higher But there are diminishing returns to scale in capital per worker, k Next, we will use this model to understand why some countries are so much richer than others What does the model say about per capita GDP? Let s use lowercases to show variables per capita. Then our model reveals that What makes a country rich or poor? Output per person is higher if the productivity parameter is higher or if the amount of capital per person is higher. Output per worker, y, is a function of productivity, A, and capital per worker, k = K/L (But there are diminishing returns to capital per worker!) Our model says countries are richer if A or k is higher What can you infer about the value of the productivity parameter or the amount of capital in poor countries? 6

7 Comparing Models with Data The model is a simplification of reality, so we must verify whether it models the data correctly. The best models: Are insightful about how the world works Predict accurately The Empirical Fit of the Production Model Development accounting: The use of a model to explain differences in incomes across countries. For now set productivity parameter = 1 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 If the productivity parameter is 1, the model overpredicts GDP per capita. 7

8 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: Simple production model with no difference in productivity across countries is misguided. We must also consider the productivity parameter. Productivity Differences: Improving the Fit of the Model The productivity parameter measures how efficiently countries are using their factor inputs. Often called total factor productivity (TFP) If TFP is no longer equal to 1, we can obtain a better fit of the model. However, data on TFP is not collected. It can be calculated because we have data on output and capital per person. 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 8

9 Output differences between the richest and poorest countries? Differences in capital per person explain about one-third 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. Technology Richer countries may use more modern and efficient technologies than poor countries. Increases productivity parameter 4.4 Understanding TFP Differences Why are some countries more efficient at using capital and labor? 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. Property rights The rule of law Government systems Contract enforcement Human Capital Human capital Stock of skills that individuals accumulate to make them more productive Education, training, etc. 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. Misallocation Misallocation Resources not being put to their best use Examples Inefficiency of state-run resources Political interference 9

10 Case Study: A Big Bang or Gradualism? Economic Reforms in Russia and China When transitioning from a planned to a market economy, the change can be sudden or gradual. A big bang approach is one where all old institutions are replaced quickly by democracy and markets. A gradual approach is one where the transition to a market economy occurs slowly over time. 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. Russia followed a big bang approach, yet GDP per capita has declined since the transition. China has seen accelerated economic growth using the gradual approach. 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. Summary Per capita GDP varies by a factor of 50 between the richest and poorest countries of the world. The key equation in our production model is the Cobb-Douglas production function: Output Productivity parameter Inputs 10

11 The exponents in this production function: One-third of GDP is paid out to capital. Two-thirds is paid to labor. The production model implies that output per person in equilibrium is the product of two key forces: Total factor productivity (TFP) Capital per person Exponents sum to 1, implying constant returns to scale in capital and labor. The complete production model consists of five equations and five unknowns: - Output Y - Capital K - Labor L - Wage rate w - Rental rate r Assuming the TFP is the same across countries, the model predicts that income differences should be substantially smaller than we observe. Capital per person actually varies enormously across countries, but the sharp diminishing returns to capital per person in the production model overwhelm these differences. The solution to this model is called an equilibrium. The prices w and r are determined by the clearing of labor and capital markets. The quantities of K and L are determined by the exogenous factor supplies. Y is determined by the production function. Making the production model fit the data requires large differences in TFP across countries. Economists also refer to TFP as the residual, or a measure of our ignorance. 11

12 Understanding why TFP differs so much across countries is an important question at the frontier of current economic research. Differences in human capital (such as education) are one reason, as are differences in technologies. These differences in turn can be partly explained by a lack of institutions and property rights in poorer countries. This concludes the Lecture Slide Set for Chapter 4 Macroeconomics Second Edition by Charles I. Jones W. W. Norton & Company Independent Publishers Since

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