ProblemSet2-Solution

ProblemSet2-Solution - Econ326 Problem Set#2 Fall 2011...

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Unformatted text preview: Econ326 Problem Set #2 Fall 2011, Instructor: Ginger Jin Due Date: October 28 Name:______________ Section #: ____________ Section Time: _____________ 1. Richard's father gives him a lottery ticket. The probability of winning payoffs is as follows: Probability 0.5 0.25 0.2 0.05 Payoff $0 $1 $2 $20 Please keep your calculation up to four decimal points. (a) What is the expected value of the lottery ticket? What's the standard deviation? EV=0.5*0+0.25*1+0.2*2+0.05*20=1.65. stdev = 0.5 * (0 − 165)^ 2 + 0.25(1 − 165)^ 2 + 0.2 * (2 − 165)^ 2 + 0.05 * (20 − 165)^ 2 = 4.281063 . . . . (b) Richard's utility function is U ( x ) = ln( X + 1) . Is he risk averse, risk neutral or risk loving? Explain why. He is risk averse because the utility function is increasing (U'(x)>) but concave (U''(x)<0). 1 U ' ( x) = >0 x +1 1 U ' ' ( x) = − < 0. ( x + 1) 2 (c) Under this utility function, calculate the minimum price at which Richard is willing to sell the ticket. Note that the minimum price that he is willing to sell for the ticket should make him indifferent between (a) selling the ticket and keeping the money for sure and (b) waiting for the lottery outcome. Richard has no other income or savings. Will Richard sell the ticket at price $1? Write out your steps. If he holds on to the ticket, the ticket will generate expected utility EU = 0.5 * U (0) + 0.25 * U (1) + 0.2 * U (2) + 0.05 * U (20) = 0.545235. If he sells it for a minimum price P, the expected utility from P for sure is equal to the expected utility from the ticket. Solve for P: U ( P) = ln( P + 1) = EU = 0.54235 ==> P = 0.725014. He will sell the ticket for $1. (d) Richard has a friend Eric. Eric's utility function is U ( x ) = x 0.75 . If Eric faces the same situation, will Eric sell the lottery ticket at price $1? Write out your steps. EU = 0.5 * U (0) + 0.25 * U (1) + 0.2 * U (2) + 0.05 * U (20) = 1059229 > U (1). He will not sell the . ticket for $1. 1 2. Do the following production functions exhibit increasing, constant, or decreasing returns to scale? Explain how you get your answers. In general, a production function f( L, K) will exhibit Returns to Scale IRS CRS DRS Criteria Interpretation If f(h*L, h*K) > h * f(L,K) If I double the inputs, the output I can produce(LHS) is more than double of what I was previously producing(RHS) If f(h*L, h*K) = h * f(L,K) If I double the inputs, the output I can produce(LHS) is exactly double of what I was previously producing(RHS) If f(h*L, h*K) < h * f(L,K) If I double the inputs, the output I can produce(LHS) is less than double of what I was previously producing(RHS) Where h is any constant > 1(The interpretations are provided for h =2) a. q 3L 2 K f ( L,K) = 3 L + 2K LHS f(h*L, h*K) = 3 (hL) + 2 (hK) = h ( 3L + 2K) RHS h * f(L,K) = h (3L + 2K) LHS = RHS so it exhibits CRS b. q (2 L 2 K )1/2 f ( L,K) = (2 L + 2K) 1/2 1/2 1/2 1/2 LHS f(h*L, h*K) = (2 (hL) + 2(hK)) = (h (2 L + 2K)) = h (2 L + 2K)) 1/2 RHS h * f(L,K) = h (2 L + 2K)) LHS < RHS (because h > 1 => h 1/2 <h) so it exhibits DRS 1/2 1 c. q 3LK 2 Here, we can use the shortcut for cobb douglas functions of the form q = L α K β Since α + β = 1 + 2 = 3 > 1, its exhibits IRS d. q L1/2 K 1/2 Here, we can use the shortcut for cobb douglas functions of the form q = L α K β Since α + β = 1/2 + 1/2 = 1 , its exhibits CRS e. q 4 L1/2 4 K f ( L,K) = 4 L 1/2 + 4K 1/2 1/2 1/2 LHS f(h*L, h*K) = 4 (hL) + 4(hK) = 4 h L + 4hK 1/2 RHS h * f(L,K) = h (4 L + 4K) = 4 h L 1/2 + 4hK LHS – RHS = 4 L 1/2 (h 1/2 - h ) < 0 LHS < RHS So it exhibits DRS (f) q 16 L1/3 K 1/2 Here, we can use the shortcut for cobb douglas functions of the form q = A L α K β Since α + β = 1/3 + 1/2 <1 , its exhibits DRS 2 1/2 3/2 3. Suppose the production function is Cobb-Douglas and the output is f ( x1 , x2 ) x1 x2 . (a) Write an expression for (1) the marginal product of x1 at the point ( x1 , x2 ) , (2) the marginal product of x2 , and (3) the technical rate of substitution between x2 and x1 . Does this technology have diminishing technical rate of substitution? Explain. Marginal Product of x1 at the point ( x1 , x2 ) = δ f /δx1 = (½) x1 -1/2 x2 3/2 Marginal Product of x2 at the point ( x1 , x2 ) = δ f /δx1 = (3/2) x1 1 / 2 x2 1//2 MRTS = MP x1 / MP x2 = (1/3) x2 / x1 Since this MRTS decreases with an increase in x1 ( inversely related) , so this technology has diminishing technical rate of substitution (b) Does this technology demonstrate increasing, constant, or decreasing returns to scale? Explain. Here, we can use the shortcut for cobb douglas functions of the form q = L α K β Since α + β = 1/2 + 3/2 = 4/2 = 2 > 1 , its exhibits IRS (c) Suppose the input price for x1 is w1 and the input price for x2 is w2. How much x1 and x2 should you hire to minimize the cost of producing y units of output? (Hint: your answer should be a function of w1, w2, and y.) At this optimal choice, write down your cost of production as a function of w1, w2, and y. We solve the following problem : Min Cost = w1 x1 + w2 x2 Subject to x1 1/2 x2 3/2 = y ( As we have to minimize the cost of producing ‘y’ units of output) 3 L = w1 x1 + w2 x2 + λ (x1 1/2 x2 3/2 - y) FOC : x1 : w1 = (½) x1 -1/2 x2 3/2 x2 : w2 = (3/2) x1 1 / 2 x2 1//2 Dividing the 2 equations, we get w1 /w2 = MRTS = (1/3) x2 / x1 So 3 w1 x1 = w2 x2 Moreover, x1 1/2 x2 3/2 = y Substituting, we get ( w2 x2/ 3 w1) 1/2 x2 3/2 = y Or x2 4/2 = x2 2 = y ( 3w1 / w2) 1/2 x2 = y 1/2 ( 3w1 / w2) 1/4 x1 w2 x2/ 3 w1 = (w2/ 3 w1) y 1/2 ( 3w1 / w2) 1/4 = y 1/2 (w2/ 3 w1) 3/4 = 4 Question4. Consider a production function with one single variable input. Fill in the gaps in the table below. T ips: Denote input as X , output as Y , production function as Y = f (X ) the main formulas useful for this exercise are MP = ￿Y f (X2 ) − f (X1 ) Y f (X ) = and AP = = ￿X X2 − X1 X X Answer: Denote input as X, and output as Y. Line2: ￿Y Y X=1, M P = ￿X = 225−0 = 225, AP = X = 225 = 225 1 1 Line3: X = 2, T P = AP · X = 300 · 2 = 600, M P = (600 − 225)/1 = 375 Line4: X = 3, T P = f (2) + M P · ￿X = 600 + 300 · 1 = 900, AP = 900/3 = 300 Line5: X = 4, M P = (1140 − 900)/1 = 240, AP = 1140/4 = 285 Line6: X = 5, T P = f (4) + M P · ￿X = 1140 + 225 · 1 = 1365, AP = 1365/5 = 273 Line7: X = 6, T P = AP · X = 225 · 6 = 1350, M P = (1350 − 1365)/1 = −15 We can see from the chart/table that we do not always have diminishing marginal returns since marginal product is increasing when we change variable input from 1 to 2. Also, we do not always have diminishing average product since average product is increasing when input change from 1 to 2. 1 (&##$ (%##$ ("##$ (###$ +,$ '##$ -,$ &##$ .,$ %##$ "##$ #$ ($ "$ )$ %$ *$ &$ !"##$ Question5. Fill in the blanks of the following table: Answer: Column F ixed Cost: F C = 100 since when output is zero, the firm has already incurred a cost of 100, which according to definition, it is the fixed cost. Column V ariable Cost: V C = T C − F C 2 Column M arginal Cost: M C = ￿T C/￿U nits of Output Column Average F ixed Cost: AT C = F C/U nits of Output Column Average V ariable Cost: AT C = V C/U nits of Output Column Average T otal Cost: AT C = T C/U nits of Output or you can also use formula AT C = AF C + AV C Draw marginal cost (MC), average variable cost (AVC) and average total cost (ATC) in the graph below. Please label each curve clearly. '$!" '#!" '!!" &!" ,-" ./-" %!" .0-" $!" #!" !" '" #" (" $" )" %" *" &" +" '!" Question6. Nadine sells user-friendly software. Her firm’s production function is f (x1 , x2 ) = x1 +2x2 , where x1 is the amount of unskilled labor and x2 is the amount of skilled labor that she employs. In this technology, the two inputs are perfect substitutes. Suppose Nadine faces factor prices (w1 , w2 ) for the two inputs. Answer: 3 (a) In the graph below, use black ink to draw a production isoquant representing input combinations that will product 20 units of output, and another isoquant representing input combinations that will produce 40 units of output. Answer: Since the two inputs are perfect substitutes, we can infer that the isoquants of the production function are a straight lines. To draw the isoquant of output 20, we only need to find two points on this isoquant, and connect them to get the isoquant line. For example, pick points (x1 , x2 ) = (0, 10) and (x1 , x2 ) = (20, 0) and connect them gives the isoquant of output level of 20. (in black ink) Similarly, to draw the isoquant of output 40, we may pick points (x1 , x2 ) = (0, 20) and (x1 , x2 ) = (40, 0) and connect them gives the isoquant of output level of 40. (in blue ink) (b) Does this production function exhibit increasing, decreasing or constant returns to scale? Explain why. Answer: The production function exhibits constant returns to scale since doubling the inputs also doubles the output. (c) If Nadine uses only unskilled labor, how much unskilled labor would she need in order to produce y units of output? (hint: your answer should be an expression of y .) What’s the cost of hiring these unskilled labor? (hint: your answer should be an expression of y and w1 ). Explain how you get your answers. Answer: In order to produce y units of output with unskilled labor x1 only, we can set 4 x2 = 0, and hence the units of x1 needed is: x1 = y or x1 = y And the cost of this level of input is: C = x1 · w 1 = y · w 1 (d) If Nadine uses only skilled labor to produce output, how much skilled labor would she need in order to produce y units of output? What’s the cost of hiring these skilled labor? Explain how you get your answer. Answer: In order to produce y units of output with skilled labor x2 only, we can set x1 = 0, and hence the units of x2 needed is: 2x2 = y or x2 = y/2 And the cost of this level of input is: C = x2 · w 2 = y · w2 2 (e) What will be the minimal cost of producing y units of output? (hint: your answer may depend on the relative magnitude of w1 and w2 . You can discuss each scenario separately and express the cost function for each scenario.) Answer: Because the production function exhibits perfect substitutability, Nadine’s cost minimization input choice will be using only one of the input. We have computed the cost of using only one of the input in part (d) and (e). So we can do the following comparison. If it is cheaper to use unskilled labor only, we produce the condition on w1 and w2 that Cunskilled only < Cskilled only or y · w1 < w1 1 < w2 2 5 y · w2 2 If it is cheaper to use skilled labor only, we produce the condition on w1 and w2 that Cunskilled only > Cskilled only or y · w1 > w1 1 > w2 2 y · w2 2 If using unskilled labor only and using skilled labor only cost the same, it will cost the same to use any combination of x1 and x2 , and this produces the condition on w1 and w2 that Cunskilled only = Cskilled only or y · w1 = w1 1 = w2 2 6 y · w2 2 ...
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