MIDTERM EXAM ANSWER KEY

Transcription

1 MIDTERM EXAM ANSWER KEY ECON 10 PROFESSOR GUSE Instructions. You have (at least) hours to complete the exam. There are a total of 75 points on the exam. The exam is designed to take about 1 minute per point. You are allowed to have a single page of handwritten notes, -sided. You may not use any other notes, books or aids of any kind, be they human, electronic or mechanical. Calculations may be left in expression form for full credit. There is space provided for each question. If you need additional space, an extra sheet is provided at the end. You may also attach additional sheets. Please justify and explain your answers where needed and show your work. Please write your name on the exam itself and record the time you started and the time you finished. Finally, please turn in your cheat sheet with your exam. Name: Date and Time Started: Date and Time Finished: Full Pledge and signature: Date: February 5,

2 ECON 10 PROFESSOR GUSE (1) (a) (10 Points) Betty has no savings, but she can borrow at r =.05. What is the maximum that Betty would be willing to pay for an apartment building with the following characteristics. As always, you may leave your answer in expression form - no need to make the actual calculation. It earns $10,000 per year in rental income and Betty could expect the first $10,000 to come in exactly one year. It requires $3000 in regular annual maintenance and the previous owner just recently performed all the required annual maintenance. It will require a one-time upgrade of a new roof in 3 years at a cost of $50,000. ANSWER. The most Betty should be willing to pay would be WTP = t= (1.05) t = = , The first term reflect the present value of net annual revenue generated by the property. Note that this reduces to $ by applying the the formula for calculating the PV of an infinite stream of identical annual a = a. then we must simply subtract the discounted (1+r) t r value of the one-time roof replacement cost. Since the replacement will occur in three years, we divide the cost by, making the present value of that cost approximately $43,000 for a total of roughly $97,000. (b) (5 Points - Difficult) Suppose that the roof replacement is actually expected to occur every 30 years. (So that it will need to be replaced in 3 years, again in 33 years, again in 63 years, etc). By how much, if at all, would this reduce Betty s willingness to pay? ANSWER Just summing up the present value of all the repeated liabilities, this would reduce her willingness to pay by values: t=1

3 MIDTERM EXAM ANSWER KEY (1.05) 33 (1.05) 63 (1.05) = +30n n= = 0n n=1 = = = n=1 1 0n 1 (0 ) n n=1 [ ] Explanation. We get to the third line above by simply applying the rule that x (a+b) = x a x b. In the fourth line, we factor out bringing it outside the sum (since it doesn t depend on the summation index n). The fifth line is obtained by applying the rule that x ab = (x a ) b. The sixth line uses the rule that 1 n=1 = 1 (for x > 1, which is just x n x 1 a slight variation on the infinite value stream formula). Interpretation. This works out to be about $ In other words, if someone set aside $13,000 in an account that earns 5%, in would be enough to cover a $50,000 cost in 33 years and every 30 years after that! It is worth noting that about $10,000 of that is the present value of the roof replacement in 33 years. This means that the present value of all the remaining $50,000 replacements (at years 63, 93, 13...) amounts to about $3000 in present value terms. () Tamar has $400 to spend on rocks and shells. The price of rocks is $1 per pound. The price of shells is $ each for the first 50 shells, $1 each for the next 100 shells and $0.50 each for all shell beyond the first 150. (a) (10 Points) Draw a picture of Tamar s budget for rocks and shells. AN- SWERSeeFigure1. Thebudgetlineistheboldblacklinekinkedintwo locations. The portion of the line showing bundles containing between zero and 50 shells has a slope of since she must give up pounds of Rocks for each shell when the price of shells is $ at the margin. Between

4 4 ECON 10 PROFESSOR GUSE 50 and 150 shells, the slope of her budget line is 1 since the price for each shell is reduced to $1 each in that range. Finally the slope is 1 above 150 shells, since she only has to give up a 1 pound of rock for each shell when the price drops to $0.50 each. (b) (10 Points) Suppose that Tamar s preferences over combinations of rocks and shells is given by u(r,s) = R+S - where R is the quantity of rocks ands isthequantityofshells. Whichaffordablebundleofrocksandshell makes Tamar the happiest? ANSWER. Again, see Figure 1. Tamar s MRS function is MRS(R,S) = du/ds du/dr = 1 1 = 1 Hence all of Tamar s indifferencecurves all have a slope of1as shown in Figure 1 (solid blue lines). As usual with perfect substitute preferences, we have to consider corner solutions. It s pretty clear from the figure that (S,R) = (550,0) is the optimal choice at that income level, but let s try to prove it a bit more formally. Consider each segment of her budget line. Between zero and 50 shells, Tamar is willing to give up one shell for each pound of rock, but she only has to give up a 1 shell. Therefore, the direction of improvement on that segment of her budget line (as indicated by the red arrows in Figure 1) points toward (S,R) = (0,400) which is the best she can do on that segment. Between 50 and 150 shelss, Tamar s MRS is equal to the MRT making her indifferent to any bundle along this segment including the point (S,R) = (50,300), but we have already established that (0,400) (50,300), so we can safely say that Tamar will never choose a bundle containing between 50 and 150 shells. Between 150 and 550 shells, Tamar is willing to give up a pound of rock for each shell, but she only has to give up a 1 pound. Therefore the direction of improvment along this segment always points toward (S,R) = (550,0) So far we have shown that her optimal choice is either (0,400) - the corner involving all rock and no shells - or (550,0) - the corner involving all shells and no rock. To determine which of these is better, we simply evaluate the utility at each one and find that u(s = 550,R = 0) > u(s = 0,R = 400) (since 550 > 400). QED.

5 MIDTERM EXAM ANSWER KEY 5 (c) (5 Points - Difficult) For the price structure given above, draw a picture of Tamar s income expansion path. ANSWER. From the discussion in part (b), we know that Tamar will always either choose the corner involving all Rock and no Shells or the other corner involver all Shells and no Rock. We need to figure out how those corner bundles change as income changes. First note that the all-rock corner is easy; it is always equalto(s,r) = (0,m)forallincomelevels. Theothercorneristrickier. When Tamar has less than $00, she can t buy enough shells to get the $0.50 shells. Therefore, the third segment that appears in the previous part would not exist at those lower income levels. When m > 00, that third segment starts to appear and the all-shell corner bundle would be (S,R) = (150+ m 00,0) (since she would have to $00 to get the first shells, leaving m 00 to spend on 50-cent shells.). Summarizing, Tamar s optimal choice as a function of her income level, m, would be { (0,m) m > 150+ (S (m),r m 00 m < 50 (m)) =.5 (150+ m 00,0) m 50.5 In other words, when m < 50, she buys all Rocks; when m 50 she switches to all shells. See Figure for a depiction of the Income Expansion Path. (3) Consider Harold, a consumer/worker who has preferences over bundles of leisure time (x l ) and material consumption (x c ) given by the following utility function u(x l,x c ) = x 1 l x 1 c Harold s endowment is (γ l,γ c ). Assume throughout that γ l = 100 and that Harold can choose to work (descreasing his consumption of leisure for each hour that he works) any amount between 0 and 100 hours. When he faces a constant wage equal to w per hour for all the hours he works (i.e. no overtime), Harold s partial demand equations 1 are given by 1 partial in the sense that the demand equations generate the optimal choice when the optimal choice is an interior solution.

6 6 ECON 10 PROFESSOR GUSE Rocks Indifference Curves (MRS = 1 everywhere) 400 p s = 300 p s = 1 00 Budget Line Optimal Choice p s = Shells Figure 1. Tamar s Budget and Optimal Choice. Her budget line is the kinked bold black line. The figure also depicts Tamar preferences represented by the blue indifference curves. Since she has perfect substitutes preferences, her indifference curves are straight line in Rock Shell bundle space. x l (w,γ c ) = 100w +γ c w x c (w,γ c ) = 100w +γ c Answer the following questions. Be sure to explain your steps and interpret your answers. (a) (10 Points) Suppose that γ c = 600 and w = 6. Calculate Harold s demands for leisure and consumption and draw a picture of his choice

7 MIDTERM EXAM ANSWER KEY 7 Rocks Income Expansion Path Segments Indifference Curves Budget Lines at Various income levels. 50 Arrows indicating direction of increasing utility along B.L.s. m H > 50 m L < mh 00.5 Shells Figure. Tamar s disconnected Income Expansion Path. Path is drawn with thicker brownish-orange line. For income level below $50, the income expanstion path rides along the Rocks axis. At income levels above $50, the i.e.p. switches over to the Shells axis. Budget lines for income levels below $50, exactly equal to $50 and above $50 are shown to illustrate the logic of the disconnected i.e.p. in a Leisure Consumption space diagram. Be sure to include the budget line he faces for the given endowment and wage in your diagram. ANSWER Plugging in the given values for the endowment and the wage we have the original choice See Figure 3. x 0 l = x 0 c = = = 100 = 100 = 600

8 8 ECON 10 PROFESSOR GUSE (b) (10 Points) Suppose that Harold s wage increases from 6 to 10. Add the new budget line and new choice to your diagram. ANSWER At w = 10 the new optimal choice would be x New l = x New c = = = 80 = 1600 = 800 Again, see Figure 3 (c) (10 Points) Calculate the income and substitution effects of the wage increase from 6 to 10. Add the compensated budget line and compensated choice to your diagram.answer Since the original choice is exactly the endowment, the compensated budget line is identical to the new budget line making the compensated choice identical to the new choice. Furthermore, since the income effect is defined as the difference between the new choice and the compensated choice, the income effects must be equal to zero. Therefore the total effect is entirely explained by the subsitution effect. Summarizing, we have Leisure(l) : Consumption(c) : x Comp d l IE l = x New l = x New l = 80 SE l = TE l = x New l x Comp d c IE c = x New c x Comp d l = = 0 = x New c = 800 x 0 l = 0 x Comp d c = = 0 SE c = TE c = x New c x 0 c = 00 (d) (5 Points - More Difficult) Derive the full demand equations. (Full, meaning not partial in the sense defined above. In other words, your full demand equations should describe how Harold behaves for any combination of w and γ c.) Did using the partial demand equations given above affect the calculations compared to if one had used the full demand equations? ANSWER Since the MRS approaches infinity as leisure goes to zero, there can never be corner solutions involving zero leisure. However,

9 MIDTERM EXAM ANSWER KEY 9 Harold can never have more leisure than 100 hours and he would never choose to have less material consumption than γ c ; there is no possible way for him to sell off his material goods endowment to create more time. Therefore, whenever the partial demand equations above suggest a leisure demand of greater than 100, they are suggesting something that is outside of Harold s budget set and we would find Harold at the corner which is this case, is his endowment point. In other words the full demand equations would be x l (w,γ c ) = min{100, 100w +γ c } w x c (w,γ c ) = max{γ c, 100w +γ c } For the price increase analysed above, using the full demand equations versus the partial demand equations supplied to you would not have mattered. The original budget line in that problem (when γ c = 600 and w = 6) is a borderline case where the solution is, in a way, both an interior solution and a corner solution. In other words, if the wage had been even a penny less, the partial demand equation for leisure would have suggested (incorrectly) that demand was strictly more than 100 hours, while the full demand equations would have suggest (correctly) that demand leisure is exactly 100 hours.

10 10 ECON 10 PROFESSOR GUSE Material Consumption ($) 1600 w = 10 Indifference Curves Orig. Budget (w = 6) New Budget (w = 10) Compensated Budget 100 New Choice AND Compensated Choice TE = SE w = 6 Endowment AND Original Choice Comp d Budget TE=SE Leisure (Hours) Figure 3. Harold s reaction to wage increase from 6 to 10.