1. Mathematical models in economics

1.1 Introduction
In this book we use the language of mathematics to describe situations which occur in economics. The motivation for doing this is that mathematical arguments are logical and exact, and they enable us to work out in precise detail the consequences of economic hypotheses. For this reason, mathematical modelling has become an indispensable tool in economics, finance, business and management. It is not always simple to use mathematics, but its language and its techniques enable us to frame and solve problems that cannot be attacked effectively in other ways. Furthermore, mathematics leads not only to numerical (or quantitative) results but, as we shall see, to qualitative results as well.

1.2 A model of the market
One of the simplest and most useful models is the description of supply and demand in the market for a single good. This model is concerned with the relationships between two things: the price per unit of the good (usually denoted by p), and the quantity of it on the market (usually denoted by
q). The ‘mathematical model’ of the situation is based on the simple idea of representing a pair of numbers as a point in a diagram, by means of coordinates with respect to a pair of axes. In economics it is customary to take the horizontal axis as the q-axis, and the vertical axis as the p-axis. Thus, for example, the point with coordinates (2000, 7) represents the situation when 2000 units are available at a price of $7 per unit.
How do we describe demand in such a diagram? The idea is to look at those pairs (q, p) which are related in the following way: if p were the selling price, q would be the demand, that is the quantity which would be sold to consumers at that price. If we fill in on a diagram all the pairs (q, p) related in this way, we get something like Figure 1.1.

2 Mathematical models in economics
pDq

Figure 1.1: The demand set
We shall refer to this as the demand set D for the particular good. In economics you will learn reasons why it ought to look rather like it does in our diagram, a smooth, downward sloping curve.
Suppose the demand set D contains the point (30,5). This means that when the price p = 5 is given, then the corresponding demand will be for q = 30 units. In general, provided D has the ‘right’ shape, as in Figure 1.1, then for each value of p there will be a uniquely determined value of q. In this situation we say that D determines a demand function, qD. The value written qD(p) is the quantity which would be sold if the price were p, so that qD(5) = 30, for example.
Example Suppose the demand set D consists of the points (q,p) on the straight line 6q + 8p = 125. Then for a given value of p we can determine the corresponding q; we simply rearrange the equation of the line in the form q = (125 – 8p)/6. So here the demand function is D( ) _ 125 – 8p
q p – 6 .
For any given value of p we find the corresponding q by substituting in this formula. For example, if p = 4 we get
q = qD(4) = (125 – 8 x 4)/6 = 93/6.
o

A model of the market 3
There is another way of looking at the relationship between q and p. If we suppose that the quantity q is given, then the value of p for which (q, p) is
in the demand set D is the price that consumers would be prepared to pay if q is the quantity available. From this viewpoint we are expressing p in
terms of q, instead of the other way round. We write pD(q) for the value of p corresponding to a given q, and we call pD the inverse demand function. Example (continued) Taking the same set D as before, we can now rearrange the equation of the line in the form p = (125 – 6q)/8. So the inverse demand function is
D( ) _ 125 – 6q
p q – 8 .

Next we turn to the supply side. We assume that there is a supply set S consisting of those pairs (q,p) for which q would be the amount supplied to
the market if the price were p. There are good economic reasons for supposing that S has the general form shown in Figure 1.2.
p
q
Figure 1.2: The supply set
If we know the supply set S we can construct the supply function qS and the inverse supply function pS in the same way as we did for the demand function and its inverse. For example, if S is the set of points on the line 2q – 5p = -12, then solving the equation for q and for p we get
S( ) _ 5p – 12 q p – 2 ‘ S( ) _ 2q + 12 p q – 5 .

4 Mathematical models in economics

1.3 Market equilibrium
The usefulness of a mathematical model lies in the fact that we can use mathematical techniques to obtain information about it. In the case of supply and demand, the most important problem is the following. Suppose we know all about the factors affecting supply and demand in the market for a particular good; in other words, the sets Sand D are given. What values of q and p will actually be achieved in the market? The diagram (Figure 1.3) makes it clear that the solution is to find the intersection of D and S, because that is where the quantity supplied is exactly balanced by the quantity required.
p
D s
Figure 1.3: The equilibrium set E = S n D
q
The mathematical symbol for the intersection of the sets Sand D is S n D, and economists refer to E = S n D as the equilibrium set for the given market. Fortunately, there is a simple mathematical technique for finding the equilibrium set; it is the method for solving ‘simultaneous equations’.
Example Suppose the sets D and S are, respectively, the sets of pairs (q, p) such that’
q + 5p = 40 and 2q -15p = -20.
Then a point (q*, p*) which is in the equilibrium set E = S n D must, by definition, be in both Sand D. Thus (q*, p*) satisfies the two equations
q* + 5p* = 40, 2q* – 15p* = -20.
The standard technique for solving these equations is to multiply the first one by 2 and subtract it from the second one. Working through the algebra,

Excise tax 5
we get q* == 20 and p* == 4. In other words the equilibrium set E is the single point (20,4). D It is worth remarking that in this example we get a single point of equilibrium, because we took the sets D and S to be straight lines. It is possible to imagine more complex situations, such as that we shall describe in Example 2.5, where the equilibrium set contains several points, or no points at all.

1.4 Excise tax
Using only the simple techniques developed so far we can obtain some interesting insights into problems in economics. In this section we study the
problem of excise tax. Suppose that a government wishes to discourage its citizens from drinking too much whisky. One way to do this is to impose a fixed tax on each bottle of whisky sold. For example, the government may decide that for each bottle of whisky the suppliers sell, they must pay the government $1. Note that the tax on each unit of the taxed good is a fixed amount, not a percentage of the selling price. Some very simple mathematics tells us how the selling price changes when an excise tax is imposed. Example In the previous example the demand and supply functions are given by
S 15 q (p) == 2 P – 10 ,
and the equilibrium price is p* == 4. Suppose that the government imposes an excise tax of T per unit. How does this affect the equilibrium price?
The answer is found by noting that, if the new selling price is p, then, from the supplier’s viewpoint, it is as if the price were p – T, because the supplier’s revenue per unit is not p, but p – T. In other words the supply function has changed: when the tax is T per unit, the new supply function qST is given by Of course the demand function remains the same. The new equilibrium values qT and pT satisfy the equations

6 Mathematical models in economics Eliminating qT we get
T 15 T
40 – 5p = 2(P – T) – 10.
Rearranging this equation, we obtain and so we have a new equilibrium price of The corresponding new equilibrium quantity is
qT =40-5pT =20-3T.
For example, if T = 1, the equilibrium price rises from 4 to 4.6 and the equilibrium quantity falls from 20 to 17. Unsurprisingly, the selling price has
risen and the quantity sold has fallen. But note that, although the tax is T per unit, the selling price has risen not by the full amount T, but by the fraction 3/5 of T. In other words, not all of the tax is passed on to the consumer. D

1.5 Comments
1. Economics tells us why the supply and demand sets ought to have certain properties. Mathematics tells us what we can deduce from those properties and how to do the calculations.

2. Mathematics also enables us to develop additional features of the model. In the case of supply and demand, we might ask questions such as the following:

  •  What happens if conditions change, so that the supply and demand sets are altered slightly?
  •  If the equilibrium is disturbed for some reason, what is the result?
  •  How do the suppliers and consumers arrive at the equilibrium? A typical instance of the first question is the excise tax discussed above. In this book we shall develop the mathematical techniques needed to deal withmany other instances of these questions.