Unit 8 Supply and demand: Markets with many buyers and sellers

Extension 8.6 Changes in supply and demand

If we know the demand and supply curve for a market, we can find the competitive equilibrium price and quantity by solving the two equations simultaneously for \(P\) and \(Q\). With a demand curve, \(Q=D(P)\), and supply curve, \(Q=S(P)\), the equilibrium price equalizes demand and supply:

\[D(P)=S(P)\]

If we can solve this equation to find \(P\), we can then find the corresponding equilibrium quantity by substituting back into the supply or demand curve. But what happens when one of these functions changes?

To model a demand or supply shock mathematically, we introduce parameters into the supply and demand curve to represent things other than price that affect the market for a good. Analysing how an equilibrium changes when a parameter changes is known in economics as comparative statics.

Linear supply and demand

Consider first the case that you analysed in Exercise E8.2, of a market with linear supply and demand functions:

\[D(P)=a-bP, \quad S(P)=c+dP\]

where \(a,\ b,\ c,\ d\) are constants. Assume that they are all positive, and that \(a>c\). This ensures that there is a single equilibrium with a positive price, \(P^*\), and positive quantity, \(Q^*\):

\[P^*=\frac{a-c}{b+d} \quad Q^*= a-bP^* = \frac{ad+bc}{b+d}\]

A demand shock

Suppose that in this market, there is a positive demand shock—that is, the quantity demanded is now higher at any given price. We can model a demand shock as an increase in the parameter, \(a\); if you think about the diagram, this represents a parallel rightward shift of the demand curve. It is similar to the demand shock in the market for hats shown in Figure 8.14 in the main part of this section, except that the slope of the curve doesn’t change.

One method for working out the effect is to find the new equilibrium when the demand curve is \(D(P)=a+\Delta a - bP\), and the supply curve is unchanged; \(\Delta a > 0\) represents the increase in \(a\). Then you can compare the new equilibrium price and quantity with the original ones.

But an easier way of working out what happens to price and quantity when a parameter changes is to think of \(P^*\) and quantity \(Q^*\) as functions of the parameters, and partially differentiate with respect to the parameter that changes. So for an increase in \(a\):

\[P^*=\frac{a-c}{b+d} \Rightarrow \frac{\partial P^*}{\partial a}= \frac{1}{b+d}\]

The derivative is positive; hence an increase in \(a\) results in an increase in \(P^*\). Similarly \(\frac{\partial Q^*}{\partial a}= \frac{d}{b+d}>0\), so \(Q^*\) increases too.

Of course, the derivatives tell us the effects of a very small increase in \(a\). But since they are positive wherever the equilibrium lies, we can deduce that a rise in \(a\), whether big or small, will always raise the equilibrium price and quantity.

Also, \(\frac{\partial Q^*}{\partial a}\) is less than 1. So the increase in equilibrium quantity is smaller than the increase in \(a\). The rise in \(a\) tells us how much the quantity demanded would increase if the price stayed the same. But \(P^*\) rises, so consumers buy less than they would have done without the price change.

In this example, you can easily obtain the same results diagrammatically by drawing an outward shift of the demand curve. But in other economic models, it can be difficult to be sure that your diagram captures all the possibilities, while using an algebraic method allows you to consider all the possible cases systematically.

The non-linear case

If the demand and supply curves are non-linear, it can be difficult to find an explicit solution for the equilibrium price and quantity. But it is still possible to model the effect of a shock that shifts one of the curves, and work out how it affects the equilibrium. We did this diagrammatically for the bread market example. Here we do the same thing algebraically.

We will write the demand curve for bread as \(Q=D(P,a)\), and the supply curve as \(Q=S(P,c)\). Introducing the two parameters, \(a\) and \(c\), enables us to model shifts in the curves—demand and supply shocks.

Think first about the demand curve. The quantity demanded decreases with the price, \(P\), as before; but it also depends on a parameter, \(a\), which captures consumer tastes. A high value of \(a\) represents a situation in which consumers have a strong liking for bread, so will purchase a high quantity at any given price. When \(a\) is low, less bread is demanded at each price. The dependence of demand on both \(P\) and \(a\) can be described using the partial derivatives:

\[\frac{\partial D}{\partial P} < 0; \quad \frac{\partial D}{\partial a} > 0\]

Then a positive demand shock can be represented as an increase in \(a\). It has the effect of increasing the quantity demanded at each price. When the demand curve is drawn in \((Q,P)\) space, as usual, it is drawn for a fixed value of \(a\). An increase in \(a\) shifts the demand curve to the right. Similarly, a fall in \(a\) represents a negative demand shock and shifts the curve to the left.

In the same way, a change in the parameter, \(c\), shifts the supply curve. We can think of \(c\) as representing technology. A rise in \(c\) corresponds to a technological improvement, which lowers the marginal cost of producing bread, and hence implies that bakeries will supply more bread at any given price. So both partial derivatives are positive:

\[\frac{\partial S}{\partial P} > 0; \quad \frac{\partial S}{\partial c} > 0\]

An increase in \(c\) shifts the supply curve to the right.

For any given \(a\) and \(c\), the demand curve is downward sloping and the supply curve is upward sloping in \((Q,P)\) space. Hence, there is at most one equilibrium price \(P^*\), and corresponding equilibrium quantity \(Q^*\).

In the example of linear supply and demand, we knew the exact form of the supply and demand functions, and we were able to solve the equations explicitly for the equilibrium price and quantity, in terms of the parameters. Here, that is not possible. But we know that the equilibrium price \(P^*\) (if it exists) satisfies:

\[D(P^*, a)=S(P^*, c)\]

And that the quantity, \(Q^*\), is then given by:

\[Q^*= S(P^*, c)\]

These two equations implicitly determine \(P^*\) and \(Q^*\). They will depend on the two parameters; we can think of them as functions of \(a\) and \(c\):

\[P^* = P^*(a, c), \quad Q^* = Q^*(a, c)\]

Now, we can use what we know about \(P^*\) and \(Q^*\) to work out how they change when \(a\) or \(c\) changes. Specifically, we can use the technique of implicit differentiation to find expressions for their partial derivatives with respect to \(a\) and \(c\).

First consider a change in the parameter, \(a\) (a demand shock). We differentiate the equilibrium equation with respect to \(a\), remembering that \(P^*\) depends on \(a\):

\[\begin{align*} D(P^*, a)&=S(P^*, c) \\ \Rightarrow \frac{\partial D}{\partial P} \frac{\partial P^*}{\partial a}+ \frac{\partial D}{\partial a}&=\frac{\partial S}{\partial P} \frac{\partial P^*}{\partial a} \end{align*}\]

This equation can be rearranged to write \(\partial P^*/\partial a\) in terms of the other partial derivatives:

\[\frac{\partial P^*}{\partial a} =\frac{\frac{\partial D}{\partial a}} {\frac{\partial S}{\partial P}-\frac {\partial D}{\partial P} }.\]

The denominator of this fraction is positive, because we know that \(\partial S/\partial P \gt 0\) and \(\partial D/\partial P \lt 0\). The numerator is positive too, from the way we specified the demand function, above. Hence, we can conclude that \(\partial P^*/\partial a \gt 0\). Therefore, a positive demand shock (an increase in \(a\)) leads to an increase in the equilibrium price.

To find \(\partial Q^*/\partial a\), we can use the equation:

\[Q^*=S(P^*, c)\]

again remembering that \(P^*\) is a function of \(a\). Differentiating with respect to \(a\):

\[\frac{\partial Q^*}{\partial a} =\frac{\partial S}{\partial P^*} \frac{\partial P^*}{\partial a}\]

From this expression, since \(\partial S/\partial P^*\gt 0\) and we have just shown that \(\partial P^*/\partial a\gt0\), we deduce that \(\partial Q^*/\partial a\gt0\) too. Therefore, a positive demand shock (an increase in \(a\)) leads to an increase in both the equilibrium price and the equilibrium quantity, and a negative demand shock has the opposite effects.

This result is very general. We have demonstrated that the qualitative effects of a demand shock (any shock that increases demand at each price) on the equilibrium price and quantity are the same as the ones we obtained diagrammatically in the market for hats, whatever the precise form of the supply and demand functions—provided that they have the standard properties. That is, the demand curve slopes downward, and the supply curve slopes upward.

Work through Exercise E8.5 to perform the same analysis for a supply shock.

Read more: Section 15.1 (on implicit differentiation), and Section 15.2 and the first two paragraphs of Section 15.3 (on comparative statics), of Malcolm Pemberton and Nicholas Rau. Mathematics for Economists: An Introductory Textbook (4th ed., 2015 or 5th ed., 2023). Manchester: Manchester University Press.