Problem Set #8
Suggested Answers to Problems 6.9, 7.1; 7.3,7.5 and 7.9
6.9 A utility function is termed separable if it can be written as
U(X, Y) =U1(X)+U2(Y)
Where Ui’ > 0 and Ui”< 0, and U1 and U2 need not be the same function.
a. What does separability assume about the cross partial derivative UXY? Give an intuitive discussion of what word this condition means and in what situations it might be plausible.
The cross partial derivative must equal zero. Thus, increases in the quantity of one good available do not affect the marginal utility of another. Such an assumption is plausible when goods are essentially unrelated (neither substitutes nor complements). For example, the marginal utility a consumer may derive from making charitable contributions likely does not affect their marginal utility for a private good, such as an automobile.
b. Show that if utility is separable, neither good can be inferior.
In any equilibrium the optimal consumption bundle will be Ux/Px = Uy/Py. If income increases, consumers must consume more of one of the goods. But given Uii<0 increased consumption of one good will yield lower marginal utility for that good. The consumer will be out of an equilibrium unless they also consume more of the related good.
c. Does the assumption of separability allow you to conclude definitively whether X andY are gross substitutes or gross complements? Explain
No. Even if consumers don’t directly adjust one good with the other, income effects may make the goods either gross substitutes, or gross complements. As in part b, the FONC implies Ux/Px = Uy/Py. If, for example PY goes up, then the ratio Uy/Py is out of balance. On the one hand, the individual might consume less Y thus raising Uy and restoring equality. In this process the consumer may also compensate for the price increase by economizing on x as well, making the goods gross complements. On the other hand, the consumer may respond by increasing consumption of x making them gross substitutes.
d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations
Suppose U(X,Y)=XY. This is clearly not separable (check out the cross partial derivatives). But a logarithmic transformation yields
U*(X,Y)=lnX + lnY
This utility function is clearly separable in X and Y. (Note, given that a utility specification is unique only down to a logarithmic transformation, the demand curves for U and U* will be the same. That should lend some insight into the restrictions imposed by separability)
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:
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 X = IAggPY.5/2PX.Thus,
eX,PX =[X/Px][PX/X] = - [IAgg(PY).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. That is, we can’t say anything about [X/IAgg]unless we know how IAgg changes. However, with an added assumption, we can make a calculation. For example, if all income increases by a constant percentage amount, a, then write IAgg = (a).5IAgg. Then a change in income is just a change in a, or
eX, a’ = [X/a’] [a’/X] = [.5(PY).5a-,5] /[2PX] {a/X} = 1/2
b. (i) If PX doubled, what would be the new level of X demanded? (ii) If Mr. Pauper lost his job and his income fell 50 percent, how would that affect the market demand for X? (iii) What if Ms. Rich’s income were to drop 50%? (iv) If the government imposed a 100 percent tax on Y, how would the demand for X be affected?
(i) Recall, demand equals X = IAgg PY.5/2PX.
Where IAgg = IP.5+ IB.5+ IA.5 + IR.5 and PX = PY = 1, IP = IB = 16, IA = 25 and IR = 100
Thus
X=(4 + 4+ 5+10)(1).5/(2PX). With Px = 1, X = 23/(2(1)) = 11.5 Double PX to 2 and quantity demanded becomes [23]/[2(2)], or half the previous quantity demanded (5.75)
(ii) If IP = 8 then
X’ = [22 + 4 + 5 + 10][1].5/[2(1)]
X’ = 2+2 +2 5+ 5 = 10.91
(iii) If IR = 50 then
X’ = [4 + 4 + 5 + 52][1].5/[2(1)]
= 2 + 2+ 2.5 + 2.52 = 10.03
(iv) -If Py + t = 2Py, then
X’ = IAgg 2.5PY.5/2PX. = 2X, or with our current parameters demand increases to
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.
(i) - Double PXto 2 and quantity demanded becomes [IAgg (1).5] /[2(2)], or half the previous quantity demanded (5)
(ii) 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
= 9.27
(iii) If IR falls by half, we repeat the above.
(iv) If Py + t = 2, then
X’ = 4[25(2)].5/[2(1)] = 102 = 14.1.
d. If Ms. Rich found Z a necessary complement to X, her demand function for X might be described by the function
(i) 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? (ii) What is eX,PX? eX, PY? eX, I?eX, PZ? (iii) 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.
(i) Market demand
X = [IP(PY)].5/[2(PX)] + [IBPY ].5/[2(PX)] +[IAPY].5/[2(PX)] +[IRPY]/[2(PX)(PZ)]
= [(IP.5+ IB.5+ IA.5)PZ + IR.5]PY.5/[2(PX)(PZ)] /
At current prices
X = [(16.5+16.5 +25.5)1+100.5](1) /[2(1)(1)]
= (4 + 4 + 5 + 10)/2 =23/2 = 11.5
(ii) Elasticities
eX,PX =[X/Px][PX/X].
Here let IAgg(Pz)’=[(IP.5+ IB.5+ IA.5)PZ + IR.5], thenX = IAgg(Pz)PY.5/[2PXPZ]
Then [X/Px]Px/X= - IAgg(Pz)PY.5/[2PX2PZ]{PX/X} = -1 (as before)
eX, PY =[X/Py][Py/X]
=.5IAgg(Pz)/[2PXPZ PY.5]
=.5 (again, as before)
-Income elasticity Again, income elasticity cannot be determined without knowing the distribution of an income change. However, unlike the previous case, a simple assumption of a constant percentage income changes will not suffice to calculate an aggregate income elasticity.
(iii) Finally suppose that the PZ increases to 2. Then new market demand becomes
Here let IAgg(2)’=[(IP.5+ IB.5+ IA.5)2 + IR.5], thenX = IAgg(2)PY.5/[2PX(2)]
X = [(16.5+ 16.5+25.5)2 + 100.5]1 /[2(1)(2)]
= (26+10](1)/4
= 36/4
= 9
7.3 Tom, Dick and Harry constitute the entire market for scrod.
Tom’s demand curve is given by Q1 – 100 – 2Pfor P 50. For P>50, Q1 = 0.
Dick’s demand curve is given by Q2 – 160 – 4Pfor P40. For P>40, Q2 = 0.
Harry’s demand curve is given by Q3 – 150 – 5Pfor P30. 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 / Tom (1) / Dick (2) / Harry (3) / Market50 / 0 / 0 / 0 / 0
35 / 30 / 20 / 0 / 50
25 / 50 / 60 / 25 / 135
10 / 80 / 120 / 100 / 300
0 / 100 / 160 / 150 / 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 (n.b. the right most two curves should not have kinks)
d. Use the individual demand curves and the results of part (b) to construct the total market demand curve for scrod.Summing horizontally,
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
[Q/P] [P/Q] = -X/Y
= P*/(Po-P*)
where Po is the price intercept.
Notice that -Y/Q* is the slope dP/dQ Observing that X is P, we immediately get
= -(Q*/Y)(X/Q*) = -X/Y
Given a linear demand curve, we may express demand as
Q = a – bP. Thus
= -bP/Q
= -P/[Q/b]
Now at point E, P is P*.
Solving the demand curve for P yields P = a/b – Q/b. P0 = a/b. Thus
= dQ/dP[P/Q]
= P/[Q(dP/dQ)]
More generally what does this imply about nonlinear demand curves?
It implies that for any curve at a point E, elasticity may be illustrated as the ratio –X/Y for the ray extending from E with the slope dP/dQ.
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 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.
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