# 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.

## Perfect competition

### Perfect competition

### Hotelling’s lemma

### Short-term supply function

### Long-term supply function

### Price elasticity of supply

## Pricing in repeated rounds

### Stackleberg competition

Sequential Cournot competition.

There is a first-mover advantage.

### Explicit and tacit collusion

### Monitoring and enforcing collusion

## Other

### The law of one price