# Pricing homogeneous goods

## The economic profit function

### Profit

The profit of a firm is the difference between revenue and costs.

$$\pi = pq-c$$

Where $$q$$ is the amount producted, and $$p$$ is the price, and $$c$$ is a function of production.

### Maximising profit

$$\pi = pq-c$$

The firm’s production $$q$$ affects the market price $$p$$.

$$\dfrac{\delta \pi }{\delta q}= \dfrac{\delta }{\delta q} [pq-c]$$

$$\dfrac{\delta \pi }{\delta q}= p+q\dfrac{\delta p}{\delta q}-\dfrac{\delta c}{\delta q}$$

The firm chooses $$Q$$ to maximise profits.

$$p+q\dfrac{\delta p}{\delta q}=\dfrac{\delta c}{\delta q}$$

The right side is marginal costs (MC), the left is marginal revenue.

$$p[1+\dfrac{q}{p}\dfrac{\delta p}{\delta q}]=MC$$

We know that the price elasticity of demand is: $$\epsilon = \dfrac{p}{q}\dfrac{\delta q}{\delta p}$$

So we have:

$$p[1+\dfrac{1 }{\epsilon }]=MC$$

$$p=\dfrac{\epsilon }{1+\epsilon }MC$$

### Intensive and extensive margins

$$revenue = pq$$

$$MR=p +q\dfrac{\delta p}{\delta q}$$

$$p$$ is the extensive margin.

$$q\dfrac{\delta p}{\delta q}$$ is the (negative) intensive margin.

monopoly pricing. when lower prices, gain money on extensive margin. lose money on intensive margin.

## Cournot competition

### Cournot competition

With competition, the elasticity of demand refers to the whole market, not just a single producer. Instead we have:

$$\epsilon = \dfrac{p}{Q}\dfrac{\delta Q}{\delta p}$$

$$Q=\sum_j q_j$$

We now get:

$$p[1+\dfrac{q}{Q}\dfrac{\delta Q}{\delta q}\dfrac{Q}{p}\dfrac{\delta p}{\delta Q}]=MC$$

$$p[1+\dfrac{\mu }{\epsilon }]=MC$$

Using the firm’s size elasticity: $$\mu = \dfrac{q}{Q}\dfrac{\delta Q}{\delta q}$$

With monopoly this is:

$$\mu = 1$$

## Bertrand competition

### Bertrand competition

Each player decides what price to sell at.

Firms who price above the lowest have no sales. Prices converge to cost.

## Pricing in repeated rounds

### Stackleberg competition

Sequential Cournot competition.