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Problem Set #7
Suggested Answers to Problems 7.1; 7.3, 7.5 and 7.9
7.1 Imagine a market for X composed of four individuals: Mr. Pauper (P), Ms. Broke (B),
Mr Average (A), and Ms. Rich (R). All four have the same demand function for X: It is a
function of income (I), PX, and the price of an important substitute (Y) for X:
X 
IPY
2 PX
a. What is the market demand function for X? If PX = PY = 1, IP = IB = 16, IA = 25
and IR = 100, what is the total market demand for X? What is eX,PX? eX, PY? eX, I?
X = [IP(PY)] .5/[2(PX)] + [IBPY ] .5/[2(PX)] +[ IAPY] .5/[2(PX)] +[ IRPY] .5/[2(PX)] =
X = [16(1)] .5/[2(1)] + [16(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[100(1)] .5/[2(1)] =
=2
+
2
+
2.5
+
5
= 11.5
Let IAgg denote IP.5+ IB.5+ IA.5 + IR.5. Then
eX,PX =[X/Px][PX/X] = -[IAgg (1).5]/[2PX2] { PX/ X} = -1 (recalling the definition
of X)
by similar reasoning
eX, PY = [X/Py][Py/X] = (1/2)[IAgg (PY) -.5] /[2PX] { PY/ X} = 1/2
The income elasticity cannot be computed without knowing the distribution of
income changes.
b. If PX doubled, what would the new level of X demanded? If Mr. Pauper lost
his job and his income fell 50 percent, how would that affect the market demand for X?
What if Ms. Rich’s income were to drop 50%? If the government imposed a 100 percent
tax on Y, how would the demand for X be affected?
- Double PX to 2 and quantity demanded becomes [IAgg (1) .5] /[2(2)], or half the
previous quantity demanded (5.75)
- If IP = 8 then
X’ = [16(1)] .5/[2(1)] + [8(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[100(1)] .5/[2(1)] =
X’ = 2 + 2+ 2.5 + 5 = 10.91
- If IR = 50 then
X’ = [16(1)] .5/[2(1)] + [16(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[50(1)] .5/[2(1)] =
X’ = 2 + 2+ 2.5 + 2.52 = 10.03
-If Py + t = 2, then
X’ = [16(2)] .5/[2(1)] + [16(2)] .5/[2(1)] +[25(2)] .5/[2(1)] +[50(2)] .5/[2(1)] =
X’ =11.5(2).5 = 16.26
c. If IP = IB = IA = IR =25, what would be the total demand for X? How does that
figure compare with your answer to (a)? Answer (b) for these new income levels and PX
= PY = 1.
X’ = 4[25(1)] .5/[2(1)] = 10.
- Double PX to 2 and quantity demanded becomes [IAgg (1) .5] /[2(2)], or half the
previous quantity demanded (5)
- If IP falls by half (to 12.5) then
X’ = 3[25(1)] .5/[2(1)] + [12.5(1)] .5/[2(1)] =
X’ = 7.5 +1.77
= 9.57
- If IR falls by half, we repeat the above.
-If Py + t = 2, then
X’ = 4[25(2)] .5/[2(1)] = 102 = 14.1.
d. If Ms. Rich found Z a necessary complement to X, her demand function for X
might be described by the function
IPY
2PX PZ
What is the new market demand function for X? If PX = PY = PZ= 1 and income
levels are those described by (a), what is the demand for X? What is eX,PX? eX, PY? eX, I? eX,
PZ? What is the new level of demand for X if the price of Z rises to 2? Notice that Ms.
Rich is the only one whose demand for X drops.
X
Market demand
X = [IP(PY)] .5/[2(PX)] + [IBPY ] .5/[2(PX)] +[ IAPY] .5/[2(PX)] +[ IRPY]/[2(PX) (PZ)]
At current prices
X = [16(1)] .5/[2(1)] + [16(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[100(1)] /[2(1)(1)]
=
2
+ 2
+
2.5
+ 50
=
56.5
eX,PX =[X/Px][PX/X].
Here [X/Px] =
-1[IP(PY)] .5/[2(PX)2] + -1[IB(PY)] .5/[2(PX)2]+ -1[IA(PY)] .5/[2(PX)2]+-1IRPY/[2(PX)2PZ]
and, with current parameters
= -1[16(1)] .5/[2(1)] +-1 [16(1)] .5/[2(1)] +-1 [25(1)] .5/[2(1)] +-1 [100(1)] ./[2(1)]
=-1(56.5)
= -X/PX. Thus,
eX,PX = [X/Px][PX/X] = -1
eX, PY = [X/Py][Py/X] =
Here X/Py =
.5[IP.5(PY)- .5] /[2PX] + .5[IB.5(PY)- .5] /[2PX] +.5[IA.5(PY)- .5] /[2PX] + IR /[2PX PZ]
= .5X + .5IR /[2PX PZ]
Thus
eX, PY = {.5X + IR /[2PX PZ]} [Py/X]
=[.5(56.5) + 100/2]1/56.5
= .5 + .884
= 1.38
-Income elasticity Again, income elasticity cannot be determined without knowing the
distribution of an income change
Finally suppose that the PZ increases to 2. Then new market demand becomes
X
= [16(1)] .5/[2(1)] + [16(1)] .5/[2(1)] +[25(1)] .5/[2(1)] +[100(1)] /[2(1)(2)]
=2
+
2
+ 2.5
+ 25
= 31.5
7.3 Tom, Dick and Harry constitute the entire market for scrod. Tom’s demand curve is
given by
Q1 – 100 – 2P
for P< 50. For P>50, Q1 = 0. Dick’s demand curve is given by
Q2 – 160 – 4P
for P<40. For P>40, Q2 = 0. Harry’s demand curve is given by
Q3 – 150 – 5P
for P<30. For P>30, Q3 = 0. Using this information, answer the following:
a. How much scrod is demanded by each person at P = 50? At P = 35? At P =
25? At P = 10? At P =0?
Price
50
35
25
10
0
Tom (1)
0
30
50
80
100
Dick (2)
0
20
60
120
160
Harry (3)
0
0
25
100
150
Market
0
50
135
300
410
b. What is the total market demand for scrod at each fo the prices specified in part
(a)?
See the rightmost column in the table above.
c. Graph each individual’s demand curve
Tom
60
Dick
60
50
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
0
0
200
0
0
400
Harry
60
200
400
0
200
400
d. Use the individual demand curves and the results of part (b) to construct the
total market demand curve for scrod. Summing horizontally,
Market
60
50
40
30
20
10
0
0
200
400
P
7.5 For this linear demand, show that the price elasticity of demand at any given point
(say point E) is given by minus the ratio of distance X to distance Y in the figure. How
might you apply this result to a nonlinear demand curve?
Our task is to show that
[X/P] [P/X] = -X/Y
D
D
Y
Notice first that X is the distance
from the origin to P*, or P*.
Distance Y is the distance from the
intercept to P*, or Po – P*.
E
P*
Y0
Now, a linear demand curve may
be written as
X
Q*
Q = a – bP, b>0. Inverting, P* =
a/b – Q*/b. Thus, When Q* = 0, Po
Q
= a/b. At Q*, P* = a/b – Q*/b. Differencing,
Po – P*
Thus X/Y
=
=
Q*/b
=
P*/[Q*
a/b
-
[a/b
-
Q*/b]
/b]
=
b[P*/Q*], where b = Q/P
7.9 In Example 7.2 we showed that with 2 goods the price elasticity of demand of a
compensated demand curve is given by
esX PX = -(1-sx)
where sx is the share of income spent on good X and  us the substitution elasticity. Use
this result together with the elasticity interpretation of the Slutsky equation to show that:
a. if =1 (the Cobb-Douglas case),
eX PX + eY PY = -2
b. >1 implies eX PX + eY PY < -2 and <1 implies eX PX + eY PY > -2. These results
can easily be generalized to cases of more than two goods.
Both a and b are answered similarly. The Slutsky equation implies for X and Y
that
eX PX =
esX PX + sx eX I
and
eY PY = esY PY + sY eY I
Adding the two Slutsky equations together
eX PX +
eY PY = esX PX + esY PY -
sx eX I - sY eY I
Now, by Engel’s law,
sx eX I + sY eY I = 1.
Thus
eX PX +
eY PY = esX PX + esY PY -
1
Finally, for either X or Y compensated demand may be written esX PX = -(1-sx)
or esY PY = -(1-sY). Inserting
eX PX +
eY PY
=
-(1-sx) -(1-sY)
=--1
-
1
(The latter expression since the sum of shares equals 1)
Thus  = 1 implies the sum of own price elasticities equals -2, and  <1 implies
that the sum of own price elasticities is <-2.
Finally, it is useful to consider more explicitly the definition of the elasticity of
substitution parameter, . In the utility context

= [(Y/X)/(Y/X)] / [(Py/Px)/(Py/Px)]
That is, the substitution elasticity measures the percentage change in relative
input use induced by a one percent change in relative prices. Thus, the expression
esX PX = -(1-sx) = implies that the compensated elasticity of demand for a good is
a the extent to which consumers shift away from X as the relative price of X increases,
() adjusted by the prominence of X in the consumption bundle.