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.
\(\pi = pq-c\)
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, so \(\dfrac{\delta \pi }{\delta q} = 0\).
\(0 = p+q\dfrac{\delta p}{\delta q}-\dfrac{\delta c}{\delta q}\)
\(p+q\dfrac{\delta p}{\delta q}=\dfrac{\delta c}{\delta q}\)
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:
\(p[1+\dfrac{1 }{\epsilon }]=MC\)
\(p=\dfrac{\epsilon }{1+\epsilon }MC\)
This can be plugged into the demand function to get the quantity supplied.
\(revenue = pq\)
\(MR=p +q\dfrac{\delta p}{\delta q}\)
\(p\) 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.
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:
\(p[1+\dfrac{q}{Q}\dfrac{\delta Q}{\delta q}\dfrac{Q}{p}\dfrac{\delta p}{\delta Q}]=MC\)
\(p[1+\dfrac{\mu }{\epsilon }]=MC\)
\(p=MC \dfrac{\epsilon }{\epsilon + \mu }\)
Using the firm’s size elasticity: \(\mu = \dfrac{q}{Q}\dfrac{\delta Q}{\delta q}\)
With monopoly \(\mu = 1\).
Scale benefits.
Each player decides what price to sell at.
Firms who price above the lowest have no sales. Prices converge to cost.
In the monopoly model we have:
\(\pi = pq-c\)
\(p[1+\dfrac{q}{p}\dfrac{\delta p}{\delta q}]=MC\)
The price elasticity of demand is: \(\epsilon = \dfrac{p}{q}\dfrac{\delta q}{\delta p}\)
\(p[1+\dfrac{1}{\epsilon }]=MC\)
\(\dfrac{1}{\epsilon }=\dfrac{MC}{p}-1\)
\(\dfrac{p-MC}{p}=-\dfrac{1}{\epsilon }\)
The Lerner index is:
\(\dfrac{p-MC}{p}\)
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:
\(\mu = 1\)
In this model this is:
\(\dfrac{p-MC}{p}=-\dfrac{\mu}{\epsilon }\)
The index is the sum of each firm’s market share squared.
For a monopolist this is \(1\), for a completely competitive market it is \(0\), or \(\dfrac{1}{n}\).
\(H=\sum_{i=1}^ns^2_i\)
If this is just done with the largest firms, the results will be similar to the result for the whole market, due to the quadratic element.
To normalise this between \(0\) and \(1\) we can use:
\(H*=\dfrac{H-\dfrac{1}{n}}{1-\dfrac{1}{n}}\)
Proportion of output from given firms.
For example \(CR_5\) is the proportion of output from the \(5\) largest producers.
Marginal profit. If high it suggests existing power prevents it from raising output.
\(L=\dfrac{P-MC}{P}\)
From \(0\) to \(1\).