Production and intermediate goods

Production functions

Production functions and marginal products

Production functions

A firm produces \(Q\) using inputs \(X\).

\(Q=f(X_1,...,X_n)\)

Marginal products

This is the marginal utility, adapted for the production setting.

\(MP=\dfrac{\delta }{\delta x_1}f(\mathbf x)\)

Diminishing marginal returns

This says that marginal returns decrease as the use of a factor increases.

\(\dfrac{\delta^2 }{{\delta x_1}^2}f(\mathbf x)<0\)

Marginal and average costs

Average total cost

Long-run average incremental cost

Isoquants

Isoquants are indifference curves for firms.

We have a production function: \(Q=f(X)\).

An isoquant is defined for each \(c\) \(f(X)=c\), where \(X\) is a vector.

Marginal rate of technical substitution

This is the marginal rate of substitution, adapted for firms.

\(MRTS=\)

Marginal and average costs

Average total cost

Long-run average incremental cost

Choosing production inputs

Isoquants

Isoquants are indifference curves for firms.

We have a production function: \(Q=f(X)\).

An isoquant is defined for each \(c\) \(f(X)=c\), where \(X\) is a vector.

Marginal rate of technical substitution

This is the marginal rate of substitution, adapted for firms.

\(MRTS=\)

Specific production functions

Cobb-Douglas production function

\(Q=A\sum_i X_i^{\alpha_i} \)

Leontief production function

\(Q=\sum_i X_i^{\alpha_i} \)

Constant Elasticity of Substitution (CES) production function

For some constant \(r\).

\(Q=A[\alpha_i X_i^r]^{\dfrac{1}{r}}\)

Input-output tables

Input-output tables

Specific production functions

Marginal rate of transformation

Production-Possibility Frontier

Other

Production functions

A firm produces \(Q\) using inputs \(X\).

\(Q=f(X_1,...,X_n)\)

Marginal products

This is the marginal utility, adapted for the production setting.

\(MP=\dfrac{\delta }{\delta x_1}f(\mathbf x)\)

Diminishing marginal returns

This says that marginal returns decrease as the use of a factor increases.

\(\dfrac{\delta^2 }{{\delta x_1}^2}f(\mathbf x)<0\)

Passthrough

Monopsony and monopoly power