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## 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.

## Elasticities

This is something which you will see a lot. The elasticity gives you a measure of proportionate response from one variable to another, it is commonly used for price and quantity. The elasticity of demand tells you how much quantity demanded changes when you change the price. Inelastic demand means you can raise the price and not suffer too much of a drop in output. Elastic demand means if you raise the price, people stop buying it and you suffer a larger proportionate drop in output. Obviously firms prefer to have inelastic demand. Various things affect elasticity of demand, the availability of substitutes is a big one.

Elasticity of demand is the proportional change in quantity demanded divided by the proportional change in price: $\frac{\triangle Q}{Q}\div\frac{\triangle P}{P} = \frac{\triangle Q}{Q}\frac{P}{\triangle P}$ which is generally expressed as $\frac{P}{Q}\frac{\triangle Q}{\triangle P}$. This will generally be a negative value, because when you increase the price, you decrease the quantity, so $\triangle Q$ will be negative.

As these changes tend to zero (ie at the margin) we can express the elasticity as $\epsilon=\frac{P}{Q}\frac{dQ}{dP}$.
If $-1<\epsilon<0$ then we say the demand is inelastic. If $\epsilon=1$ then it is 'unit elastic'. If $-\infty<\epsilon<-1$ then it is elastic.

Another little manoeuvre with elasticity involves using the chain rule again. Remember that a demand function expresses Q as a function of P. If we take the natural log of Q then we can differentiate it with respect to P by the chain rule:$\frac{d}{dP} ln (Q(P)) = \frac{1}{Q}\frac{dQ}{dP}$. You differentiate the outer function first (which is the natural log of Q) and multiply it by the differential of the inner function (which is Q in terms of P). At A-level you tend to get chain rule questions when you have to actually solve equations with values in them, but quite often in economics you get them written in this function notation so you have to get the hang of doing the chain rule that way.

Anyway since we just found that $\frac{d}{dP} ln (Q(P)) = \frac{1}{Q}\frac{dQ}{dP}$ we can multiply both sides by P to get $P\frac{d}{dP} ln (Q(P)) = \frac{P}{Q}\frac{dQ}{dP}$ which is the elasticity of demand again.

So $\epsilon=P\frac{d}{dP} ln Q$. This can be a useful way of calculating elasticities if the demand function is a bit complicated.

One area where this comes up is in the case of constant elasticity. Usually the elasticity will vary along different points of the demand curve (even if $\frac{dQ}{dP}$ is constant, at every point there will be a different combination of P and Q so the $\frac{P}{Q}$ part will vary. But there is a specific type of function that will give a constant elasticity all the way along the curve, it will look like this: $Q=aP^{-b}$

Now if we log both sides we will get $ln Q=ln a - b lnP$ (this is using rules of logs).
So our formula above of $\epsilon=P\frac{d}{dP} ln Q$ says that we need to differentiate that expression with respect to P, and multiply the whole thing by P. $\frac{d}{dP}(ln a - b ln P)=\frac{-b}{P}$ so $\epsilon=P\frac{-b}{P}=-b$. So the elasticity will always be the constant -b. That's why constant elasticity functions are always of that form.

## Differentiation in Economics

Most undergrad level core micro and macro involves fairly simple differentiation, you will do a lot of optimisation and use the chain rule and product rules a lot. One thing you will have to get used to in economics is seeing things written as functions and differentiating them.

You are always differentiating to find ‘marginals‘. The concept of ‘marginals’ (marginal revenue, marginal product, marginal cost) etc is about the most important concept in microeconomics, because all decisions are taken ‘at the margin’. Do you increase production by another unit or just produce at the level you are doing? Well if your marginal revenue (the amount of revenue you will earn by producing another unit of output) is higher than your marginal cost (the amount it will cost you to produce another unit) then go for it. If your marginal cost is higher then you don’t. As you produce more your MR will fall and your MC will rise so you will maximise profits by producing where MR = MC. Basic golden rule of micro!

Because MR is basically the ‘change in revenue over the change in output’ you find it by differentiating total revenue with respect to output. Total revenue is price x quantity.

So you have $TR = PQ$, $MR = \frac{d(TR)}{dQ}$ so $MR = \frac{d(PQ)}{dQ}$

Just a note here, don’t get confused by the fact that PQ is P times Q, and TR and MR are ‘total revenue’ and ‘marginal revenue’ you aren’t doing T times R or M times R, its just the abbreviation.

So if $MR = \frac{d(PQ)}{dQ}$ you use the product rule to say $MR = P\frac{dQ}{dQ}+Q\frac{dP}{dQ}$ so $MR = P+Q\frac{dP}{dQ}$

You also have to use the chain rule, and recognise how to do this in terms of the notation. Lets consider the question ‘what wage do you pay if you want to maximise profit?’

To show this example you first have to understand the concept of the production function. This will express Q as a function of K and L.

So we know that TR is a function of P and Q, and Q is a function of K and L. We have a ‘function of a function’ so the chain rule is coming up when we have to differentiate.

Now in terms of costs, we usually use w to refer to the cost (per unit) of labour (ie the wage) and r to refer to the cost of capital. If the capital was say machinery, then you aren’t having to pay the machinery a wage, but it is costing you, either you are renting it for a fee, or if you bought the machines outright, there is an ‘opportunity cost’ (ie you could have been investing the money you spent on the machines at the market interest rate, r, so if your machines are earning you less than r, you would have been better not buying them and investing your money instead)

So your total costs are going to be the wage times the number of units of labour you hire, plus the cost of capital times the number of units of capital you are using, ie $TC = wL + rK$

Your profits (usually $\pi$ is used to symbolise profit in economics, are $\pi = TR - TC$ so $\pi = TR(Q(K,L)) - wL - rK$. What this notation is saying is that total revenue is a function of Q (its also a function of P, but we don’t need that here as we are looking at labour which comes directly into the function for Q not P), and Q is a function of K and L.

To maximise profits with respect to the cost of our inputs, K and L, its an optimisation problem, we set the two partial derivatives of profit by K and L to zero to find the optimising points, ie $\displaystyle \frac{\partial \pi}{\partial K} = 0$ and $\displaystyle \frac{\partial \pi}{\partial L} = 0$.

We use the chain rule to do these partials. $\displaystyle \frac{\partial \pi}{\partial K} = \frac{d(TR)}{dQ}\frac{dQ}{dK}-r$ and $\displaystyle \frac{\partial \pi}{\partial L} = \frac{d(TR)}{dQ}\frac{dQ}{dL}-w$. When you use the chain rule you do the derivative of the outside function times the derivative of the inside function.

Remember that $\frac{dQ}{dK} = MR$ so $\displaystyle \frac{\partial \pi}{\partial K} = MR\frac{dQ}{dK}-r$ and $\displaystyle \frac{\partial \pi}{\partial L} = MR\frac{dQ}{dL}-w$.

We can also introduce a couple of concepts here called ‘marginal productivity’. The marginal productivity of capital is the amount of extra output you get by adding one more unit of capital, ie $\frac{dQ}{dK}$. The marginal productivity of labour is the amount of extra output you get by adding one more unit of labour, ie $\frac{dQ}{dL}$.

So we can now say $\displaystyle \frac{\partial \pi}{\partial K} = MR(MP_K)-r$ and $\displaystyle \frac{\partial \pi}{\partial L} = MR(MP_L)-w$.

Remember that to find the profit maximising point we had to set both partials to zero, ie $\displaystyle \frac{\partial \pi}{\partial K} = MR(MP_K)-r=0$ and $\displaystyle \frac{\partial \pi}{\partial L} = MR(MP_L)-w=0$ so $MR(MP_K)=r$ and $MR(MP_L)=w$

So the profit maximising wage is to pay each worker the multiple of his marginal productivity multiplied by the marginal revenue. In an environment of perfect competition, $P=MR$ so if we had perfect competition, $P(MP_L)=w$. This is quite an important result in micro which you use in questions to do with factor markets, because it shows that two things will drive wages up – either a rise in price of the good being produced, or an increase in marginal productivity of labour (eg workers becoming more skilled, technological advances making them more productive, or simply adding more capital so each worker has more capital to work with). Fairly logical results but that is the economics behind them.

For another example of using differentiation and algebraic manipulation, check out this little exercise in deriving the elasticity of demand for a firm in a competitive market.

Categories: Differentiation, Maths