lecture 4

GDP

The most important macro variable, highly correlated with measures of economic development like life expectancy, mortality, literacy, and human welfare.

Production Function

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

Macro Production Functions

Relate aggregate factors of production (capital and labor) to real GDP.

Micro Production Functions

Relate factors of production for individual producers to their outputs.

Gross-Output Production Function

Relates all inputs (factors and materials) to output.

Value-Added Production Function

Relates only factor inputs (capital and labor) to value-added output.

Inada Conditions

Ensure that a production function is well-behaved regarding essential inputs and marginal products.

Cobb-Douglas Production Function

A production function of the form Y=AK^αL^β, where A is total factor productivity and α, β are the elasticities of output.

Elasticity of Substitution

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

Leontief Production Function

A production function that assumes fixed proportions of inputs, represented as Y=min(aK,bL).

CES Production Function

A production function that allows for constant elasticity of substitution, Y=A[γ(A_K K)^{(σ-1)/σ}+(1−γ)(A_L L)^{(σ-1)/σ}]^{σ/(σ-1)}.

Marginal Rate of Substitution (MRS)

The rate at which one input can be substituted for another while keeping output constant.

Euler’s Theorem

States that if a function is homogeneous of degree 1, then output is fully exhausted by payments to factors of production.

Labor Share of Income

The fraction of GDP paid to labor, calculated as WL/PY.

Capital Share of Income

The fraction of GDP paid to capital, calculated as RK/PY.

Profit Share of Income

The fraction of GDP paid as profits, calculated as 1−Λ_L−Λ_K.

Income Inequality

Refers to the disparity in income distributions, often linked to factors like capital ownership and skill level.

What is the most important macro variable, and why?

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

What does a production function represent?

A production function relates inputs (capital \( K \) and labor \( L \)) to output (real GDP \( Y \)). It reflects the technology used to turn inputs into outputs.

What are the two main types of production functions?

Macro production functions*: Relate aggregate factors of production (capital \( K \) and labor \( L \)) to real GDP \( Y \).

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

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

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.

 What are the key properties of a production function

 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.

What is the Cobb-Douglas production function

The Cobb-Douglas production function is \( Y = AK^\alpha L^\beta \), where \( A \) is total factor productivity (TFP), and \( \alpha \) and \( \beta \) are the output elasticities of capital and labor.

What is the elasticity of substitution in the Cobb-Douglas production function

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

What is the Leontief production function

The Leontief production function is \( Y = \min\left(\frac{K}{a}, \frac{L}{b}\right) \), where \( a \) and \( b \) are fixed proportions. It assumes no substitutability between inputs.

What is the CES production function?

The CES (Constant Elasticity of Substitution) production function is \( Y = A \left[ \gamma (A_K K)^{\frac{\sigma-1}{\sigma}} + (1-\gamma)(A_L L)^{\frac{\sigma-1}{\sigma}} \right]^{\frac{\sigma}{\sigma-1}} \), where \( \sigma \) is the elasticity of substitution.

What are the Inada conditions?

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

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

    2. \( \lim_{K \to 0} F_K = +\infty \) and \( \lim_{K \to \infty} F_K = 0 \).

    3. \( \lim_{L \to 0} F_L = +\infty \) and \( \lim_{L \to \infty} F_L = 0 \).

What is the marginal rate of substitution (MRS)?

The MRS is the rate at which one input can be substituted for another while keeping output constant. For a production function \( F(K, L) \), \( MRS_{KL} = \frac{MP_L}{MP_K} \).

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

 Euler’s Theorem states that if a function \( F(K, L) \) is homogeneous of degree 1 (CRS), then \( Y = MP_K \cdot K + MP_L \cdot L \). This implies that output is fully exhausted by payments to factors of production (no profits).

What is the labor share of income

The labor share of income (\( \Lambda_L \)) is the fraction of GDP paid to labor: \( \Lambda_L = \frac{WL}{PY} \).

What is the capital share of income

The capital share of income (\( \Lambda_K \)) is the fraction of GDP paid to capital: \( \Lambda_K = \frac{RK}{PY} \).

What is the profit share of income?

 The profit share of income (\( \Lambda_\Pi \)) is the fraction of GDP paid as profits: \( \Lambda_\Pi = 1 - \Lambda_L - \Lambda_K \).

What is the relationship between income shares and inequality?

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 \( \frac{WL}{N_{\text{workers}}} \neq \frac{RK + \Pi}{N_{\text{capitalists}}} \).

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

Gdp