a. (4 points) What is the MRS for the point on Bobby s demand curve when the price of snacks is $0.50? Show your work.

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1 1. (11 points The figure shows Bobby's indifference map for juice and snacks. Also shown are three budget lines resulting from different prices for snacks assuming he has $20 to spend on these goods. a. (4 points What is the MRS for the point on Bobby s demand curve when the price of snacks is $0.50? Show your work. MRS = p1 I = p2 q1 I q2 = $20 40 $20 20 = = points for correct answer -2 points for showing work b. (7 points Draw Bobby s demand curve for snacks on the graph below. Be sure to label your axes and all points used to draw the curve. P -1 point for each point on curve -1 point for labeling axes correctly $2 $1 $ q

2 2. (11 points The figure below shows John's indifference map for soda and juice. B1 indicates his original budget line. B2 indicates his budget line resulting from a decrease in the price of soda. a. (2 points How much of the change in quantity is due to the substitution effect? units -2 points for answering, can give 1 point for answering 7 b. ( points How much of the change in quantity is due to the income effect? Is soda a normal or inferior good? 7 units, normal good. -2 points for answering 7, can give 1 point for answering -1 point for normal c. (6 points Write the Slutsky equation (in either of its forms and use it to show that a Giffen good must be an inferior good, but an inferior good need not be a Giffen good. The Slutsky equation may be written as dd/dp Total = dh/dp - D(dD/dY. (Or some equivalent version, where D is uncompensated and H is compensated demand. Or, it can be written ε = ε* - θξ. ( points For a Giffen good, dd/dp Total (or ε is positive, which implies that - D(dD/dY (or - θξ must be positive and large enough to offset dh/dp, (ε* which is always negative. For any inferior good, however, - D(dD/dY is positive but not necessarily large enough to make dd/dp Total positive. ( points

3 1. (16 points A Californian student consumes Internet services (I and books (B. Her preferences are represented by a Cobb-Douglas utility function of the type: U(I,B = I0.6B0.4. Initially Y = 100, PI = PB = 1. Lately, however, because of the electricity shortage, the price of the Internet services has increased to 2. The government has decided to give a transfer to the student so that she can recover her initial welfare. In order to determine the transfer the government has hired three consultants who have made the following suggestions: Consultant A: The transfer should allow the student to buy her initial bundle. Consultant B: The transfer should allow the student to get her initial level of utility. Consultant C: The government should give her a transfer of 20. a. (6 points Find the amount of the transfer implied by consultant A. (For this and all subsequent parts of the question, show your work and calculate all relevant functions, e.g. the demand and expenditure functions. To solve for the consumer s demand for both goods, must solve for the demand functions using either the Lagrangian or tangency condition. Either method gives the following: 1.5B I Plugging into the budget constraint gives: = p I p B (or the inverse of this equation p I I = 1.5p BB Y = 1.5p BB + p BB = 2.5p BB And therefore: B = 0.4Y p B = 40 I = 0.6Y p I = 60 Therefore, the transfer needed to afford her original bundle is T = $ $1 40 $100 = $60-5 points deriving quantities demanded - 2 points for calculating the amount of the transfer

4 b. (6 points Find the amount of the transfer implied by consultant B. Her original level of utility can be found from above: V = ( 0.6Y 0.6 ( 0.4Y = ( p I p B ( = (11.67(4.7 = 51 To figure out the amount of income needed to maintain this level of utility at the new prices, one could derive the expenditure function directly from the expenditure minimization Lagrangian or from the principle of duality. You should get: E = U p I 0.6 p B 0.4 This then gives And the transfer should be E = (20.6 (1 0.4 = $151 T = $151 $100 = $51 - points for finding correct utility - 2 points for figuring out amount of income needed to achieve that utility at new prices (this can be done less formally than the solution above - 1 point for calculating transfer c. (4 points Determine whether the consumer is better or worse off from Consultant C's suggestion than before the price increases. Definitely worse off since $20<$51, which is the amount needed to get consumer back to her original utility after the price increase. Credit: A clear discussion of this is worth full credit, the correct answer with no discussion is worth one point, and the correct answer with a not very clear explanation is worth two points. Students could also attempt to calculate the consumer s utility with $120 at the new prices and should find that it is less than 51. This is worth full credit if done correctly and worth 2 points if done incorrectly, regardless of the answer.

5 . (17 points Ed's utility from vacations (V and meals (M is given by the function U(V,M = V2M. His purchase of both are constrained by his income. a. (9 points Calculate the income and substitution effects of a change in meal prices. (For this and all subsequent parts of the question, show your work and calculate all relevant functions, e.g. the demand and expenditure functions. i Calculate the uncompensated demand: Max U(V,M=V 2 M s.t. Y=pvV+pmM Either use Lagrangian or tangency condition to get to: Plugging into the budget constraint gives: And therefore:m = Y 2M V = p v p v V = 2M Y = 2M + M = M ii Calculate compensated demand, either by using the expenditure minimization Lagrangian or the duality between V and E. Lagrangian Method: L = p v V + M + λ(u V 2 M This gives the same tangency condition as above: 2M V = p v Plugging in to the utility constraint: Duality Method: U = ( 2M p v 2 M = M = (2 p 2 M M p v U 1 ( p 2 v From part i find indirect utility V (not to be confused with V the demand for vacations Transforming this: V = ( 2Y 2 Y = p v 4Y 27p v 2

6 U = 4E 27p 2 v E = 2 1 U 1 p 4 1 v And then the compensated demand can be found from Shephard s Lemma: M = de = 1 U 1 d 4 1 ( p v p M With these two demands, you can easily calculate the substitution and income effects. Substitution Effect (note they should use compensated demand, may denote it with an H: M = Or, in elasticity form: ( U 1 ( p 2 v 2 = ( U 1 p v p 5 M ε = M M = 2 ( U 1 p v p 5 M ( 1 U ( p v p M 2 1 = 2 Income effect (note they should use uncompensated demand, may denote with D: Or, in elasticity form: θξ = M Y M M Y = Y 1 = Y 9 2 M Y Y M = p M M Y = p 1 M = 1 Note: Please don t penalize them for leaving the minus sign off here, although it should show up in the substitution effect. -4 points for income effect (including derivation of demand -5 points for substitution effect (including derivation of uncompensated demand Should give most partial credit for correct derivation of demand fns, since they are also used below

7 Last year, the price of vacations was $200 and the price of meals was $50. This year, the price of meals rose to $75, the price of vacations remained the same. Both years, Ed had an income of $1500. b. (4 points What is the compensating variation for the price change in meals? The CV is the amount of money needed to offset a consumer's harm from a price increase. Ed's utility before the price change is based on his optimal consumption bundle. M1 = 1500/*50 = 10 and V1 = 2*1500/*200 = 5. His utility is U(5,10 = 5210 = 250. Solving for M = 7.6. Solve for V = The expenditure required to purchase this bundle is: 75M + 200V = 75( (5.72 = Thus the CV is $1,500 - $1, = $ (approximately. Students can also solve using the integral of the compensated demand over the range of prices: 75 2 dpm = (approximately 75 Md 50 = U 1 ( p v -1 point for correct utility -2 points for correct CV function (either expenditure or integral -1 point for correct answer (negative ok c. (4 points What is the change in consumer surplus resulting from the change in P M? M* = 500/pM The change in consumer surplus is found from the integral: 75 CS = 500 dp = 500[ln(75 ln(50] = (approximately M 50 - points for correct CS function -1 point for correct answer (negative ok

8 4. (12 points The production function for hamburgers can be written as q = 0.1X + 0.1Y, where X is Canadian ground beef and Y is U.S. beef, both measured in pounds. a. (4 points Draw the isoquant for q=10 hamburgers on the graph below. Be sure to label axes and all relevant points. Y points for straight line -1 point each for correct intercepts 100 X b. (5 points Calculate the MRTS and elasticity of substitution for this production function. Show your work. MRTS = f x f y = = 1 -Note: If X and Y are flipped on the graph above, the inverse function would be the MRTS. σ = d k l MRTS = d k l d MRTS k 0 l MRTS k l = -Can also give some sort of reasoning about the MRTS not changing along the isoquant to get full credit provide they wrote out the initial formula. Also, don t penalize them for leaving off the absolute value brackets, as long as the answer is positive. -2 points for MRTS (1 for function and 1 for correct derivatives & answer. - points for ES (1 for function and 2 for correct exectution & answer.

9 c. ( points Does this function exhibit constant, increasing, or decreasing returns to scale? Show why. Constant. F(X,Y=0.1X+0.1Y F(2X,2Y (or F(tX,tY = 0.1*2X+0.1*2Y = 2(0.1X+0.1Y=2F(X,Y -1 point for constant -2 points for proof