Problem Set 2

Problem Set 2 - Econ326 Problem Set #2 Spring 2011,...

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Unformatted text preview: Econ326 Problem Set #2 Spring 2011, Instructor: Ginger Jin Due Date: March 30 Name:______________ Section #: ____________ TA Name: _____________ 1. Richard is deciding whether to buy a state lottery ticket. Each ticket costs $1, and the probability of winning payoffs is given as follows: Probability 0.5 0.25 0.2 0.05 Payoff $0.00 $1.00 $2.00 $7.50 (a) What is the expected value of Richard's payoff if he buys a lottery ticket? What's the standard deviation? • E(X) = Σpi * xi = .5 * 0 + .25* 1 + .2* 2 + .05 * 7.5 = 1.025 • Var ( X) = Σpi * (xi – E (x) )2 = 5 * (0- 1.025) 2 + .25* (1-1.025) 2 + .2* (2- 1.025) 2 + .05 * (7.5-1.025) 2 = 2.8119 • Standard Deviation (x) = [Var (X)] 0.5 = 1.6769 (b) Richard derives utility U=W0.5 from wealth W. For simplicity, let us assume Richard has $100 before buying the ticket and this lottery is the only other income/wealth Richard may have. Would he buy the ticket? In order to see whether or not he will buy the ticket, Richard will compare his utility from buying the ticket to his utility from not buying the ticket. If he does NOT buy the ticket, he has W = $ 100 for sure, so his Utility from not buying UNB = (100).5 = 10 If he buys the ticket, he does not know beforehand how much wealth and therefore utlity he will get for sure, so he computes the EXPECTED UTILITY from BUYING the ticket E(UB ) = Σpi * ui where ui is his utility from the different outcomes. There are 4 possibilities: i. With probability .5, the following happens : He has $100 at the start. He pays $1 for the ticket, he gets $ 0 , so his net payoff is 100 -1 + 0 = 99. So his utility is (99).5 ii. With probability .25, the following happens : He has $100 at the start. He pays $1 for the ticket, he gets $ 1 , so his net payoff is 100 -1 + 1 = 100. So his utility is (100).5 1 Econ326 Problem Set #2 Name:______________ Spring 2011, Instructor: Ginger Jin Section #: ____________ Due Date: March 30 TA Name: _____________ iii. With probability .2, the following happens : He has $100 at the start. He pays $1 for the ticket, he gets $ 2 , so his net payoff is 100 -1 + 2 = 101. So his utility is (101).5 iv. With probability .05, the following happens : He has $100 at the start. He pays $1 for the ticket, he gets $ 7.5, so his net payoff is 100 -1 + 7.5= 100. So his utility is (106.5).5 So E(UB ) = .5 * (99).5+ .25 *(100).5 + .2 *(101).5 +.05 * (106.5).5 E(UB ) = 10.0009065 > UNB = 10 so he will buy the ticket (c) How much is the maximum amount that Richard is willing to pay for the ticket? There are many ‘acceptable’ answers/ methods to go about this part. If you used any of the following, you will get full credit MOST CORRECT ANSWER: The maximum willingness to pay price is such that the person is indifferent between buying or not buying. If you don’t buy, you still have the $100 wealth, which gives utility UNB = 10. Suppose the maximum willingness to pay is ‘c’, then ‘c’ must be such that E(UB ){given that the price of the ticket is c}= UNB = 10 Following the logic stated in part (b) , if he buys at price c, his expected utility is E(UB ) = .5 * (100 – c + 0 ).5+ .25 *(100 – c + 1).5 + .2 *(100 – c + 2 ).5 +.05 * (100 - c +7.5).5 = .5 * (100 – c).5+ .25 *(101 – c).5 + .2 *(102– c ).5 +.05 * (106.5 - c ).5 To make it equal to 10, c should be equal to 1.018. Note however, this equation is hard to solve analytically. METHOD 2 : An alternative thought is if the person already has the lottery at what price will he be indifferent between selling it or holding it? If we consider the $100 wealth and suppose he has incurred $1 cost of lottery, SO he has $ 99 with him. Suppose the price of selling is c. Utility from selling is (99+c) 0.5, utility from holding is 99.5*0.5+100.5*0.25+101.5*0.2+106.5.5*0.05=10.00091. Making the two equal yields x=1.018. This is complicated as well METHOD 3 : 2 Econ326 Problem Set #2 Spring 2011, Instructor: Ginger Jin Due Date: March 30 Name:______________ Section #: ____________ TA Name: _____________ You could have simply shown the calculation of risk premium while ignoring the $100 wealth. Although this is not entirely correct, at this point we will give you full credit for attempting to answer it this way: If the person already has the lottery and this is his only wealth, at what price will he be indifferent between selling it or holding it. Utility from selling at price c is c0.5. Utility from holding is 00.5*0.5+10.5*0.25+20.5*0.2+7.50.5*0.05=0.66977. Setting the two equal, we get c=0.4486. The resulting risk premium is 1.025-0.4486=0.5764. (d) In the long run, given the price of the lottery tickets ($1) and the probability/payoff table, how much net revenue can the state expect to get from each ticket? (Hint: net revenue for the state = price at which the state sells a ticket – average payoff that the state has to pay if the ticket holder wins.) Net Revenue = Revenue - E(payoff) that state has to make = Cost of ticket - E (X) = 1 -1.025 = - .025 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 Criteria Interpretation IRS 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) CRS DRS Where h is any constant > 1(The interpretations are provided for h =2) 3 Econ326 Problem Set #2 Spring 2011, Instructor: Ginger Jin Due Date: March 30 Name:______________ Section #: ____________ TA Name: _____________ a. 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. f ( L,K) = (2 L + 2K) 1/2 LHS f(h*L, h*K) = [2 (hL) + 2(hK)] RHS h * f(L,K) = h (2 L + 2K) 1/2 = [h (2 L + 2K)] 1/2 =h 1/2 (2 L + 2K) 1/2 1/2 LHS < RHS (because h > 1 => h 1/2 <h) so it exhibits DRS c. 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. 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 4 Econ326 Problem Set #2 Spring 2011, Instructor: Ginger Jin Due Date: March 30 Name:______________ Section #: ____________ TA Name: _____________ e. 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) 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 5 Econ326 Problem Set #2 Spring 2011, Instructor: Ginger Jin Due Date: March 30 Name:______________ Section #: ____________ TA Name: _____________ 3. Suppose the production function is Cobb-Douglas and the output is . (a) Write an expression for (1) the marginal product of at the point , (2) the marginal product of , and (3) the technical rate of substitution between and . Does this technology have diminishing technical rate of substitution? Explain. Marginal Product of at the point Marginal Product of x2 at the point = δ f /δx1 = (½) x1 -1/2 x2 3/2 = δ f /δx2 = (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) 6 Econ326 Problem Set #2 Spring 2011, Instructor: Ginger Jin Due Date: March 30 Name:______________ Section #: ____________ TA Name: _____________ 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 ( 3 w1 / w2) 1/2 x2 = y 1/2 ( 3 w1 / w2) 1/4 x1 3 w1 = (w2 / 3 w1) y 1/2 ( 3 w1 / w2) 1/4 (w2 / 3w1) 3/4 = w2 x2/ 1/2 =y Cost of production is hence: Cost = w1 x1 + w2 x2 = w1 y 1/2 (w2 / 3w1) 3/4+ w2 y 1/2 ( 3 w1 / w2) ¼ = y 1/2 w1 1/4 (w2 / 3) 3/4+ y 1/2 ( 3 w1) 1/4 w2 3/4 7 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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This note was uploaded on 09/07/2011 for the course ECON 326 taught by Professor Hulten during the Spring '08 term at Maryland.

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Problem Set 2 - Econ326 Problem Set #2 Spring 2011,...

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