lecture 4

1. Basic Q&A Cards

• Q: What is the most important macro variable, and why?

  • A: GDP is the most important macro variable because it highly correlates with measures of economic development like life expectancy, mortality, literacy, and human welfare.

  • Tag: #GDP #EconomicDevelopment

• Q: What does a production function represent?

  • A: A production function relates inputs (capital KK and labor LL) to output (real GDP YY). It reflects the technology used to turn inputs into outputs.

  • Tag: #ProductionFunction #InputsOutputs

• Q: What are the two main types of production functions?

  • A:

    1. Macro production functions: Relate aggregate factors of production (capital KK and labor LL) to real GDP YY.

    2. Micro production functions: Relate factors of production for individual producers to their outputs.

  • Tag: #MacroVsMicro #ProductionFunctions

• Q: What is the difference between gross-output and value-added production functions?

  • A:

    • Gross-output production functions: Relate all inputs (factors and materials) to output.

    • Value-added production functions: Relate only factor inputs (capital and labor) to value-added output.

  • Tag: #GrossOutput #ValueAdded

• Q: What are the key properties of a production function?

  • A:

    1. Twice-continuously differentiable: Smooth and differentiable.

    2. Positive marginal products: More inputs increase output.

    3. Diminishing marginal products: Additional inputs increase output by less and less.

    4. Returns to scale: How output scales with inputs (CRS, DRS, IRS).

    5. Inada conditions: Ensure the function is well-behaved.

  • Tag: #ProductionFunctionProperties #MarginalProducts

• Q: What is the Cobb-Douglas production function?

  • A: The Cobb-Douglas production function is Y=AKαLβY=AKαLβ, where AA is total factor productivity (TFP), and αα and ββ are the output elasticities of capital and labor.

  • Tag: #CobbDouglas #ProductionFunction

• Q: What is the elasticity of substitution in the Cobb-Douglas production function?

  • A: The elasticity of substitution (σσ) in the Cobb-Douglas production function is 1, meaning capital and labor are neither perfect substitutes nor perfect complements.

  • Tag: #ElasticityOfSubstitution #CobbDouglas

• Q: What is the Leontief production function?

  • A: The Leontief production function is Y=min⁡(Ka,Lb)Y=min(aK​,bL​), where aa and bb are fixed proportions. It assumes no substitutability between inputs.

  • Tag: #Leontief #ProductionFunction

• Q: What is the CES production function?

  • A: The CES (Constant Elasticity of Substitution) production function is Y=A[γ(AKK)σ−1σ+(1−γ)(ALL)σ−1σ]σσ−1Y=A[γ(AK​K)σσ−1​+(1−γ)(AL​L)σσ−1​]σ−1σ​, where σσ is the elasticity of substitution.

  • Tag: #CES #ProductionFunction

• Q: What are the Inada conditions?

  • A: The Inada conditions ensure that a production function is well-behaved:

    1. F(K,0)=0F(K,0)=0 and F(0,L)=0F(0,L)=0 (essential inputs).

    2. lim⁡K→0FK=+∞limK→0​FK​=+∞ and lim⁡K→∞FK=0limK→∞​FK​=0.

    3. lim⁡L→0FL=+∞limL→0​FL​=+∞ and lim⁡L→∞FL=0limL→∞​FL​=0.

  • Tag: #InadaConditions #ProductionFunction

• Q: What is the marginal rate of substitution (MRS)?

  • A: The MRS is the rate at which one input can be substituted for another while keeping output constant. For a production function F(K,L)F(K,L), MRSKL=MPLMPKMRSKL​=MPK​MPL​​.

  • Tag: #MRS #MarginalRateOfSubstitution

• Q: What is Euler’s Theorem, and how does it relate to income distribution?

  • A: Euler’s Theorem states that if a function F(K,L)F(K,L) is homogeneous of degree 1 (CRS), then Y=MPK⋅K+MPL⋅LY=MPK​⋅K+MPL​⋅L. This implies that output is fully exhausted by payments to factors of production (no profits).

  • Tag: #EulersTheorem #IncomeDistribution

• Q: What is the labor share of income?

  • A: The labor share of income (ΛLΛL​) is the fraction of GDP paid to labor: ΛL=WLPYΛL​=PYWL​.

  • Tag: #LaborShare #IncomeDistribution

• Q: What is the capital share of income?

  • A: The capital share of income (ΛKΛK​) is the fraction of GDP paid to capital: ΛK=RKPYΛK​=PYRK​.

  • Tag: #CapitalShare #IncomeDistribution

• Q: What is the profit share of income?

  • A: The profit share of income (ΛΠΛΠ​) is the fraction of GDP paid as profits: ΛΠ=1−ΛL−ΛKΛΠ​=1−ΛL​−ΛK​.

  • Tag: #ProfitShare #IncomeDistribution

• Q: What is the relationship between income shares and inequality?

  • A: Income shares (labor, capital, profits) are related to income inequality. For example, if capitalists own capital and firms while workers supply labor, inequality arises if WLNworkers≠RK+ΠNcapitalistsNworkers​WL​=Ncapitalists​RK+Π​.

  • Tag: #IncomeShares #Inequality

2. Cloze Deletion Cards

• Q: The most important macro variable is __________, which correlates with measures like life expectancy and literacy.

  • A: GDP.

  • Tag: #GDP #EconomicDevelopment

• Q: A production function relates __________ and __________ to output YY.

  • A: capital KK, labor LL.

  • Tag: #ProductionFunction #InputsOutputs

• Q: The Cobb-Douglas production function is .

  • A: AKαLβAKαLβ.

  • Tag: #CobbDouglas #ProductionFunction

• Q: The elasticity of substitution in the Cobb-Douglas production function is __________.

  • A: 1.

  • Tag: #ElasticityOfSubstitution #CobbDouglas

• Q: The Leontief production function assumes __________ substitutability between inputs.

  • A: no.

  • Tag: #Leontief #ProductionFunction

• Q: The CES production function allows for __________ elasticity of substitution.

  • A: constant.

  • Tag: #CES #ProductionFunction

• Q: The Inada conditions ensure that a production function is __________.

  • A: well-behaved.

  • Tag: #InadaConditions #ProductionFunction

• Q: The labor share of income is calculated as .

  • A: WLPYPYWL​.

  • Tag: #LaborShare #IncomeDistribution

• Q: The capital share of income is calculated as .

  • A: RKPYPYRK​.

  • Tag: #CapitalShare #IncomeDistribution

• Q: The profit share of income is calculated as .

  • A: 1−ΛL−ΛK1−ΛL​−ΛK​.

  • Tag: #ProfitShare #IncomeDistribution

3. Reverse Cards

• Term: GDP

  • Definition: The most important macro variable, representing the total value of goods and services produced in an economy.

  • Tag: #GDP #EconomicDevelopment

• Term: Production Function

  • Definition: A function that relates inputs (capital and labor) to output (real GDP).

  • Tag: #ProductionFunction #InputsOutputs

• Term: Cobb-Douglas Production Function

  • Definition: A production function of the form Y=AKαLβY=AKαLβ, where AA is TFP, and αα and ββ are output elasticities.

  • Tag: #CobbDouglas #ProductionFunction

• Term: Elasticity of Substitution

  • Definition: A measure of how easily one input can be substituted for another in production.

  • Tag: #ElasticityOfSubstitution #CobbDouglas

• Term: Labor Share of Income

  • Definition: The fraction of GDP paid to labor, calculated as WLPYPYWL​.

  • Tag: #LaborShare #IncomeDistribution

• Term: Capital Share of Income

  • Definition: The fraction of GDP paid to capital, calculated as RKPYPYRK​.

  • Tag: #CapitalShare #IncomeDistribution

• Term: Profit Share of Income

  • Definition: The fraction of GDP paid as profits, calculated as 1−ΛL−ΛK1−ΛL​−ΛK​.

  • Tag: #ProfitShare #IncomeDistribution

4. Image Occlusion Cards

• Q: Label the components of the Cobb-Douglas production function: .

  • A: AKαLβAKαLβ.

  • Tag: #CobbDouglas #ProductionFunction

• Q: Label the components of the labor share of income: .

  • A: WLPYPYWL​.

  • Tag: #LaborShare #IncomeDistribution

5. Study Prompts

• Q: What is the relationship between GDP and economic development?

  • A: GDP per capita highly correlates with measures of economic development like life expectancy, literacy, and human welfare.

  • Tag: #GDP #EconomicDevelopment

• Q: How does the Cobb-Douglas production function differ from the Leontief production function?

  • A: The Cobb-Douglas function allows for substitutability between inputs (σ=1σ=1), while the Leontief function assumes no substitutability (σ=0σ=0).

  • Tag: #CobbDouglas #Leontief

• Q: What is the significance of the Inada conditions in production functions?

  • A: The Inada conditions ensure that a production function is well-behaved, with essential inputs and diminishing marginal products.

  • Tag: #InadaConditions #ProductionFunction
    ### ECO 3302 – Intermediate Macroeconomics

    #### Lecture 4: National Income—How It Is Earned

    Instructor: Luis Pérez

    Email: luisperez@smu.edu

    Dates: January 31 & February 3–10, 2025

    ---

    ### Table of Contents

    1. Introduction

    2. Production Functions

    - Properties of production functions

    - Popular production functions

    3. The Decision-Making of Firms

    4. The National Distribution of Income

    - Calculating Income Shares: Data and Models

    5. Inequality

    6. Taking Stock

    ---

    ### 1. Introduction

    - GDP is the most important macroeconomic variable, highly correlated with measures of economic development such as life expectancy, mortality, literacy, and the Human Development Index (HDI).

    - Correlation with Economic Development:

    - Life Expectancy: Positively correlated with GDP per capita.

    - Child Mortality: Negatively correlated with GDP.

    - Human Development Index (HDI): Positively correlated with GDP.

    - Focus of the Lecture: Understanding what determines a nation’s income and who receives it, starting with the circular flow chart of income distribution.

    ---

    ### 2. Production Functions

    - Output in an economy depends on available technologies and quantities of production factors (capital and labor).

    - Production Functions: Relate inputs (capital and labor) to outputs (GDP).

    - Macro vs. Micro Production Functions:

    - Macro: Relates aggregate capital (K) and labor (L) to real GDP (Y).

    - Micro: Relates inputs of individual producers to their outputs.

    - Gross-Output vs. Value-Added Production Functions:

    - Gross-Output: Includes all production inputs (factors and materials).

    - Value-Added: Relates factor inputs to value-added output.

    #### Properties of Production Functions

    1. Twice-Continuously Differentiable: Smooth and differentiable with respect to inputs.

    2. Positive Marginal Products:

    - \( F_K(K, L) > 0 \), \( F_L(K, L) > 0 \)

    - Output increases with more inputs.

    3. Diminishing Marginal Products:

    - \( F_{KK}(K, L) < 0 \), \( F_{LL}(K, L) < 0 \)

    - Additional inputs yield smaller increases in output.

    4. Homogeneity of Degree \( k \):

    - \( F(\lambda K, \lambda L) = \lambda^k F(K, L) \)

    - Constant Returns to Scale (CRS): \( k = 1 \) (doubling inputs doubles output).

    - Decreasing Returns to Scale (DRS): \( k < 1 \).

    - Increasing Returns to Scale (IRS): \( k > 1 \).

    5. Inada Conditions:

    - \( F(K, 0) = 0 \), \( F(0, L) = 0 \) (essential inputs).

    - Marginal products approach infinity as inputs approach zero and zero as inputs approach infinity.

    #### Popular Production Functions

    1. Cobb-Douglas:

    - \( Y = AK^\alpha L^\beta \)

    - Key Properties:

    - Hicks-neutral technological progress.

    - Constant Elasticity of Substitution (CES) between K and L (σ = 1).

    - Returns to scale = \( \alpha + \beta \).

    - Positive and diminishing marginal products.

    - K and L are q-complements.

    2. Leontief (Fixed Proportions):

    - \( Y = \min\left(\frac{K}{a}, \frac{L}{b}\right) \)

    - Key Properties:

    - No substitutability between K and L.

    - Output determined by the limiting input.

    - Constant Returns to Scale (CRS).

    3. Constant Elasticity of Substitution (CES):

    - \( Y = A[\gamma(A_K K)^{\frac{\sigma-1}{\sigma}} + (1-\gamma)(A_L L)^{\frac{\sigma-1}{\sigma}}]^{\frac{\sigma}{\sigma-1}} \)

    - Key Properties:

    - Elasticity of substitution = σ.

    - Returns to scale = ν.

    - K and L may be q-complements (σ ≤ 1) or q-substitutes (σ > 1).

    4. Stone-Geary:

    - \( Y = A(K - \underline{K})^\alpha (L - \underline{L})^\beta \)

    - Key Properties:

    - Minimum input requirements (K and L).

    - Similar to Cobb-Douglas once minimum inputs are met.

    ---

    ### 3. The Decision-Making of Firms

    - Firms make production and pricing decisions to maximize profits or minimize costs.

    - Pricing Rules:

    - Competitive Markets: Firms take prices (P, W, R) as given.

    - Market Power: Firms set prices above marginal cost (monopoly) or below marginal product (monopsony).

    - Profit Maximization:

    - \( \Pi = PY - WL - RK \)

    - First-Order Conditions (FOCs):

    - \( W = P \times MPL \)

    - \( R = P \times MPK \)

    ---

    ### 4. The National Distribution of Income

    - Accounting Identity:

    - \( PY = WL + RK + \Pi \)

    - Dividing by PY:

    - \( \frac{WL}{PY} + \frac{RK}{PY} + \frac{\Pi}{PY} = 1 \)

    - Labor Share: \( \Delta_L = \frac{WL}{PY} \)

    - Capital Share: \( \Delta_K = \frac{RK}{PY} \)

    - Profit Share: \( \Delta_\Pi = \frac{\Pi}{PY} \)

    - Calculating Income Shares:

    - Approach 1: Use National Accounts data to compute labor share and assume competitive economy (ΔΠ = 0).

    - Approach 2: Estimate capital share using imputed rental rate (R).

    - Approach 3: Use micro data and economic theory to estimate income shares.

    ---

    ### 5. Inequality

    - Types of Inequality: Income, consumption, wealth.

    - Income Shares and Inequality:

    - Representative Agent: No inequality if everyone owns factors equally.

    - Capitalists vs. Workers: Inequality if capitalists own capital and firms.

    - High-Skilled vs. Low-Skilled Workers: Inequality due to skill differences.

    - US College Wage Premium:

    - College-educated workers earn more than high-school graduates.

    - CES production function explains wage premium with skill-biased technological change.

    ---

    ### 6. Taking Stock

    - GDP per capita is highly correlated with economic development.

    - Production Functions relate inputs (K, L) to output (Y) and have key properties like marginal products, returns to scale, and homogeneity.

    - Income Shares can be calculated using data and models, with labor share declining in the US.

    - Inequality is influenced by ownership of factors and skill differences.

    ---

    ### Study Questions

    1. What are the key properties of a neoclassical production function?

    2. How does the Cobb-Douglas production function differ from the CES production function?

    3. Explain the concept of diminishing marginal products with an example.

    4. How are income shares calculated in a competitive economy?

    5. What factors contribute to income inequality between high-skilled and low-skilled workers?

    ---

    ### Key Formulas

    1. Cobb-Douglas Production Function:

    - \( Y = AK^\alpha L^\beta \)

    2. CES Production Function:

    - \( Y = A[\gamma(A_K K)^{\frac{\sigma-1}{\sigma}} + (1-\gamma)(A_L L)^{\frac{\sigma-1}{\sigma}}]^{\frac{\sigma}{\sigma-1}} \)

    3. Income Shares:

    - \( \Delta_L = \frac{WL}{PY} \), \( \Delta_K = \frac{RK}{PY} \), \( \Delta_\Pi = \frac{\Pi}{PY} \)

    ---

    ### Visuals

    1. Circular Flow Chart: Illustrates the flow of income between households, firms, and markets.

    2. Cobb-Douglas Production Function Graph: Shows output as a function of capital and labor.

    3. Income Shares Over Time: Graphs showing labor, capital, and profit shares in the US.

    ---

    This document provides a comprehensive overview of Lecture 4, covering all key topics, formulas, and concepts. Let me know if you need further revisions or additional details!