# 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.

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

### 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.

$$MRTS=$$

## 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 $$r$$.

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

## Specific production functions

### 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.

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