EC2010-Intermediate Economics Trinity College Dublin

Microeconomics Module Department of Economics

Lecturer: Martín Paredes Hilary Term 2007

Solutions For Assignment # 3

1.  Chapter 3, Review Question # 10

a)

b)

The indifference curve in this case will be convex toward the origin. The marginal rate of substitution is measured as the absolute value of the slope of a line tangent to the indifference curve. As can be seen in the graph above, this slope becomes less negative as we move down the indifference curve, implying a diminishing MRS.

c) If the MRS was constant, this would imply that at any consumption level the consumer would be willing to trade a fixed amount of one good for a fixed amount of the other. This occurs with perfect substitutes.

d) If the consumer wishes to always consume goods in a fixed ratio, then the goods are perfect complements. In this case, the indifference curves will be L-shaped.

2.  Chapter 3, Problem # 3.19

a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive.

b) The marginal utility of increases as the consumer buys more .

c)

d) As the consumer substitutes for , the will diminish.

e) Since it is possible to have U > 0 if either x = 0 (and y > 0) or y = 0 (and x > 0), the indifference curves intersect both axes.

f) The slope of a typical indifference curve at some basket is the . At , . Note that this holds regardless of the value of . Therefore, the slope of any indifference curve at will be .

3.  Chapter 4, Review Question # 1

Relative to any point on the budget line, when the consumer has a positive marginal utility for all goods she could increase her utility by consuming some basket outside the budget line. However, baskets outside the budget line are unaffordable to her, so she is constrained (as in “constrained optimization”) to choosing the most preferred basket that lies along the budget line.

4.  Chapter 4, Review Question # 6

At an interior optimum, the slope of the budget line must equal the slope of the indifference curve. This implies

This can be rewritten as

which is known as the “bang for the buck” condition. If this condition does not hold at the chosen interior basket, then the consumer can increase total utility by reallocating his spending to purchase more of the good with the higher “bang for the buck” and less of the other good.

5.  Chapter 4, Problem # 4.2

a) If Ann is spending all of her income then

b)

c) Yes, the indifference curves are convex, i.e., bowed in toward the origin. Also, note that they intersect the F-axis.

d)

e)  The tangency condition requires that

Plugging in the known information yields

Substituting this result into the budget line, results in

Finally, plugging this result back into the tangency condition implies that . At the optimum the consumer chooses 5 units of clothing and 12 units of food.

f) The marginal rate of substitution is equal to the price ratio.

g) Yes, the indifference curves do exhibit diminishing . We can see this in the graph in part c) because the indifference curves are bowed in toward the origin. Algebraically, . As increases and decreases along an isoquant, diminishes.

6.  Chapter 4, Problem # 4.3

This question cannot be solved using the usual tangency condition. However, you can see from the graph below that the optimum basket will necessarily lie on the “elbow” of some indifference curve, such as (5, 3), (10, 6) etc. If the consumer were at some other point, he could always move to such a point, keeping utility constant and decreasing his expenditure. The equation of all these “elbow” points is 3x = 5y, or y = 0.6x. Therefore the optimum point must be such that 3x = 5y.

The usual budget constraint must hold of course. That is, . Combining these two conditions, we get (x, y) = (20, 12).

7.  Chapter 4, Problem # 4.5

See the graph below. The fact that Helen’s indifference curves touch the axes should immediately make you want to check for a corner point solution.

To see the corner point optimum algebraically, notice if there was an interior solution, the tangency condition implies (S + 10)/(C +10) = 3, or S = 3C + 20. Combining this with the budget constraint, 9C + 3S = 30, we find that the optimal number of CDs would be given by which implies a negative number of CDs. Since it’s impossible to purchase a negative amount of something, our assumption that there was an interior solution must be false. Instead, the optimum will consist of C = 0 and Helen spending all her income on sandwiches: S = 10.

Graphically, the corner optimum is reflected in the fact that the slope of the budget line is steeper than that of the indifference curve, even when C = 0. Specifically, note that at (C, S) = (0, 10) we have PC / PS = 3 > MRSC,S = 2. Thus, even at the corner point, the marginal utility per dollar spent on CDs is lower than on sandwiches. However, since she is already at a corner point with C = 0, she cannot give up any more CDs. Therefore the best Helen can do is to spend all her income on sandwiches: (C, S) = (0, 10). [Note: At the other corner with S = 0 and C = 3.3, PC / PS = 3 > MRSC,S = 0.75. Thus, Helen would prefer to buy more sandwiches and less CDs, which is of course entirely feasible at this corner point. Thus the S = 0 corner cannot be an optimum.]

8.  Chapter 4, Problem # 4.7

a)  The budget line will have a kink where round trips = 10 and other goods = 5,000. Northwest of the kink, the budget line’s slope will be –500 . Southeast of the kink, the slope will be –200.

b) With the indifference curves drawn on the above graph, Toni is better off with the frequent flyer program (at point B) than she would be without it (at point C). Without the frequent flyer program the best she could achieve is point C, which lies on the hypothetical budget line where the price of round trips is always $500.

c) With the indifference curves drawn on graph below, Toni is no better off with the frequent flyer program than she would be without it (at point A). At this point, her indifference curve is tangent to a portion of the budget line where the frequent flyer program does not apply (less than 10 round trips).

9.  Chapter 4, Problem # 4.8

a)

This utility function has a diminishing marginal rate of substitution since declines as H increases and M decreases.

b)

Since it is possible to have U > 0 if either H = 0 (and M > 0) or M = 0 (and H > 0), the indifference curves will intersect both axes.

c)  We know from the tangency condition that

Substituting this into the budget line, , yields

Finally, plugging this back into the tangency condition implies . At the optimum the consumer will choose 4 hamburgers and 16 milkshakes. This can be seen in the graph above.

10.  Chapter 4, Problem # 4.12

a) In this case, . If Jack neither borrows nor lends, then MRSx,y = 1050/(2*1000) = 0.525. Recall that if the interest rate is r, the slope of the budget line is –(1+r) = –1.05. Thus, if he neither borrows nor lends it will be the case that MRSx,y < 1 + r. That is, the “bang for the buck” for spending this month (good x) is less than that for spending next month (good y). Thus, Jack should lend some of his income this month (so x < 1000) in order to earn interest and have higher spending next month (y > 1050).

b) Now MRSx,y = 2y/x. If Jack neither borrows nor lends, MRSx,y = 2.1 > (1 + r). Thus, Jack could increase his utility by borrowing in the first month (so that x > 1000 and y < 1050).

c) Now MRSx,y = y/x. If Jack neither borrows nor lends, MRSx,y = 1.05 = (1 + r). Thus, Jack’s utility is maximized when he neither borrows nor lends, simply spending all of his income in each month: (x, y) = (1000, 1050).

11.  Chapter 4, Problem # 4.13

The utility function implies that MRSC1,C2 = C2 / C1. At point A, MRSC1,C2 = 1.10, which lies between (1 + rL) = 1.05 and (1 + rB) = 1.12. Therefore Meg will neither borrow nor lend and will simply spend her entire income each month.

If the borrowing rate falls to 8%, then the lower part of the budget line pivots outward, as depicted in the graph below. Then at point A, MRSC1,C2 > (1+ rB) > (1 + rL) since 1.10 > 1.08 > 1.05. So Meg can increase her utility by moving away from point A to a point like B, where she borrows money, spending more than her income this month (C1 > 2000) and less than her income next month (C2 < 2200).

12.  Chapter 4, Problem # 4.17

Let x denote the number of phone calls, and y denote spending on other goods. Under Plan A, Darrell’s budget line is JLM. Under Plan B, it is JKLN. These budget lines intersect at point L, or about x = 67.

If we know that Darrell chooses Plan A, his optimal bundle must lie on the line segment JL. No point between L and M would be optimal under this plan because then Darrell could have chosen a point under Plan B, between L and N, that would have given him more minutes, and left him with more money to buy other goods. However, we cannot exclude point L itself (Darrell could, for instance, have perfect complements preferences with an “elbow” at point L). Thus, if Darrell chooses Plan A his optimal basket could be anywhere between J and L, including either of these points.

Similarly, if he chose Plan B then his optimal basket must lie between L and N. Any point between L and K (but not including point L) would be dominated by a point under Plan A between L and J. Thus, if Darrell chooses Plan B his optimal basket could be anywhere between L and N, including either of these points.

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