Home > Differentiation, Micro concepts > Deriving the demand elasticity for a competitive firm

## Deriving the demand elasticity for a competitive firm

This is a neat little exercise in using differentiation in economics and getting used to the notation involved, and using tricks of algebraic manipulation such as multiplying top and bottom of a term in an equation by the same value.

The goal here is to find an expression for the elasticity of demand for a firm in a competitive market, in terms of the market elasticity of demand and supply.

The basic concept here is one of residual demand curves. When there are a number of firms in the market, you will have a market demand curve which tells you how much of the good consumers will produce at a given price, and each firm will face a residual demand curve, which is the demand that is left over for them that is not met by other sellers. We are going to assume here that all the firms are identical, face identical marginal costs and produce the same output.

You can write it like this $D^r(p) = D(p) - S^o (p)$ (1)

This notation basically says the residual demand function equals the market demand function minus the supply function of all the other firms. These demand and supply firms are really in terms of output, but if you used the notation Q(p) for both you would get mixed up later on in the algebra. The reason p is in brackets is that all of these functions are written in terms of p.

Small letters, p and q, get used for a firm’s price and quantity, while capital letters P and Q will refer to market price and quantity. Of course as we are talking about a competitive market here, p and P will be the same, as all firms are ‘price takers’, they face the same price. $q=\frac{Q}{n}$ if there are n identical firms all producing the same amount of output. The output produced by others is $Q_o=(n-1)q$ which is basically saying there are (n-1) ‘other’ firms (aside from the firm we are looking at), each producing q output.

If we want to express this in terms of elasticities, it’s useful to think of some definitions before we start.

The market elasticity of demand is $\epsilon = \frac{p}{Q} \frac{dQ}{dp}$. Now in equation (1) above the D(p) is really Q(p) as the market demand function is in terms of Q, the notation D and S just gets used so you don’t get mixed up with which Q is for what later one. So with respect to our equation, $\epsilon = \frac{p}{Q} \frac{dD}{dp}$.

The elasticity of demand which a firm faces is $\epsilon_{firm} = \frac{p}{q}\frac{dD^r}{dp}$. This time we are using q instead of Q because it refers to the firm’s output not the market output, and we are using the differential of the residual demand curve rather than the market demand curve.

The elasticity of supply from other firms is $\eta_{o} = \frac{p}{Q_o}\frac{dS^o}{dp}$. This time you are thinking in terms of the quantity supplied by other firms, and the differential of the supply function of other firms.

Now given these definitions we can work towards them starting by differentiating equation (1) with respect to p.

$\frac{dD^r}{dp} = \frac{dD}{dp} - \frac{dS^o}{dp}$
Again this is the type of differentiation notation you will get used to in economics. You aren’t actually given a function to differentiate here, you are just expressing what the differential is!

Now for some tricks to manipulate this equation. First multiply throughout by $\frac{p}{q}$

$\frac{p}{q}\frac{dD^r}{dp} = \frac{p}{q}\frac{dD}{dp} - \frac{p}{q}\frac{dS^o}{dp}$
This has allowed us to write the left hand side in terms of the firm’s demand elasticity, from the definition above

$\epsilon_{firm} = \frac{p}{q}\frac{dD}{dp} - \frac{p}{q}\frac{dS^o}{dp}$

Now you might be able to see where we are going with this. That first term on the right hand side is nearly the market elasticity of demand, except on the denominator we have q instead of Q. So multiply top and bottom of that term, by Q (which is just multiplying by 1 as Q divided by Q is 1, so it doesn’t change it).

$\epsilon_{firm} = \frac{p}{Q}\frac{dD}{dp}\frac{Q}{q} - \frac{p}{q}\frac{dS^o}{dp}$

So now it is in terms of the market elasticity of demand.

$\epsilon_{firm} = \epsilon \frac{Q}{q} - \frac{p}{q}\frac{dS^o}{dp}$.

That second term on the right hand side is nearly the elasticity of supply from other firms, but again we have the wrong denominator, we need to get $Q_o$ in there, so lets multiply top and bottom by $Q_o$

$\epsilon_{firm} = \epsilon \frac{Q}{q} - \frac{p}{Q_o}\frac{dS^o}{dp}\frac{Q_o}{q}$.

Now we have the elasticity of supply from other firms in there.

$\epsilon_{firm} = \epsilon \frac{Q}{q} - \eta_{o}\frac{Q_o}{q}$.

Remember above we said that $q=\frac{Q}{n}$ and $Q_o=(n-1)q$. This means our equation becomes

$\epsilon_{firm} = \epsilon \frac{Q}{\frac{Q}{n}} - \eta_{o}\frac{(n-1)q}{q}$

So this cancels down to

$\epsilon_{firm} = n\epsilon - \eta_{o}(n-1)$

That’s how you express the elasticity of demand for a competitive firm in terms of the market elasticity of demand, the elasticity of supply of other firms, and the number of firms in the market.

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