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Unformatted text preview: ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission University of Toronto, Department of Economics, ECO 204, Summer 2009. Ajaz Hussain Midterm Test Solutions PLEASE FILL OUT THE INFORMATION BELOW Please write your name as it appears in ROSI: LAST NAME: FIRST NAME: MIDDLE NAME: U Toronto ID #: SIGNATURE: SCORES Question 1 2 3 4 5 6 7 Points 5 15 15 20 10 5 30 TOTAL SCORE OUT OF 100 Score This test consists of 7 questions. For your convenience, there is a worksheet at the end of this test. Keep your answers brief. Good luck! 1 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 1 (5 points) Suppose a consumer has the utility function U = Q11/4 Q23/4. Does the utility function: U = {Q11/4 Q23/4}2 represent the same preferences as U = Q11/4 Q23/4? Explain briefly. Answer No. The utility function U = {Q11/4 Q23/4}2 is obtained from U = Q11/4 Q23/4 by raising the latter to the power of 2. Since x2 is a decreasing function (i.e. if x > y then x2 < y2), the utility function U = {Q11/4 Q23/4}2 is not a monotonic transformation of U = Q11/4 Q23/4 and, since utility is an ordinal concept, it does not represent the same preferences. Question 2 (15 points) You're trying to figure out a consumer's preferences and notice that regardless of income or prices, she always spends 10% of her income on good 1, 20% of her income on good 2, 40% of her income on good 3 and the remainder of her income on good 4. (a) (5 points) Can you guess what her utility function is? Explain your answer. Answer: That the consumer always spends a constant fraction of her income on goods is a familiar result: we know that if someone has CobbDouglas preferences, then the expenditure on goods is always a constant fraction of income. For example, if: U = Q1Q2 then we know from the UMP that: Q1 = {/( + )} Y / P1 P1 Q1 = {/( + )} Y Q2 = {/( + )}Y / P2 P2Q2 = {/( + )}Y Now if ( + ) = 1 these expressions become: 2 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission P1 Q1 = Y P2Q2 = Y That is, the consumer always spends a fraction of income on good 1 and a fraction of income on good 2. Therefore we can guess that if someone always spends 10% of her income on good 1, 20% of her income on good 2, 40% of her income on good 3 and the remainder of her income (30%) on good 4 that her utility function must be: U = Q10.1 Q20.2 Q30.4 Q40.3 Of course, we could only venture this guess because in this question the percentage of income spent on each good adds up to 100%. Otherwise we couldn't answer the question. (b) (5 points) What is the demand equation for good 3? Answer: Given that the consumer always spends 40% of income on good 3, it must be true that: P3 Q3 = 0.4 Y Q3 = 0.4 Y / P3 (c) (5 points) What are the income and price elasticity of good 3? Answer: From: Q3 = 0.4 Y / P3 Q3 = 0.4 Y P31 We see that this is a constant elasticity demand function with income elasticity 1 and price elasticity 1. Intuitively, if someone always spends 40% of her income on a good then a 1% increase in income must result in a 1% increase in consumption because otherwise expenditure wouldn't be constant at 40%. Likewise, if price increases by 1% then consumption must fall by 1% to ensure that expenditure remains constant. 3 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission Question 3 (15 points) A consumer perceives four "good" goods as complements. You can assume that all prices are uniform and, along with Y, are given. (a) (5 points) Do the UMP and derive the demand function for good 2. Explain your answer and show (any) calculations. Answer: Her utility function is: U = min (Q1, Q2, Q3, Q4) Because the utility function is not differentiable everywhere we cannot use the Lagrangian method for the UMP. Instead, we recognize that at the optimum: Q1 = Q2 = Q3 = Q4 The budget constraint is: P1 Q1 + P2 Q2 + P3 Q3 + P4 Q4 = Y Since we're interested in the demand for good 2 this can be expressed as: P1 Q2 + P2 Q2 + P3 Q2 + P4 Q2 = Y (P1 + P2 + P3 + P4 )Q2 = Y Q2 = Y/(P1 + P2 + P3 + P4 ) That is, the demand for good 2 or for that matter any other good is income divided by total price of all 4 goods. (b) (5 points) What is the elasticity of good 2 with respect to the total price of all goods? Answer: We had: Q2 = Y/(P1 + P2 + P3 + P4 ) Q2 = Y (P1 + P2 + P3 + P4 )1 Observe how this is a constant elasticity function in Y and total price of all 4 goods where the elasticity of good 2 with respect to the total price of all goods is 1. 4 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission (c) (5 points) Suppose P1 = $10, P2 = $100, P3 = $40 and P4 = $50. What is impact on the consumption of good 2 if the price of good 2 increases by 10%? Explain your answer and show any calculations. Answer: We want to know: E = % Q2 / % P2 Because then: % Q2 = E (% P2) % Q2 = E (10%) Now we had: Q2 = Y/(P1 + P2 + P3 + P4 ) Q2 = Y (P1 + P2 + P3 + P4 )1 We have to use the point E formula: E = {dQ2 /dP2}{P2/Q2} E = {Y (P1 + P2 + P3 + P4 )2} {P2/Q2} Substituting Q2 = Y (P1 + P2 + P3 + P4 )1 we get: E = {Y (P1 + P2 + P3 + P4 )2} {P2/ Y (P1 + P2 + P3 + P4 )1} E = P2/ (P1 + P2 + P3 + P4 ) If P1 = $10, P2 = $100, P3 = $40 and P4 = $50 then: P1 + P2 + P3 + P4 = $200 And therefore: E = $100/$200 E = 0.5 Thus: 5 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission % Q2 = E (% P2) % Q2 = E (10%) % Q2 = 0.5 (10%) % Q2 = 5% In response to a 10% increase in price of good 2, this consumer will consume 5% less of good 2. Question 4 (20 points) A company uses labor (L), capital (K) and materials (M) to produce output (Q). The production function is: Q = L K M where = , = and = . Assume that PK =$10 , PL = $20 and PM = $20. (a) (5 points) Describe this company's returns with respect to labor, capital and materials. Show all calculations. Answer: Ceteris paribus there are decreasing returns with respect to each input. For example, suppose labor is doubled: Output with doubled labor = (2L) K M Output with doubled labor = 2 L K M Output with doubled labor = 2 Original output Since < 1, this implies that doubling labor less than doubles output: Output with doubled labor < Original output A similar argument holds for capital and materials. (b) (10 points) Now suppose capital is fixed at k = 100. Derive the short run cost function and show that it exhibits the returns in part (b). Show all calculations clearly. Answer: The CMP is: given k, choose L and M to min cost of producing target output q. The Lagrangian 6 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission is: L = PL L + PK k + PM M [L k M q]. I will do the calculations algebraically and numerically side by side: Algebra Answer The FOCs are: L/L = 0 PL L1 k M = 0 PL = L1 k M L/M = 0 PM L k M1 = 0 PM = L k M1 L/ = 0 L k M = q Dividing the 1st and 2nd FOCs and solving, we get: PL / PM = { L1 k M}/{ L k M1 } PL / PM = {/}{M/L} L = (PM /PL) {/} M Substitute this in the 3rd FOC: L k M = q [(PM /PL) {/} M] k M = q [(PM /PL) {/}] k M+ = q M+ = q/{[(PM /PL) {/}] k} Numerical Answer The FOCs are: L/L = 0 20 (1/4) L3/4 (100)1/2 M1/4 = 0 20 = (1/4) L3/4 10 M1/4 L/M = 0 20 (1/4) L1/4 (100)1/2 M3/4 = 0 20 = (1/4) L1/4 10 M3/4 L/ = 0 L1/4 k1/2 M1/4 = q Dividing the 1st and 2nd FOCs and solving, we get: 1 = {(1/4) L3/4 10 M1/4}/{ (1/4) L1/4 (100)1/2 M3/4 } 1 = {M/L} L = M Substitute this in the 3rd FOC: L1/4 k1/2 M1/4 = q [M]1/4 10 M1/4 = q 10 M1/2 = q M1/2 = q/10 7 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission M = {q/{[(PM /PL) {/}] k}} 1/(+) M = {q/10} 1/(1/2) M = {q/10} 2 M = q2/100 Since L = M this implies that: L = q2/100 Now the cost function is: C = PL L + PK k + PM M Substituting prices and k = 100 yields: C = 20 L + 10 (100) + 20 M C = 1,000 + 20(L + M) C = 1,000 + 20({q2/100} + {q2/100}) C = 1,000 + 20(q2/50) C = 1,000 + (2/5)q2 Observe that the TVC is: TVC = (2/5)q2 Now: AVC = TVC/q AVC = (2/5)q This is increasing in q implying there are decreasing returns. (c) (5 points) Will the company's cost affected more by an increase in wages or the price of materials? Show all calculations. Answer: This is a direct application of the envelope theorem. Given target output the company seeks to minimize cost: 8 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission L = PL L + PK k + PM M [L k M q] If wages increase, the impact on cost is simply: L/PL = L But L = q2/100. Thus: L/PL = q2/100 If price of materials increase, the impact on cost is simply: L/PM = M But M = q2/100. Thus: L/PM = q2/100 Thus the company is impacted equally whether wages or the price of materials increase (by the same amount). Question 5 (10 points) An individual lives for two periods (T = 1 and 2). Her real income (measured in actual output, i.e. not in $) when she's "young" in T = 1 is Y1 and her income (measured in actual output, i.e. not in $) when she's "old" in T = 2 is Y2. Let T = 1 be the base period so that P1 = 1 and allow for the possibility of inflation (i.e. it may be that P1 P2). Denote the nominal interest rate by i. Suppose this individual has CobbDouglas preferences over consumption in T = 1 and in T = 2: U = C1 C2 This individual recently won a lottery (measured in actual output). She can choose to receive the entire prize either when she's young or when she's old. That is, she can choose to have Y1 or Y2 . Which option would she choose? Show all calculations clearly. It's OK to just give not derive the relevant intertemporal budget constraint. Hint: She'll choose whichever option makes her happier. Answer: She will choose that option which makes her happier. Hence we need to calculate the impact 9 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission on her happiness measured by utility from either an increase in income when she's young or when she's old. This is a simple envelope theorem application after all, we're being asked to examine the impact on the objective (utility subject to intertemporal budget constraint) from a change in a parameter (income). To begin, write down her intertemporal budget constraint given that there may be inflation: C2 = Y2 + (Y1 C1)(1 + r) C1(1 + r) + C2 = Y1(1 + r) + Y2 Note: You could've shown this from 1st principles too. The intertemporal budget constraint is: (Expenditure on Consumption at T = 2) = (Nominal Income at T = 2) + (Savings from T = 1)(1 + i) The budget constraint can be rewritten as: P2C2 = P2Y2 + (P1Y1 P1C1)(1 + i) Notice that we have multiplied consumption levels by prices to obtain expenditure and real incomes by prices to get nominal incomes. P2C2 = P2Y2 + (P1Y1 P1C1)(1 + i) C2 = Y2 + (P1Y1 P1C1)(1 + i)/P2 This can be simplified by using the formula that connects nominal and real interest rates with the rate of inflation: = (P2 P1)/P1 P1 = P2 P1 P2 = P1 (1 + ) P2 = (1 + ) Substituting in budget constraint: C2 = Y2 + (P1Y1 P1C1)(1 + i)/(1 + ) Now recall that the real interest rate is defined as: 1 + r = (1 + i)/(1 + ) From which the budget constraint becomes: C2 = Y2 + (P1Y1 P1C1)(1 + r) 10 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission Now substitute P1 = 1: C2 = Y2 + (Y1 C1)(1 + r) To gauge the impact of Y1 or Y2 on this individual's utility, write down the Lagrangian for this individual and simply compare the derivative of the Lagrangian with respect to Y1 versus the derivative with respect to Y2. The Lagrangian is: L = C1 C2 [C1(1 + r) + C2 Y1(1 + r) Y2] L = C1 C2 C1(1 + r) C2 + Y1(1 + r) + Y2 The impact on utility from Y1 or Y2 is simply: L/Y1 = (1 + r) L/Y2 = She will choose the option which most increases her utility. Thus we need to know which of these derivatives is larger. First note that > 0. This can be proved in two ways. Note how is the Lagrange multiplier on the constraint {C1(1 + r) + C2 = Y1(1 + r) + Y2} and therefore measures the impact of an increase in income on the constrained utility. Since higher income raises happiness, it must be that > 0. The other way is to actually do the UMP and solve for . If you do this, you'll see that it is positive. Now: L/Y1 > L/Y2 when: (1 + r) > (1 + r) > 1 r > 0 Thus, ceteris paribus she'll be happier with an increase in income when she's young when r > 0. Using the Fisher approximation (r i ) r > 0 implies that i > . Thus, if nominal interest rates are greater than the rate of inflation, she will prefer getting the lottery prize in her youth to be wasted on bottles of 11 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission Champagne at Circa, friends who are leeches and expensive bags. L/Y1 < L/Y2 when: (1 + r) < (1 + r) < 1 r < 0 Thus, ceteris paribus she'll be happier with an increase in income when she's old when r < 0. Using the Fisher approximation (r i ) r < 0 implies that i < . Thus, if nominal interest rates are lower than the rate of inflation, she will prefer getting the lottery prize in her old age to be wasted on bottles of brandy, family who are leeches and health care. Question 6 (5 points) A company uses labor and capital as complements to produce output with the production function q = min(L, K). Under what conditions will this company have a Ushaped long run AC (average cost) curve? Show all calculations and explain your answer. Answer: First note that a Ushaped AC curve will not arise from increasing RTS at low levels of output followed by constant RTS and decreasing RTS at high levels of output. This is because the production function q = min(L, K) always has constant returns to scale. We can see this in two ways. First, compare the output with doubled inputs versus twice the output with initial level. The initial output is: q = min(L, K). The output with double inputs is: New q = min(2L, 2K) New q = 2 min(L, K) New q = 2 Old q Doubling inputs always doubles output: hence, there are constant returns to scale. Secondly, we can show that AC is always constant and therefore there are always constant RTS. At the optimum point L = K which implies that: 12 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission L = q K = q Thus cost is: C = PL L + PK K C = PL q + PK q C = (PL + PK )q AC = (PL + PK) This is a constant and therefore the company has constant RTS. Given the company always has constant returns to scale, we cannot use a returns to scale argument to generate a Ushaped AC curve. One other explanation is that input prices wages and/or the price of capital at first decline with output only to later increase with output. For example, this may happen because as the company grows, the company initially has bargaining power over wages (so that wages are depressed) but eventually labor unions have greater bargaining power (so that wages rise). Question 7 (30 points) A company uses labor and capital as complements to produce output with the production function q = min(L, K) + ( notice the "" here). (a) (5 points) What "returns to scale" does the company have? Answer: First, compare the output with doubled inputs versus twice the output with initial level. The initial output is: q = min(L, K) + The output with double inputs is: New q = min(2L, 2K) + New q = 2 min(L, K) + To express this in terms of old q let's play trick: add and subtract : 13 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission New q = 2 min(L, K) + + New q = 2 min(L, K) + 2 New q = 2 [min(L, K) + ] New q = 2 old q Now: If = 0: New q = 2 old q That is, there are constant RTS. If > 0: New q = 2 old q New q < 2 old q That is, there are decreasing RTS. If < 0: New q = 2 old q New q > 2 old q That is, there are increasing RTS. (b) (5 points) Solve for the optimal (long run) labor and capital demands. Show all calculations. Answer: At the optimum point, L = K. Now: q = min(L, K) + 14 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission Suppose we use L = K, then we have: q = min(L, L) + q = L + L = q Since L = K this implies: K = q (c) (5 points) Derive the long run cost function. Answer: We had: L = q K = q Thus cost is: C = PL L + PK K C = PL (q ) + PK (q ) C = (PL + PK )q (PL + PK ) (d) (15 points) Plot the AC curve. Hint: there are 3 different curves corresponding to increasing, constant and decreasing returns to scale. Answer: AC = C/Q AC = (PL + PK ) (PL + PK ) /q Examine each case separately: Case 1: = 0 or constant RTS. AC = (PL + PK ) (PL + PK ) /q AC = (PL + PK ) 15 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission Observe how AC is constant (as it should be). This is depicted below. Case 2: > 0 or decreasing RTS. AC = (PL + PK ) (PL + PK ) /q Since > 0 it means the term: (PL + PK ) /q < 0 When q = 0, this means that: (PL + PK ) /q so that AC When q then: (PL + PK ) /q 0 so that AC AC = (PL + PK ) The fact that AC goes from and approaches a constant (PL + PK ) means it has to cross the x axis at some point. Where? Set AC = 0: (PL + PK ) (PL + PK ) /q = 0 q = Hence, AC is increasing in q (as it should be with decreasing RTS) and is: 16 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission Case 3: < 0 or increasing RTS. AC = (PL + PK ) (PL + PK ) /q Since < 0 it means the term: (PL + PK ) /q > 0 When q = 0, this means that: (PL + PK ) /q so that AC When q then: (PL + PK ) /q 0 so that AC AC = (PL + PK ) Hence, AC is decreasing in q (as it should be with increasing RTS) and is: 17 ECO 204, Summer 2009, Midterm Test Solutions This test is copyright material and may not be used for commercial purposes without prior permission Note how AC never crosses the xaxis. Can we confirm this? Yes. If the AC curve crosses the xaxis then AC = 0: (PL + PK ) (PL + PK ) /q = 0 q = But < 0 so that if the AC curve crossed the xaxis, q < 0 which is impossible. Thus, the graph is right. ____________________________________________________________________________________________ THE END 18 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto Toronto.
 Fall '08
 HUSSEIN
 Economics, Microeconomics

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