## 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

### 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 \(r\).

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

## Specific production functions

### 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