Econ 604
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
[pic]
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
Solving dY/dX = -Y/X so 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)
3.5 Graph 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 implies dY/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 implies dY/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 implies dY/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
Solving dY/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 U110. 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 ................
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