Problem Set Chapter 3 & 4 Solutions

Transcription

1 Problem Set Chapter 3 & 4 Solutions. Graph a tpical indifference curve for the following utilit functions and determine whether the obe the assumption of diminishing MRS: a. (, ) Slope -3 3 Since the indifference curves are not bowed towards the origin, the do not obe the assumption of diminishing MRS. b. (, ) 4 4 Since the indifference curves are bowed towards the origin, the do obe the assumption of diminishing MRS. Alternativel, we know M and M are both positive. So when quantit of X increases, quantit of Y must decrease. The MRS Y/X. So as X increase, the denominator gets bigger and MRS decreases. As X increase, Y decreases and the numerator gets smaller so MRS decreases. Both these effects work so that as X increase MRS decreasing.

2 c. (, ) Since the indifference curves are not bowed towards the origin, the do not obe the assumption of diminishing MRS. Alternativel, we know M is positive and M is positive, so as quantit of X increases, the quantit of Y must decrease. The MRS X/Y. So as X increases the numerator increases so MRS increases. As X increases Y decreases so denominator is getting smaller and MRS increases. Both these effects work in the same direction so MRS is increasing not diminishing. d. (, ) /3 / Since the indifference curves are bowed towards the origin, the do obe the assumption of diminishing MRS.

3 e. (, ) min(x, 3Y) This is an eample of perfect complements. The MRS is undefined. But lets graph the indifference curve, remember the L shaped. We need to find the corner point. To do this set the two elements of in the utilit function equal to each other so there is no etra X or Y being consumed that gives no etra utilit. X3Y rearrange YX/3 so ra from original which goes through all the corners of the L has to have the slope /3. The indifference curve is for when utilit is 6. Ra from the origin slope is / X

4 . Suppose a consumer s preferences for two goods can be represented b the Cobb- Douglas utilit function (, ) A, where A,, and are positive constants. a. What is MRS,? We begin b calculating the marginal utilities with respect to and : A (, ) M A (, ) M We can then use these marginal utilities to obtain MRS, : MRS, M M A A.

5 b. Is MRS, diminishing, constant, or increasing as the consumer substitutes for along an indifference curve? To determine this, we need to substitute for using the equation of the indifference curve so as to have MRS, epressed solel in terms of. The equation of the indifference curve is A, where represents a constant level of utilit. Solving this equation for gives us A A A Substituting for in our epression for MRS, ields +, A A MRS Since A,, and are positive constants, the first two terms in the equation above are also positive and constant. Moreover, the eponent on, +, is also positive and constant. Therefore, as increases, MRS, decreases. That is, MRS, is diminishing.

6 c. On a graph with on the horizontal ais and on the vertical ais, draw a tpical indifference curve. Indicate on our graph whether the indifference curve will intersect either or both aes. We know more is better because M and M are both positive; therefore, the indifference curves must be downward sloping. Moreover, we determined in part b that MRS, is diminishing; therefore, the indifference curves must be bowed in towards the origin. And finall, recall that the equation of a tpical indifference curve is given b A, where represents a constant level of utilit. Since for an > 0, it cannot be the case that either or equals zero, the indifference curves do not intersect either ais. These three observations indicate that the indifference map must be as follows:

7 3. Ch 3, Problem 3.6 For the following sets of goods draw two indifference curves, and, with >. Draw each graph placing the amount of the first good on the horizontal ais. a. Hot dogs and chili (the consumer likes both and has a diminishing marginal rate of substitution of hot dogs for chili) Chili Hot Dogs b. Sugar and Sweet N Low (the consumer likes both and will accept an ounce of Sweet N Low or an ounce of sugar with equal satisfaction) Sweet N Low Slopes - Sugar

8 c. Peanut butter and jell (the consumer likes eactl ounces of peanut butter for ever ounce of jell) Jell 4 Peanut Butter d. Nuts (which the consumer neither likes nor dislikes) and ice cream (which the consumer likes) Ice Cream Nuts

9 4. Julie has preferences for food, F, and clothing, C, are derived b a utilit function (F,C)FC. Food costs $ a unit and clothing costs $ a unit. Julie has $ to spend on food and clothing. a. Write the equation for Julie s budget line. What is the slope of the budget line? Julie s budget equation is: F + C, slope - P c P f b. Graph Julie s budget line. Place food on the vertical ais and clothing on the horizontal ais. c. On the same graph, draw an indifference curves that is tangent to his budget line. F 6 C

10 d. Julie is a utilit maimizer, write the objective function. e. Write down the full optimization problem with the objective function and the constraint. Her objective function is her utilit function (F,C)FC. The full optimization problem is Ma (F,C) FC C,F s.t. F+C f. sing calculus and algebra, find the basket of food and clothing that maimizes Julie s utilit (i.e. solve the maimization problem ou wrote down in e) (Assume Julie can purchase fractional amounts of both goods.) Ma (F,C) FC C,F s.t. F+C Step : using the budget constraint solve for F or C: F- C Step : Substitute the budget constaint into the utilit function so the utilit function is a function of one good. MAX (C)(-C)CC-4C C Step 3: Now we need to maimize with respect to C. 4C 0 C C 3 Step 4: Sub C back into budget constraint to figure out optimal F F+(3) so F6 The basket which optimizes Julie s utilit funtion is C3 and F6.

11 5. Each da Peter, who is in the third grade, eats lunch at school. He onl likes liver (L) and onions (N), and these provide him a utilit of ( L N ) ln( LN),. Liver costs $4.00 per serving, onions cost $.00 per serving, and Peter s mother gives him $8.00 to spend on lunch. a. Give the equation for Peter s budget line. What is the slope of the budget line? Peter s budget line is 4L + N 8. The slope of the budget line is P N /P L /. b. Graph Peter s budget line. Place the number of liver servings on the vertical ais and the number of onion servings on the horizontal ais. L Slope - ½ 4 N c. On the same graph, draw several of Peter s indifference curves, including one that is tangent to his budget line. L 3 4 N

12 d. sing calculus and algebra, find the basket of liver and onions that maimizes Peter s utilit. (Assume Peter can purchase fractional amounts of both goods.) Mark this basket on our graph. REMEMBER YO MAY NOT SOLVE THE PROBLEM THE WAY THE BOOK DOES, YO MST SET P THE MAXIMIZATION PROBLEM AND SOLVE THAT. Set up the Maimization Problem Ma ( L, N) ln( LN) LN, st.. 4L+ N 8. Rewrite budget constraint so N is in terms of L and L in terms of N N 4 - L 3. Sub the rewritten budget constraint back into the objective function and rewrite the maimization. ( ) Ma ( L) ln L*(4 L) L 4. Maimize the utilit function with respect to L. To do this ou need to find the slope of the utilit function (take the derivative w.r.t L) and set the slope equal to ). Hint before ou do this ou want to epand the utilit function to be ln( 4L -L ). *(4 4 L) 0 L L(4 L) 4 4L 0 L(4 L) 4 4L 0 * L 5. To find N*, sub L* back into the budget constraint. N () N* Therefore the optimal bundle is (L*, N* ) (, ). Therefore, the optimal basket is ( L, N ) (, ) the figure below:. The optimal basket is marked in

13 L Optimal Basket 3 4 N

14 6. Ch 4, Problem 4.6 Jane likes hamburgers (H) and milkshakes (M). Her indifference curves are bowed in and toward the origin and do not intersect the aes. The price of a milkshake is $ and the price of a hamburger is $3. She is spending all her income at the basket she is currentl consuming, and her marginal rate of substitution of hamburgers for milkshakes is. Is she at an optimum? If so, show wh. If not, should she bu fewer hamburgers and more milkshakes, or the reverse? Note: here we have to use the tangenc condition since we are not given enough information for consumer optimization. From the given information, we know that P H 3, P M, and MRS H,M. Comparing the MRS H,M to the price ratio, PH 3 MRS H, M <. P Since these are not equal, Jane is not currentl at an optimum. In addition, we can sa that which is equivalent to P P H H > MRS H, M, M M M H M M M M M > H. P M That is, the bang for the buck from milkshakes is greater than the bang for the buck from hamburgers. So Jane can increase her total utilit b reallocating her spending to purchase fewer hamburgers and more milkshakes. P

15 7. Ch 4, Problem 4.3 Toni likes to purchase round trips between the cities of Pulmonia and Castoria and other goods out of her income of $0,000. Fortunatel, Pulmonian Airwas provides air service and has a frequent-fler program. A round trip between the two cities normall costs $500, but an customer who makes more than 0 trips a ear gets to make additional trips during the ear for onl $00 per round trip. a. On a graph with round trips on the horizontal ais and other goods on the vertical ais, draw Toni s budget line. (Hint: This problem demonstrates that a budget line need not alwas be a straight line.) Let Y represent the quantit of other goods Toni purchases and let P Y represent the price of a unit of other goods. If Toni spends her entire income of $0,000 on other goods, she will obtain 0,000/P Y units of other goods. This gives us the Y intercept of the budget line. But what happens if Toni spends her entire income on round trip flights (X )? Toni has to pa $500 per flight for the first 0 flights that she bus, so those 0 flights will cost her $5,000, leaving her with $5,000. How man flights can she bu with the $5,000 she has left over? After the tenth flight, she can bu additional flights at $00 per flight. So, she can bu 5,000/00 5 additional flights at $00 a piece. In sum, if Toni spends her entire income of $0,000 on flights, she will obtain flights. This gives us the X intercept of the budget line. But what happens in between the two intercepts? We know that the slope of the budget line is P X /P Y. So, when 0 X 0, the slope of the budget line is 500/ P Y, but when 0 < X 35, the slope of the budget line is 00/ P Y. The change in slope generates the kink in the budget line at X 0. Other Goods (Y) 0,000/P Y 5,000/P Y Budget Line 0 35 Round Trips (X)

16 b. On the graph ou drew in part a, draw a set of indifference curves that illustrates wh Toni ma be better off with the frequent-fler program. Other Goods (Y) 0,000/P Y Optimal Bundle 5,000/P Y 0 35 Round Trips (X) With the indifference curves drawn on the above graph, Toni is better off with the frequent fler program than she would be without it. If there were no frequent fler program, the top portion of the budget line would etend straight down to the X ais instead of becoming flatter towards the bottom, and Toni would be unable to afford the optimal bundle shown in the graph above. c. On a new graph draw the same budget line ou found in part a. Now draw a set of indifference curves that illustrates wh Toni might not be better off with the frequent-fler program. Other Goods (Y) 0,000/P Y Optimal Bundle 5,000/P Y 0 35 Round Trips (X) With the indifference curves drawn on the graph above, Toni is no better off with the frequent fler program than she would be without it. If there were no frequent fler program, the top portion of the budget line would etend straight down to the X ais instead of becoming flatter towards the bottom but given the indifference curves drawn, this wouldn t affect Toni s choice of bundle.