Econ 604. Problem Set #2. Chapter 3, Problems 3.2, 3.4, 3.5, 3.7

Econ 604. Problem Set #2. Chapter 3, Problems 3.2, 3.4, 3.5, 3.7

3.2. Suppose the utility function for two goods, X and Y, has the Cobb-Douglas form

utility=U(X,Y)= (XY)1/2

a. Graph the U=10 indifference curve associated with this utility function

b. If X = 5, what must Y equal to be on theU=10 indifference curve? What is the MRS at this point?

X / Y / U
5 / 20 / 100
10 / 10 / 100
15 / 6.666667 / 100
20 / 5 / 100

Given U =10, we can write (XY)1/2=10 implies XY = 100. Solving, Y = 100/X. Thus

MRS = -dY/dX = 100/X2 With X=5, MRS = 100/25 = 4.

c. In general, develop an expression for the MRS for this utility function. Show how this can be interpreted as the ratio of the marginal utilities for X and Y.

U(X,Y)= (XY)1/2. Taking the total differential

dU=UXdX+UYdY=.5(Y/X).5dX +.5(X/Y).5dY=0

Solving the middle equality

dY/dX=-UX/UY

Solving the right most expression

.dY/dX=-Y/X so the MRS = Y/X (the opposite of dY/dX)

. Consider a logarithmic transformation of this utility function

U’=logU

Where log is the logarithmic function to base 10. Show that for this transformation the U’=1 indifference curve has the same properties as the U=10 curve calculated in parts (a) and (b). What is the general expression for the MRS of this transformed utility function?

U’=log (XY)1/2=.5logX + .5logY

Plotting ordered pairs when U’=1 yields

X / Y / U / logX / logY / U'
5 / 20 / 100 / 0.69897 / 1.30103 / 1
10 / 10 / 100 / 1 / 1 / 1
15 / 6.666667 / 100.000 / 1.176091 / 0.823909 / 1
20 / 5 / 100 / 1.30103 / 0.69897 / 1

Obviously indifference curves are the same for each utility function.

One can totally differentiate U’to obtain the same general expression for the MRS as before:

dU’=(.5/X)dX+ (.5/Y)dY=0

SolvingdY/dX=-Y/Xso the MRS = Y/X

3.4 For each of the following expressions, state the formal assumption that is being made about the individual’s utility function.

a.It (margarine) is just as good as the high-price spread (butter).

MRSmb=1, where m = margarine and b = butter.

b.Peanut butter and jelly go together like a horse and carriage

Peanut butter and jelly are perfect complements. That is

U(peanut butter, jelly) =min{peanut butter, jelly}

Where the terms “peanut butter”and “jelly”refer to servings of each product.

c.Things go better with Coke.

Coca Cola is a complement for all goods. That is, for any good x

Ux, coca cola>0

d.Popcorn is addictive – the more you eat, the more you want.

Popcorn consumption exhibits increasing marginal utility, e.g.,

Upopcorn >0.

e.Mosquitoes ruin a nice day at the beach.

Let the utility of the day at the beach be U(beach)>0. Then the utility of a day at the beach with mosquitoes is U(beach, mosquitoes) = 0. Thus, it must the case the that the marginal utility of a day at the beach just equals the marginal disutility of mosquitoes.

f.A day without wine is like a day without sunshine. The marginal (more precisely the incremental) utility of a “wine”just equals the marginal (incremental) utility of sunshine in a day.

g. It takes two to tango. “tango”dancing and a partner are perfect complements in consumption. U(tango, partner) = min(tango, partner)

3.5Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether they obey the assumption of a diminishing MRS)

a.U = 3X + Y

Here the MRS = -dY/dX = -3. The MRS is a constant, and does not exhibit diminishing MRS

b. U = (XY).5

X / Y / U / MRS
5 / 20 / 10 / 4
10 / 10 / 10 / 1
15 / 6.6666667 / 10 / 0.444444
20 / 5 / 10 / 0.25

Here MRS is –dY/dX = X/Y. As seen in the rightmost column of the above table, this does exhibit diminishing MRS

c.U=(X2 + Y2).5

Suppose we confine attention to constant increments of X and a utility level of 28.28.

X / Y / U / MRS
20 / 20 / 28.284271 / 1
15 / 23.976 / 28.281594 / 0.625626
10 / 26.455 / 28.28192 / 0.378
5 / 27.836 / 28.281494 / 0.179624

Here the utility function is obviously concave, implying an increasing MRS. More formally,

dU = X(X2 + Y2)-.5dX+ Y(X2 + Y2)-.5dY =0 impliesdY/dX=- X/Y. Values are shown in the rightmost column of the above table. Notice that the MRS moves directly with X (Constant increments of X require giving up increasing increments of Y)

d. U=(X2 - Y2).5

Plotting some points

X / Y / U / MRS
20 / 17.315 / 10.009534 / -1.15507
15 / 11.175 / 10.005967 / -1.34228
12 / 6.63 / 10.002155 / -1.80995
11 / 4.55 / 10.014864 / -2.41758
10 / 0 / 10 / #DIV/0!

Graphically

Here, notice the Y is a “bad.”Thus, the slope of the MRS is positive. This does not exhibit diminishing MRS More formally,

dU = X(X2 - Y2)-.5dX- Y(X2 - Y2)-.5dY =0 impliesdY/dX=X/Y. Values are shown in the rightmost column of the above table.

e. U = X2/3Y1/3

X / Y / U / MRS
20 / 2.5 / 10 / 0.25
15 / 4.45 / 10.004165 / 0.593333
10 / 10 / 10 / 2
5 / 40 / 10 / 16

This is another variant of a Cobb-Douglas function. The function does exhibit diminishing MRS Formally,

dU = (2/3)X-1/3Y1/3)dX+ (1/3) X2/3Y-1/3)dY =0 impliesdY/dX=-2Y/X. Values are shown in the rightmost column of the above table.

f. U = log X + log Y. We analyzed this function in problem 3.2(d). Looking the table shown below, it is obvious that the MRS for this function is the same as for 3.5(b).

X / Y / U / MRS / logX / logY / U'
20 / 5 / 10 / 0.5 / 1.30103 / 0.69897 / 1
15 / 6.6666667 / 10 / 0.888889 / 1.1760913 / 0.8239087 / 1
10 / 10 / 10 / 2 / 1 / 1 / 1
5 / 20 / 10 / 8 / 0.69897 / 1.30103 / 1

Formally,dU=dX/X+dY/Y=0

SolvingdY/dX=-Y/X

3.7. Consider the following utility functions. Show that each of these has a diminishing MRS, but that they exhibit constant, increasing and decreasing marginal utility, respectively. What can you conclude.

a. U(X,Y)=XY

MRS:dU=YdX+XdY=0

Implies that dY/dX=-Y/X. This is diminishing.

Utility. Observe that the second order condition for concavity with a two variable function is U11<0 and U11U22 – U122 >0. In the above function U11 = 0, U22 = 0 which implies that utility increases at a constant rate.

b. U(X,Y)=X2Y2

MRS: dU=2XY2dX+2YX2dY=0

Implies that dY/dX=-Y/X. This is the same as above, diminishing.

Utility. In the above function U11 =2Y2 0, U22 = 2X2 and U12 = 4XY. Thus

U11 >0, and U11 U22 - U122 =4X2Y2 -16X2Y2 <0. Thus, this curve is not convex. (I suspect that it satisfies quasi-concavity). Notice, however, that it increases in both X and Y.

c. U(X,Y)=lnX + lnY

MRS: dU=dX/X+dY/Y=0

Implies that dY/dX=-Y/X. This is the same as above, diminishing.

Utility. In the above function U11 =-dX/X2 0, U22 = -dY/Y2 and U12 = 0. Thus

U11 <0, and U11 U22 - U122 =1/X2Y2 >0. This implies diminishing marginal utility.

Result: Convexity in X1 X2 space does not imply concavity in U(X1 X2 ) space.