Exercises 1
Exercise 1.1 Ifx + y = 3 and x – 2y = -3, what are x and y ?
Exercise 1.2 Solve the following simultaneous equations.
2x + y = 9,
x – 3y = 1.
Exercise 1.3 Suppose the market for a commodity is governed by supply and demand sets defined as follows. The supply set S is the set of pairs (q,p) for which q – 6p = -12 and the demand set D is the set of pairs (q,p) for which q + 2p = 40. Sketch S and D and determine the equilibrium set E = S n D, the supply and demand functions qS, qD, and the inverse supply and demand functions pS, pD.
Exercise 1.4 Suppose that the government decides to impose an excise tax of T on each unit of the commodity discussed in Exercise 1.3. What price will the consumers end up paying for each unit of the commodity?
Exercise 1.5 Find a formula for the amount of money the government obtains from taxing the commodity in the manner described in Exercise 1.4.
Determine this quantity explicitly when T = 0.5.
Exercise 1.6 The supply and demand functions for a commodity are If an excise tax of T is imposed, what are the selling price and quantity sold, in equilibrium?
Exercises 2
Exercise 2.1 For each of the following functions f sketch the graph y = f (x) in lR~. Decide whether the inverse function f- l exists, and if it does exist write down the formula for f-l(y).
f(x)=5x; f(x)=x 2 -4x+8; f(x)=x 5•
Exercise 2.2 Each of the following definitions specifies a subset K oflR 2• (In this question x is the name of the coordinate measured on the horizontal axis and y is the name of the coordinate measured on the vertical axis.)
{(x,y) I 3x + 4y = 12}, {(x,y) I 3x + 4y ~ 12}, {(x,y) I x2 + y2 = 4},
{(x,y) I x 2 + y2 ~ 4}, {(x,y) I x 2 = 4y}, {(x,y) I y2 = 4x}.
In each case sketch the set K and indicate on your sketch the set K n lR~.
Exercise 2.3 Suppose that the supply and demand sets for a particular market
are S = {(q,p) I 3p – q = 5}, D = {(q,p) I 3p + q2 + 2q = 9}.
Sketch Sand D and determine the equilibrium set E = S n D. Comment briefly on the interpretation of the results.
Exercise 2.4 The functions f, g, h are given by f(x) = x 2 + 1, g(x) = 1/x 2, hex) = JX. Find formulae for the compositions fg, gf, hf, fh, hfg·
Exercise 2.5 Suppose that the supply and demand sets for a good are given
by S = {(q,p) I q – 3p = -1}, D = {(q,p) I q + p = 2}.
Determine the supply function, qS, the demand function, qD, the inverse supply function, pS, and the inverse demand function, pD. Verify that for any p and q, (pSqS)(p) = P and (pDqD)(p) = p.
Exercise 2.6 For which values of a has the equation
x 2 + ax + 1 = 0
no solutions, exactly one solution, or two solutions? Determine the solutions in the second and third cases.
