Chapter 4 - A Model of Production PDF

Chapter 4 - A model of production mercredi 25 septembre 2024 09:29 Chapter 4 I. Introduction This chapter explains how to set up and solve a macroeconomic model, a mathematical framework used to study economic phenomena. Key concepts include the production function, which helps explain differen...

Chapter 4 - A model of production mercredi 25 septembre 2024 09:29 Chapter 4 I. Introduction This chapter explains how to set up and solve a macroeconomic model, a mathematical framework used to study economic phenomena. Key concepts include the production function, which helps explain differences in GDP per capita by analyzing the role of capital per person and technology in economic growth. It also covers returns to scale and diminishing marginal products. Macroeconomists use real-world data to build and test models to better understand and predict economic outcomes. II. A model of production Even with vast oversimplifications, models can still offer valuable insights. For example, consider a model with a single, closed economy producing one consumption good. The production process involves labor and capital, and the production function shows how much output (Y) can be generated from varying inputs. Despite its simplicity, such a model helps analyze fundamental economic relationships. Then the function of production is : Output growth (Y) changes through three main factors :changes in capital stock (K), labor force (L), or the ability to produce with existing resources (K, L), often driven by technological advances (A). In the Solow model, Total Factor Productivity (TFP) is assumed to be exogenous. The Cobb-Douglas production function is a specific production function that takes the form where alpha is assumed to 1/3. Key points: is increasing in both capital (K) and labor (L), meaning more inputs lead to more output. The partial derivatives with respect to K and L are both positive: The function exhibits constant returns to scale, meaning that doubling both inputs (K and L) will exactly double the output (Y). Mathematically, this is expressed as: To analyze output per worker, we divide total output (Y) by the number of workers (L). In this notation, lowercase letters represent per capita values, so output per person is rewritten as. Macroeconomic theory Page 1 Where: (output per worker) (capital per worker). Sum of exponents Result Sum to 1 Constant returns of scale Sum to > 1 Incresing returns to scale Sum to < 1 Deacresing returns of scale Then profit maximization for a firm is represented by the equation: Where: = profits = rental rate of capital = wage rate Under perfect competition: The rental rate and wage rate are taken as given. Firms hire capital until the marginal product of capital (MPK) equals. Firms hire labor until the marginal product of labor (MPL) equals. For simplicity, the price of output is normalized to one, which is referred to as a "numéraire good." The Marginal Product of Labor (MPL) is the additional output produced when one more unit of labor is added, holding all other inputs constant. It is given by: The Marginal Product of Capital (MPK) is the additional output produced when one more unit of capital is added, holding all other inputs constant. It is given by: Macroeconomic theory Page 2 Formally, the second-order partial derivatives of the production function with respect to capital (K) and labor (L) are both negative: This means there are diminishing returns to both capital and labor. Example: Suppose we start with one unit of capital (K) and one unit of labor (L), which results in one unit of output (Y). If a second unit of labor is added: Initially, each unit of labor had one unit of capital to work with. Now, each unit of labor has 1/2 unit of capital to work with. The assumption is that adding more labor without increasing capital makes each unit of labor less productive, so output will increase but not double, reflecting diminishing marginal returns. We have five endogenous variables Output (Y) The amount of capital (K) The amount of labor (L) The wage (w) The rental price of capital (r) With this variables we can construct 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 A solution to the model involves finding a set of equations that express the five unknowns in terms of the model's parameters and exogenous variables. This solution leads to general equilibrium, where all markets in the economy clear simultaneously, not just a single market. General equilibrium ensures that supply equals demand across all factors and goods in the model. 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 Macroeconomic theory Page 3 Supply equals the demand for capital The solution to the model implies the fact that the firms employ all the supplied capital and labor in the economy. Also the production function is evaluated with the given supply of inputs: The wage rate is equal to the marginal product of labor (MPL), evaluated at the equilibrium values of output (Y), capital (K), and labor (L). The rental rate is equal to the marginal product of capital (MPK), evaluated at the equilibrium values of Y, K, and L. The equilibrium wage is proportional to output per worker, and the equilibrium rental rate is proportional to output per unit of capital: Output per worker: Output per unit of capital: In the United States, empirical evidence shows that the two-thirds (2/3) of production is paid to labor and one-third (1/3) of production is paid to capital. The factor shares of payments correspond to the exponents on capital and labor in the Cobb-Douglas production function: and This reflects how output is divided between labor and capital inputs. In this model all income is paid either to capital or labor. This results in zero profit for firms, which verifies the assumption of perfect competition. It also confirms that production equals spending, which equals income. The equation summarizing this is: III. Analyzing the Production Model Development accounting uses a model to explain income differences across countries. The output per worker is expressed as: By setting the productivity parameter to 1, the equation simplifies to: Macroeconomic theory Page 4 This approach helps understand the role of capital and productivity in explaining differences in income levels between countries. If the productivity parameter is set to 1, the model tends to overpredict GDP per capita. Key implications of diminishing returns to capital: Countries with low capital (K) will have a high marginal product of capital (MPK), meaning additional capital can significantly increase output. Countries with high capital (K) will have a low MPK, meaning they cannot significantly raise GDP per capita through more capital accumulation alone. This explains why increasing capital has more impact in less capital-rich economies and why simply adding more capital in wealthier nations does not yield large increases in output. If the marginal product of capital (MPK) is higher in poorer countries with low capital (K), why doesn’t capital flow to those countries? As a short answer, we can say that the simple production model that assumes no differences in productivity across countries is flawed. We must also consider the productivity parameter (A), as differences in productivity levels across countries can influence capital flows. Even if MPK is high, low productivity in some countries can prevent capital from flowing there. The productivity parameter measures how efficiently countries use their factor inputs (capital and labor). Total factor productivity (TFP) is not directly collected but can be calculated using data on output and capital per person. TFP is often referred to as the "residual" because it captures factors affecting output beyond labor and capital inputs. A lower TFP means workers produce less output for a given level of capital per person. If TFP ≠ 1, it results in a better model, as it accounts for productivity differences between countries, improving accuracy in predicting economic outcomes. IV. Understanding TFP differences Output differences between the richest and poorest countries are explained by: Differences in capital per person, which account for about one-third of the variation. Total factor productivity (TFP) explains the remaining two-thirds. Thus, rich countries are wealthier because: They have more capital per person. More importantly, they use labor and capital more efficiently. Reasons some countries are more efficient at using capital and labor include: Human Human capital refers to the skills individuals gain through capital education and training to boost productivity. Returns to education reflect wage increases from additional schooling. By factoring in human capital, the productivity gap between countries is reduced significantly, narrowing the unexplained differences from a factor of 11 to 6. Technology Richer countries use more advanced technologies, raising productivity (TFP) and leading to higher output and more Macroeconomic theory Page 5  productivity (TFP) and leading to higher output and more efficient use of labor and capital. Institutions Strong institutions in rich countries—like property rights, rule of law, government systems, and contract enforcement—enable human capital development and technological progress, fostering growth. Misallocation Misuse of resources lowers productivity, often due to inefficient of resources state management and political interference, which distort resource allocation and hinder economic growth. V. Evaluating the Production Model Per capita GDP is higher when capital per person is higher or when actors are used more efficiently (higher TFP). With constant returns to scale, output per person can be expressed as a function of capital per person. However, capital per person experiences diminishing returns, as its exponent in the production function is much less than one. Without accounting for TFP, the production model incorrectly predicts income differences between countries. The model does not explain why countries have different TFP levels, leaving this variation unexplained. Macroeconomic theory Page 6