Pricing homogeneous goods

The economic profit function


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


The law of one price