Baldani

Instructor’s Instructor’s Manual on Disk

to accompany

Mathematical Economics by Jeffery Baldani James Bradfield Robert Turner

Disk 1 ( Solutions to problems from Chapters 2, 3, 4, and 5 )

Copyright © 1996 by Harcourt Brace & Company. All rights reserved. Subject to the restrictions hereof, permission is hereby granted until further notice, to duplicate this diskette without alteration onto another diskette or the hard disk drive of a computer for use in connection with a course for which Mathematical Economics, First Edition, by Jeffery Baldani, James Bradfield and Robert Turner has been adopted, and not for resale resale,, provid provided ed the the copie copiess are made made from from this this maste masterr diske diskette tte only, only, and and provid provided ed that that the follow following ing copyrig copyright ht notic noticee appea appears rs on the label label of all copie copiess in disket diskette te form: form: © 1996 1996 by Harcourt Brace & Company. The program may not be merged into another program or modified in any way. Copies may not be made of copies. Problems and solutions may be displayed and may be reproduced in print form for instructional purposes only, provided a proper copyright notice appears on the last page of each print-out. Except as previously stated, no part of the computer program embodied in this diskette may be reproduced or transmitted in any form or by any means, electronic or mechanical, including input into or storage in any information system, without permission in writing from the publisher. Produced in the United States ISBN: 0-03-011578-7

Copyright © 1996 by Harcourt Brace & Company. All rights reserved. Subject to the restrictions hereof, permission is hereby granted until further notice, to duplicate this diskette without alteration onto another diskette or the hard disk drive of a computer for use in connection with a course for which Mathematical Economics, First Edition, by Jeffery Baldani, James Bradfield and Robert Turner has been adopted, and not for resale resale,, provid provided ed the the copie copiess are made made from from this this maste masterr diske diskette tte only, only, and and provid provided ed that that the follow following ing copyrig copyright ht notic noticee appea appears rs on the label label of all copie copiess in disket diskette te form: form: © 1996 1996 by Harcourt Brace & Company. The program may not be merged into another program or modified in any way. Copies may not be made of copies. Problems and solutions may be displayed and may be reproduced in print form for instructional purposes only, provided a proper copyright notice appears on the last page of each print-out. Except as previously stated, no part of the computer program embodied in this diskette may be reproduced or transmitted in any form or by any means, electronic or mechanical, including input into or storage in any information system, without permission in writing from the publisher. Produced in the United States ISBN: 0-03-011578-7

Table of Contents Chapter 1

Chapter 4

no problems Chapter 2

4.1

30 – 35

4.2

36 & 37

4.3

38 & 39

2.1

4

4.4

40 & 41

2.2

5

4.5

42 & 43

2.3

6

4.6

44 – 46

2.4

7

4.7

47 & 48

2.5

8

4.8

49 & 50

2.6

9

2.7

10

Chapter 5

2.8

11

5.1

51 – 54

2.9

12

5.2

55 & 56

2.10

13 & 14

5.3

57

2.11

15

5.4

58 – 60

2.12

16

5.5

61 & 62

2.13

17

5.6

63 – 65

5.7

66 – 68

5.8

69 & 70

Chapter 3

3.1

18

3.2

19

3.3

20

3.4

21

3.5

22

3.6

23

3.7

24

3.8

25

3.9

26

3.10

27

3.11

28

3.12

29

Remaining chapters are on subsequent disks.

2.1 From the text, the solutions to the four cases are:

and

(a) so when

decreases, all solutions except

decrease.

(b) so when

decreases,

and

decrease while the others do not change.

2-1 Harcourt Brace & Company items and derived items copyright © 1996 by Harcourt Brace & Company .

2.2

(a)

α

is exogenous: its value is not determined within the model.

is the relative weight put on the goal of income maximization as opposed to rent maximization. α

(b)

(c)

When

When

the solution for rent maximization.

the solution for income maximization.


2.3 with K fixed

(a)

From FOC,

(b)


2.4 (a)

For

(b)

(i)

so

(ii)


2.5

(a) which holds when

(b)


2.6

Yes, these answers are the same.


2.7 (a)

(b)

(c)

(d)

(e)

The first-order condition for choosing t to maximize T is

which is cubic in t , with no easy solution.


2.8


2.9

(a) by symmetry

(b)


2.10 (a)

Solutions from text: equation (2.70):

equation (2.71):

equation (2.72):

(b)

equilibrium n is when

so

Substituting into solutions for


Note also that

(c)

The first-order condition for choosing t to maximize T is


2.11 (a)

From the text,

and

But in this example, c = 0.

(b)

which is the monopoly price.


2.12 (equation (2.84))


2.13

(a) g > 0 is sufficient for

(b)

to be positive and finite.

, which is smaller than the usual multiplier,

(c)

When g is larger, there are two possible effects: government spending increases by more when equilibrium GDP is lower than the target level, but government spending decreases by more when equilibrium GDP is higher than the target level. So if the target level is greater than equilibrium GDP,

and a higher value of g will make equilibrium GDP higher. But if the target level of GDP is lower than equilibrium GDP,

and a higher value of g will make equilibrium GDP lower. Either way, a higher value of g will move equilibrium GDP closer to the target level.


3.1 (a)

(b) (b)

( 1 x 3 ) row vecto ectorrs are :

( 2 x 1 ) column vectors are :


3.2 (a)

(b)

(c)


3.3 (a)

(b)

(c)


3.4 (a)

(b)

BA is not defined. Since A is a 4x2 matrix and B is a 2x1 vector, the vector products we would need to calculate to multiply B by A are not defined.


3.5 In matrix form, the given system of equations is:


3.6

Similarly,


3.7


3.8 (a)

(b)


3.9 (a)


3.10 The cofactor matrix for matrix A is :

Now,

Similarly,

Now,

(b)

From 3.9, | A | = -40. So,


3.11 The given system of equations can be written as :

Now, the matrix of cofactors is :

We have,

So, solution to the given system of equations is : x = 3 / 11 ,

y = 2 / 11,

z = - 1 / 11.


3.12 Using Cramer’s Rule :

Now,


4.1

Now,

so,

we know,

so,

(1)

We already derived in eq (4-24) that (2)

To get the equal prices in the presence of a unit subsidy b 1 granted to producers of good 1, following the reasoning in page 106, we just need to substitute

s 1 + s11 b1 for

s1

in

equations (1) and (2). This gives us the following equilibrium expressions:

Now, (a)

(b)

P.T.O.

Now,

so,

so, (iii)

Now,

(iv)

From (iii),

so,

(c)

From (iv),

and then taking partial derivative with respect to b 1, we get,

(d)

Interpretations: If two goods are substitutes,

d 12 , d21 will be positive, and

s12 , s21

will be positive,

and

d11 , d22

always negative,

and

s11 , s22

always positive.

Thus,

Thus, a unit increase in subsidy for good A will reduce the price of good A.

Thus, a unit increase in subsidy for good A will increase the price of good B.

Thus, a unit increase in the subsidy for good A will increase the quantity of good A produced. We can similarly analyze the effects of a unit increase in subsidy on the quantity of good B produced. If two goods are complements, we can interpret equations (a), (b), (c), and (d) in a similar fashion as above. We can do similar analyses if two goods are neither complements nor substitutes.

4.2 The equation (4.36) gives us:

For the equilibrium quantities of output of firm 1 to be positive, the signs of the numerator and the denominator of the R.H.S. of equation (1) has to be the same. First, assume that they are both negative. Then,

for inequality (3) to be true. One way of (3) being true is to have: . But if: , then price is less than both marginal and average cost. In that case, the firm will stop operating. Therefore, such a case is unlikely.

Now, assume that both the numerator and the denominator are positive. Then, and,

(4)

One way of (4) being true is to have: . That is to say that when

Firm 1's average cost will lie below price. But, when Firm 2's average cost will lie above price. The economics of this is quite clear. Firm 2 is simply not posing any competition to firm 1. Thus firm 1 can make a positive quantity of goods and sell them at a price that will let the firm make positive profits. The case where:

are both positive is discussed in section 4.4.2. We can interpret other cases when

can be positive in a similar fashion. The conditions for

to be positive and the interpretations of those conditions should be a mirror image of those for .

4.3 Let’s assume we need a tax of rate t . Then,

We can get equal quantities by substituting in 4.36. Thus,

From (4.28),

Now,

Let

Then,

or,

(1)

If the value of the R.H.S. of equation (1) is positive, then we need a tax of that amount. Otherwise, we need a subsidy of that amount.

4.4

Assume also

Now,

(i)

(ii) and we also know, In matrix form, equations (i), (ii), and (iii) are:

To solve this system, we use Cramer’s Rule.

(iii)

Now, let’s assume, k = 0.5

P.T.O

Then,

If we let c2 = c, then we see that the price in this case will be higher than in equation 4.48. Thus, everything else being equal, a higher degree of conjectural variation leads to a higher price in a duopoly.

4.5 The demand function is: . Given is:

Now,

(i)

Similarly, (ii) And, (iii)

The matrix form of these conditions is:

We can get the solution of this system from (4.56) by changing the constant c:

(iv)

(v) Comparing (iv) and (v) with (4.56) and (4.57) respectively, we find that the aggregate triopolist output and the triopolist price is the same as those of a monopoly with no taxes or subsidies.

4.6 We are given that

and

so,

So, from (4.65)

(i) Similarly, (ii) (iii) So,

a)

Now,

So, for a unit upward shift of the inverse demand function, price will change by . Thus, if b = 1, price will go up by .

b)

In this case,

From (4.65),

(iv)

(v)

Similarly,

(vi) Since equilibrium total quantity is unchanged, equilibrium price for the industry will be unchanged. Comparing (i), (ii), (iii) with (iv), (v), (vi) , we find that the equilibrium quantity of firm 1 decreases, the equilibrium quantity of firm 2 increases, and the equilibrium quantity of firm 3 remains unchanged.

4.7

If

then Equation (4.66) becomes,

Thus, the system in matrix form is:

Now,

a)

By Cramer’s Rule: (1)

b)

The multiplier for autonomous spending: (2)

c)

If budget is balanced,

Now, from (1)

(3) d)

The multiplier for autonomous spending in the case of a balanced budget is: (4)

Equation (1) shows that the equilibrium income will be higher than the income in Equation (4.70). Similarly, Equation (2) shows that the multiplier is larger. Similarly, the balanced budget income and multiplier as specified by Equations (3) and (4) are higher than those in equations (4.73) and (4.74).

4.8

From (4.82), we know,

So,

(i)

(ii) (iii)

From Equation (1), we find that an increase in b increases C*. Thus, an increase in marginal propensity to consume will increase the equilibrium level of consumption. A reduction in tax rate t will have similar effect on C* as an increase in MPC. Similarly, we can interpet the roles of other parameters on C*. The equilibrium values and the rates of changes of

can be calculated in a process

similar to that of the calculation of Equations (i), (ii), and (iii).

5.1 (a)

all third derivatives equal 0. (b)

all third derivatives equal 0. (c)

(d)

(e)

all second- and third-order cross partial derivatives equal 0

(f)

Now let and

Then

so

and

Now let

and

Then

and

, so

5.2 (a)

(b)

(c)

(d)

(e)

(f)

5.3 (a)

(b)

(c)

(d)

(e)

(f)

5.4 (a)

at the particular values of x and y that are of interest.

OR

(b)


OR

(c)


OR

(d)


OR

(e)


OR

(f)

both exist so exists as long as


OR

5.5 (a)

(b)

(c)

5.6 (a)

the Jacobian determinant

given in the answer to problem 5.5(a),


(b) all exist as long as the Jacobian determinant

given in the answer to problem 5.5(b),

at the particular values of x, y, and z that are of interest.

(c) all exist as long as the Jacobian determinant

given in the answer to problem 5.5(c),

at the particular values of x, y, and z that are of interest.

5.7 (a)

Let

. If y =1, then , or

Note that since two values of x exist for one value of y, x is not a function of y, except in the locality of particular combinations of x and y .

Evaluated at x = 1 and y = 1,

Evaluated at and y = 1,

So when x is positive the slope of the level curve becomes more negative when y increases but if x is negative the slope of the level curve becomes less negative when y increases.

(b)

Let


Note that since two values of x exist for one value of y, x is not a function of y, except in the locality of particular combinations of x and y .

Evaluated at x = 1/4 and y = 1,

Evaluated at and y = 1,

So when x is positive the slope of the level curve becomes more negative when y increases but if x is negative the slope of the level curve becomes less negative when y increases. (c)

Let


when evaluated at

. So the slope of the level curve becomes less negative when y increases.

(d)

Let


when evaluated at


(e)

Let

.

If y = 1, then , or

when evaluated at


(f)

Let

.

If y = 1, then , or

when evaluated at


5.8 (a)

so homogeneous of degree 2

(b)


(c)


(d)

so homogeneous of degree

(e)

not homogeneous

Baldani

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