(0.9S)Yt-1. During year t, N tonnes of new grain are added, and so the size
of the grain mountain at the end of t years is given by
We are given that there are initially 30000 tonnes, so Yo = 30000.
Thus we have a r
27. First-order differential equations
In Chapter 23 we looked at the dynamics of the economy, using what are
known as discrete-time models. This means that the time-periods involved
were taken to be successive calendar years (
Now, p(O) = 6 and so when t = 0, 24 - (8/3)p = 8. This shows that we must
take the positive square-root, and that 8 =: J3/16c. Therefore, 16c = 3/64
24- -p =
16t + 3/64
Rearranging, we obtain the following for
25. Areas and integrals
The consumer surplus
A typical downward-sloping demand set D is illustrated in Figure 25.1. As
the number of units of the good increases, the price consumers are prepared
to pay for each unit decreases.
Figure 25.1: A ty
Example Let X, Y,Z denote the sets defined above, that is
Then we have, for example
Xu Y = cfw_1,2,3,4, 5,7, 8,9,
Y uZ = cfw_1,2,3,4,5, 7,9,
X n Y = cfw_5,7,9,
X n (Y nZ)
We also have relatio
Integration by parts
+ 2x + 5
-2- - - d x .
The difficulty here arises from the complicated denominator. So, a possible
substitution is x 2 + 2x + 5 = t, which implies (2x + 2) dx = dt. For the indefinite
integral, we ha
We now return to the dynamics of the simplified national economy, as
described by the 'multiplier-accelerator' equations in Section 23.2. As we shall
see, the solution to this model can exhibit oscillatory behaviou
Example 23.1 Find the general solution of the recurrence equation
Yt - 6Yt-l
+ 5Yt-2 = 0.
Solution: The auxiliary equation is z2 - 6z + 5 = 0, that is (z - 5)(z - 1)
with solutions 1 and 5. The general solution is there
A consumer purchases quantities of two commodities, fruit
and chocolate, each month. The consumer's utility function is
for a bundle (Xl, X2) of Xl units of fruit and X2 units of chocolate. The
consumer has a total of
Example Find the general solution of the equation
Find also the solution for which y(O)
+ 13y = 0.
= 1 and y' (0) = o.
The auxiliary equation is z2 - 4z + 13 = 0, which has no solutions. We have
= 4/2 = 2
Exercise 28.1 Find the general solutions of the following equations.
+ 4y =
+ 2 dt + 2y =
Exercise 28.2 Find a particular solution, in the form y
(At + B)e 3t , of the
the cobweb model in general terms
how to derive a recurrence equation when demand and supply are linear
solving this recurrence to find the sequences of prices and quantities
analysing the stability of the cobweb model
The logarithm function
General properties of the logarithm function can be deduced from the corresponding properties of the exponential function. For example, from the rule
exp(x) exp(y) = exp(x + y) it is easy to deduce (Example 7.5) that
Use the derivative to find the approximate change in the
x 4 when x changes from 3 to 3.005. Compare this with the
Solution: The derivative of f is f'(x) = 4x 3 and we therefore have the
Example 4.3 An amount of $1000 is invested and attracts interest at a rate
equivalent to 10% per annum. Find the total after one year if the interest is
compounded (a) annually, (b) quarterly, (c) monthly, (d) daily. (Assume the
Main topics/Key terms
+ 2q + 27 -
+ 5q2 + 10q = 0,
that is 6q2 + 12q - 48 = 0.
Dividing by 6, we get q2 + 2q - 8 = 0, which factorises as (q + 4)(q - 2) = O.
The solutions are q = -4 and q = 2 and the corresponding values of p, given
Answers to selected exercises
21.3 The cost function is C(q)
= 2-/3q2. The optimal production level is
L U cfw_(L,p) I p> 5J6L3/ 2 .
= cfw_(q,5J6q3/2) I q
21.5 S = cfw_(O,p) I p
Z U cfw_(L,p) I p > Z.
Example 25.1 Find the area enclosed by the lines t
the graph of the function f(t) = et .
= 2, the t-axis, and
Solution: The required area is equal to the definite integral A = J1 et dt. Now,
J et dt is one of t
The price ratio and the tangency condition
Eliminating A we get
This says that the price ratio Pi/Pj is equal to another ratio, known to
economists as the marginal rate of substitution, evaluated at the optimal point.
In the case n = 2
Figure 21.4: the case of decreasing returns to scale
Example 21.1 A firm's weekly output is given by the production function
q(k,l) = k 3/ 4 11/ 4, and the unit costs for capital and labour are v =
21. Constrained optimisation
The elementary theory of the firm
You will recall that a firm can be described in terms of a function whose
inputs are amounts k (capital) and I (labour), and whose output is a quantity
q (production). The value of q depe
Find recurrence equations for an, bn, Cn, dn, and solve them to determine an
explicit formula for An.
Exercise 15.5 Find a general formula for B n, where
Exercise 15.6 Suppose that the matrix
of Section 15.3 is
Exercise 11.1 Find the first and second partial derivatives of the following
Exercise 11.2 If f(x,y) = x
note that x y2 = exp (y2 In x) .
find % and 0,. [Hint: to help calculate
Exercise 11.3 Suppose that f(x,y)
Choosing optimal bundles
In order to see what this means, consider Figure 14.4. Here a and bare
bundles on the same indifference curve u(x) = c, and the assumption requires
that every point on the straight line segment joining them is in Uc . This
!k- I / 211/ 2
lk l / 2 1- 1/ 2
The gradient at (1, 1) is, therefore -1/1
Similarly, the derivative dl / dk on an isoquant v(k, I)
= c is
which is 1 at (1,1).
Example 12.4 The notatio
11. Partial derivatives
Functions of several variables
Recall that a function f may be thought of as a 'black box', which accepts
an input x and produces an output f(x). In this chapter we shall look at
functions for which the input consists of a pai
Figure 8.2: A function with four critical points
We can decide the nature of a given critical point by considering what happens
to f' in its vicinity. Looking at the point a in Figure 8.2 we observe that the
gradient (the deri
Main topics/Key terms
revenue and profit
finding critical points
classifying critical points using second derivative
optimisation in an interval
Key terms, notations and formulae
revenue, R(q) = qP(q)
profit function, I1(q)
Exercise 9.1 Calculate the elasticity of demand when the demand function is
qD(p) = 70 - 4p.
For what range of values ofp is your expression valid, and for which of these
values is the demand inelastic?
Exercise 9.2 Show t