Econ. 410
Spring 2008
Tauchen
Answers to Practice Problem -- Applications of Consumer Theory
Questions 9-14
9. Let’s again consider the standard labor leisure choice model. The wage that Howard receives depends upon whether he works part-time or full-time. If he works part-time, defined as six hours per day or less, he earns a wage of $6 per hour. If he works full time, defined as more than six hours per day, he earns a wage of $12 per hour. Construct his budget set. Show an example of preferences for which he works part time and preferences for which he works full-time.
Answer: The graphs below show the budget line (dark) and a portion of the indifference map. The different wages for part-time and full-time work produce a kink in the budget line. For the graph on the left, the individual selects to work full-time. For the graph on the right, the individual selects to work part time.
Compare the marginal rates of substitution between leisure and good Y for the two individuals at a bundle such as 18 hours of leisure and 100 units of good Y. The Marginal Rate of Substitution between leisure and good Y at the bundle (18,100) is much smaller for the individual on the left than for the individual on the right. ( The individual on the left has a much flatter indifference curve at this bundle than does the individual on the right.) The person on the right must be offered a great deal of good Y in order to give up an additional unit of leisure. Even the full-time wage is insufficient to induce the individual to give up an additional hour of leisure. The individual selects to work only part time. A person in his/her mid 60s who has already put his/her children through college, owns his/her house, and has some minor health problems might require a large amount of good Y to give up an hour of leisure. (After all, we have to be sure that we see every rerun of Law and Order before biting the dust.) A younger person, who is saving for a first house might be willing to give up an hour of leisure in return for fewer units of good Y. The younger person has preferences as shown in the graph on the left and works full-time.
10. As shown previously, the three quarterbacks have different utility functions but the same preferences. Faced with the same budget lines, all of them make the same choices, and thus their demand functions are identical.
To verify that all three have the same demand schedules, we will derive the demand curves for each.
Eli: His utility is U(x,y)=xy. For this utility function, MUx(x,y) =y, MUy(x,y)=x, and MRSxy(x,y)=y/x.
The conditions satisfied at the optimum are
MRSxy(x,y) = px /py and px x + py y = I.
The above are two equations in two unknowns, and we use standard algebra techniques to solve.
Since MRSxy(x,y)=y/x, the equality MRSxy(x,y) = px/py implies that y/x = px/py. Solving for y yields,
y= x px/py. Substitute this equality into the budget constraint,
pxx + pyx px/py = I.
The last step is to solve this equality for x or x=I/(2px), which is the demand curve for good X. To find the demand curve for good Y, use the relationship between x and y derived above: y= x px/py .
Substituting from the demand curve for good x, we obtain
Peyton: His utility is U(x,y)=x2 y2 . For this utility function, MUx(x,y) =2 xy2, MUy(x,y)= 2xy2, and MRSxy(x,y)=y/x. Although Eli’s and Peyton’s utility functions differ, the conditions satisfied at the optimum are identical: MRSxy(x,y) = px /py and px x + py y = I. These two equations are identical to the two equations that we solved above and lead to the same demand functions.
Tom: His utility function is U(x,y) = x.5 y.5 . For this utility function, MUx(x,y )=.5 x-.5 y.5,
MUy(x,y)=.5 x.5 y-.5, and MRSxy(x,y)=y/x. The conditions which hold at the optimum are the same as for Eli and Peyton. All three have the same demand functions.
11. Holden’s utility function is U(x,y) = a lnx + y. For this utility function, MUx(x,y) = a/x, MUy(x,y)=1, and MRSxy(x,y) = a/x.
The two equalities satisfied at an optimum (for which the individual consumes positive amounts of good X and Y), are
MRSxy(x,y) = px /py and px x + py y = I.
Since MRSxy(x,y) = a/x for Holden’s utility function, the two equalities satisfied at an optimum are
a/x = px/py and px x + py y = I.
Solving the first equation for x,
x=apy/px.
This is the demand equation for good X. For this utility function, the demand for good X does not depend upon income.
To find the demand equation for good Y, substitute the above expression for x into the budget constraint to obtain
apy + pyy = I.
Thus,
y = (I-apy)/py .
The reason for the assumption that I-apy is now apparent. This assumption is required in order to be sure that the demand for good Y is not negative.
12. Holden’s demand is U(x,y) = a x1/4 +y1/4. For this utility function, MUx(x,y) = ax-3/4/4,
MUy(x,y) = y-3/4/4, and MRSxy(x,y) = ay3/4/x3/4.
The two equalities satisfied at an optimum (for which the individual consumes positive amounts of good X and Y), are
MRSxy(x,y) = px /py and px x + py y = I.
Since MRSxy(x,y) = ay3/4/x3/4, the above two equalities are equivalent to
ay3/4/x3/4 = px/py and px x + py y = I.
The first equation may look a little complicated but we use standard algebra in order to express it in a form such that we can solve for y. Raise both sides of the first equality above to the 4th power to obtain,
a4 y3/x3 = (px/py)4.
Next, raise both sides of the equality to the 1/3 power,
a4/3 y/x = (px/py)4/3.
Tus, the two equalities satisfied at an optimum are
a4/3 y/x = (px/py)4/3 and px x + py y = I.
The remainder of the algebraic work is very similar to that for other problems. Solve the first equality above for y to obtain
y = (px/py)4/3 x / a4/3.
Substitute this expression for y into the budget constraint,
px x + py (px/py)4/3 x / a4/3 = I.
Although the above equation is a bit messy, the equation is linear in x. Solving for x,
Since y = (px/py)4/3 x / a4/3, the demand function for good Y is
13. a. The initial BL is BL-0 and the new budget line at the higher price for good X is BL-N. The initial optimum is the bundle A0=(2,6) and the new optimum is the bundle AN=(1,3) . The hypothetical budget line permits the individual to obtain the same utility as at (2,6) with the new higher price for good X. The bundle AH=(2,6).
The SE is the change from A0 to AH. With Leontief preferences, there is no substitution effect. The IE is the movement from AH to AN. The total effect of the price increase is the movement from A0 to AN, which is the income effect. There is no substitution effect and all of the changes in the consumption of good X and of good Y are attributable to the income effect.
b. The initial BL is BL-0 and the new budget line at the higher price of good X is BL-N. The initial optimum is the bundle A0=(2,10) and the new optimum is (approx.) the bundle AN=(1.2,8.4) . To find the income and substitution effects construct the hypothetical BL. The tangency of the hypothetical BL and the indifference curve for the new optimum is (approximately) the bundle AH=(1.2,11.2) . The SE for good X is the movement from (2,10) to (1.2, 11.2). The income effect is the movement from (1.2,11.2) to (1.2,8.4). With the parallel indifference curves, there is no income effect for good X.
Indeed, we can show that the income effect of an increase in the price of good X on the consumption of good X is zero for any preferences with parallel indifference curves (parallel with respect to the horizontal axis). In reasoning through this result, we will refer to the indifference curve that A-0 lies on as IC-0, the indifference curve that A-N lies on as IC-N, and the indifference curve that A-H lies on as IC-H.
In determining the IE and SE, we know that the new and hypothetical BLs have the same slopes. The new budget line is tangent to IC-N at A-N and the hypothetical BL is tangent to IC-H at A-H. Thus, the slope of the indifference curve IC-N at A-N is the same as the slope of indifference curve IC-H at the point A-H. Since the ICs are parallel, two ICs have the same slope only at the same X values. Thus, the amount of good X must be the same for the bundles A-N and A-H and the income effect for good X is zero.
14. a. The individual consumes the bundle (40,40).
b. The budget line is marked BL on the graph. It is the farthest out of the three budget lines. With this budget line, the individual selects the bundle (18,62) and is better off than when the water use costs were covered through the tax system and there was no explicit cost for the use of water.
c. With the $5 fee, the budget line is the middle budget line. The individual can obtain a higher IC than for the bundle (40,40).
d. As the metering cost increases and the fixed fee for water increases, the budget line shifts back. At the lowest budget line shown on the graph, the individual is equally well off with water metering as at the bundle (40,40), which is the outcome with no metering. If the metering cost were even higher, the individual can obtain higher utility without metering.
e. The issue of free blue books at the book store is exactly analogous. When blue books are free, students might pick up more than one blue book at a time and not keep track of the extras to use for the next several exams. (Faculty would be the same.) The “metering cost” for the blue books is the cost of having a clerk ring up the sale.