Chapter 4: Appendix
CHAPTER 4 APPENDIX
DEMAND THEORY – A MATHEMATICAL TREATMENT
EXERCISES
1. Which of the following utility functions are consistent with convex indifference curves,
and which are not?
a.
U(X, Y) = 2X + 5Y
b.
U(X, Y) = (XY)0.5
c.
U(X, Y) = Min(X, Y), where Min is the minimum of the two values of X and Y
The three utility functions are presented in Figures 4A.1.a, 4A.1.b, and 4A.1.c. The first
may be represented as a series of straight lines; the second as a series of hyperbolas;
and the third as a series of “L’s.” Only the second utility function meets the definition
of a strictly convex shape.
To graph the indifference curves which represent the preferences given by
U(X,Y)=2X+5Y, set utility to some given level U0 and solve for Y to get
Y
U0 2
 X.
5 5
Since this is the equation for a straight line, the indifference curves are linear with
intercept
U0
2
and slope  .
5
5
Y
U2
5
U1
5
U0
5
U1
U0
U0
2
U1
2
Figure 4A.1.a
58
U2
U2
2
X
Chapter 4: Appendix
To graph the indifference curves that represent the preferences given by
U(X,Y)  (XY)0.5 , set utility to some given level U0 and solve for Y to get
Y
U02
.
X
By plugging in a few values for X and solving for Y, you will be able to graph the
indifference curve U0, which is illustrated in Figure 4A.1.b, along with the indifference
curve U1.
Y
U1
U0
X
Figure 4A.1.b
To graph the indifference curves which represent the preferences given by
U(X,Y)  Min(X,Y) , first note that utility functions of this form result in indifference
curves which are L-shaped and represent a complementary relationship between X and
Y. In this case, for any given level of utility U0, the value of X and Y will also be equal
to U0. As X increases and Y does not change, utility will also not change. If both X and
Y change, then utility will change and we will move to a different indifference curve.
See the following table.
X
10
10
10
11
9
9
Y
10
11
9
10
10
9
59
U
10
10
9
10
9
9
Chapter 4: Appendix
Y
U1
U1
Uo
U0
U0
X
U1
Figure 4A.1.c
2. Show that two utility functions given below generate the identical demand functions for
goods X and Y:
a.
U(X, Y) = log(X) + log(Y)
b.
U(X, Y) = (XY)0.5
The Appendix discusses how to derive demand functions from utility functions. If we
show that the two utility functions are equivalent, then we know that the demand
functions derived from them are identical. We may show their equivalence by
identifying a suitable transformation from one set of numbers into another set without
changing their order.
Taking the logarithm of U(X, Y) = (XY)0.5 we obtain:
logU(X, Y) = 0.5log(X) + 0.5log(Y).
Now multiply both sides by 2:
2(logU(X,Y) = log(X) + log(Y).
Therefore, the two utility functions are equivalent and will yield identical demand
functions. However, we will solve for the demand functions in both cases to show that
they are the same.
a. To find the demand functions for X and Y, corresponding to U(X, Y) = log(X) + log(Y),
given the usual budget constraint, write the Lagrangian:
 = log(X) + log(Y) - (PXX + PYY - I)
Differentiating with respect to X, Y, , and setting the derivatives equal to zero:
 1

 PX  0
X X
 1
  PY  0
Y Y

 PX X  PY Y  I  0.

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Chapter 4: Appendix
The first two conditions imply that PX X 
The third condition implies that
1


1

Substituting this expression into PX X 
1

 I  0 , or  
I
and PY Y 

1
and PY Y 

.
2
.
I
I
gives the demand functions:

0.5
0.5
X   I and Y   I .
PX 
PY 
Notice that the demand for each good depends only on the price of that good and on
income, not on the price of the other good.
b. To find the demand functions for X and Y, corresponding to U(X,Y) = (XY)0.5 given
the usual budget constraint, first write the Lagrangian:
 = 0.5(logX) + (1 - 0.5)logY - (PXX + PYY - I)
Differentiating with respect to X, Y,  and setting the derivatives equal to zero:
 0.5

 PX  0
X
X
 0.5

 PY  0
Y Y

 PX X  PYY  I  0.

The first two conditions imply that PX X 
0.5

and PY Y 
Combining these with the budget constraint gives:
Now substituting this expression into PX X 
functions:
0.5

0.5


0.5

0.5

.
 I  0 or  
and PY Y 
0.5

1
.
I
gives the demand
0.5 
0.5
X   I and Y   I.
 PX 
 PY 
3. Assume that a utility function is given by Min(X, Y), as in Exercise 1(c). What is the
Slutsky equation that decomposes the change in the demand for X in response to a change
in its price? What is the income effect? What is the substitution effect?
The Slutsky equation is
X
X

PX PX
X 
 X 
I ,
U U *
where the first term represents the substitution effect and the second term represents
the income effect. Because there is no substitution as prices change with this type of
utility function, the substitution effect is zero. The income effect is the shift from U1 to
U2.
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Chapter 4: Appendix
Y
L2
L1
U2
L3
U1
New Budget,
New Utilility
New Budget,
Old Utility
Old Budget,
Old Utility
X
Figure 4A.3
4. Sharon has the following utility function:
U(X,Y) 
X Y
where X is her consumption of candy bars, with price PX=$1, and Y is her consumption of
espressos, with PY=$3.
a.
Derive Sharon’s demand for candy bars and espressos.
Using the Lagrangian method, the Lagrangian equation is
X  Y  (PX X  PY Y  I).

To find the demand functions, we need to maximize the Lagrangian equation with
respect to X, Y, and , which is the same as maximizing utility subject to the budget
constraint. The necessary conditions for a maximum are
 1  12
 X  PX   0
X 2
(1)
 1  12
 Y  PY   0
Y 2

 PX X  PY Y  I  0.

(2)
(3)
Combining necessary conditions (1) and (2) results in

1

2PX X
1
1
2PY Y
1
PX X 2  PY Y 2
(4)
P 2 
X   Y2 Y.
PX 
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Chapter 4: Appendix
You can now substitute (4) into (3) and solve for Y. Once you have solved for Y, you can
substitute Y back into (4) and solve for X. Note that algebraically there are several
ways to solve this type of problem, and that it does not have to be done exactly as we
have done here. The demand functions are
b.
Y
PX I
I
or Y 
P  PY PX
12
X
PY I
3I
or X  .
P  PY PX
4
2
Y
2
X
Assume that her income I=$100. How many candy bars and espressos will Sharon
consume?
Substitute the values for the two prices and income into the demand functions to find
that she consumes X=75 candy bars and Y=8.3 espressos.
c.
What is the marginal utility of income?
From part a

1
2PX X

1
. Substitute into either part of the equation to
2PY Y
find that =0.058. This is how much utility would increase by if Sharon had one
more dollar to spend.
2
2
5. Maurice has the following utility function: U(X,Y)  20X  80Y  X  2Y , where X is
his consumption of CD’s, with a price of $1, and Y is his consumption of movie videos,
with a rental price of $2. He plans to spend $41 on both forms of entertainment.
Determine the number of CD’s and video rentals that will maximize Maurice’s utility.
Using the Lagrangian method, the Lagrangian equation is
  20X  80Y  X  2Y  (X  2Y  41).
2
2
To find the optimal consumption of each good, we need to maximize the Lagrangian
equation with respect to X, Y, and , which is the same as maximizing utility subject to
the budget constraint. The necessary conditions for a maximum are
(1)
(2)
(3)

 20  2X    0
X

 80  4Y  2  0
Y

 X  2Y  41  0.

Combining necessary conditions (1) and (2) results in
  20  2X  40  2Y
(4)
2Y  20  2X.
You can now substitute (4) into (3) and solve for X. Once you have solved for X, you can
substitute this value back into (4) and solve for Y. Note that algebraically there are
several ways to solve this type of problem, and that it does not have to be done exactly
as we have done here. The optimal bundle is X=7 and Y=17.
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