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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain ECO 204 20082009 Ajaz Hussain HW 8 Solutions Question 1 In lectures 8 and 9 we saw that if a company uses an imperfect substitutes inputs technology with capital fixed and labor variable: The short run production function is: q = f(L, k) = L k The short run cost minimizing amount of labor is: L = f1(q, k) = [q/k] 1/ The short run cost function is: C(q) = PK k + PL L = PK k + (PL /k/)q1/ Suppose the company has constant returns (not the same as constant returns to scale) a) What is the short run production function? Answer: With capital fixed and labor variable, this is: q = f(L, k) = L k If there are constant returns, it means that (holding k constant) doubling L doubles q. Thus = 1 and: q = L k b) What is the elasticity of output with respect to labor? Does your answer make sense give the company has constant returns? 1 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain Answer: We have from part (a): q = L k We want the elasticity of q with respect to L = % q / % L. Now: % q / % L { q/q}*100 / { L/L}*100 { q/q} / { L/L} For small changes this is: {dq/q} / {dL/L} We can derive this in two ways: (i) Note: {dq/q} / {dL/L} = d ln q / d ln L Thus from q = L k ln q = ln L + ln k d ln q / d ln L = 1 (ii) Note: {dq/q} / {dL/L} = {dq/dL} {L/q} From q = L k {dq/dL} {L/q} = k L / L k = 1 Either way: % q / % L = 1. Put simply: if L increases by x% then q will also increase by x%. This makes sense given that we have constant returns (increasing inputs by x% 2 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain increases q by x% too). c) Suppose the company doubles labor. Without using the equation for C(q) what is the impact on total variable cost? Average variable cost? Average cost Answer: Total variable cost TVC(q) is: PL L. If the company doubles labor, then TVC(q) must also double. Now: AVC(q) = TVC(q)/q. As the company doubles inputs because it has constant returns it doubles output. Thus, when the company doubles inputs, the ratio TVC(q)/q remains constant. On the other hand: AC(q) = C(q)/q AC(q) = {TFC + TVC(q)}/q AC(q) = TFC/q + TVC(q)/q AC(q) = AFC(q) + AVC(q) With constant returns, as labor is doubled, AVC(q) remains constant but AFC(q) declines with q. Thus, AC(q) declines. d) Suppose the company doubles labor. Using the equation for C(q) what is the impact on total variable cost? Average variable cost? Average cost Answer: The cost equation is: C(q) = PK k + PL L = PK k + (PL /k/)q1/ With constant returns = 1 and: C(q) = PK k + PL L = PK k + (PL /k)q = TFC + TVC(q) Now: as L doubles from q = L k q doubles. Thus, TVC(q) = (PL /k)q doubles. 3 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain Now, AVC(q) = TVC(q)/q = (PL /k)q/q = (PL /k), a constant. If a company had a constant AVC(q) it would make contribution margin analysis easy. Now AC(q) = C(q)/q = {PK k + (PL /k)q}/q = (PK k)/q + (PL /k), which declines with quantity. Suppose the company has decreasing returns (not the same as decreasing returns to scale) e) What is the short run production function? Answer: With capital fixed and labor variable, this is: q = f(L, k) = L k If there are decreasing returns, it means that (holding k constant) doubling L less than doubles q. Thus < 1 and: q = L k f) What is the elasticity of output with respect to labor? Does your answer make sense give the company has constant returns? Answer: We have from part (e): q = L k We want the elasticity of q with respect to L = % q / % L. I will, for your benefit, derive this using the two techniques shown in part (b). Now: % q / % L { q/q}*100 / { L/L}*100 { q/q} / { L/L} For small changes this is: 4 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain {dq/q} / {dL/L} We can derive this in two ways: (i) Note: {dq/q} / {dL/L} = d ln q / d ln L Thus from q = L k ln q = ln L + ln k d ln q / d ln L = < 1 (ii) Note: {dq/q} / {dL/L} = {dq/dL} {L/q} From q = L k {dq/dL} {L/q} = L1 k L / L k = < 1 Either way: % q / % L = < 1. Put simply: if L increases by x% then q will increase by less than x%. This makes sense given that we have decreasing returns (increasing inputs by x% increases q by less than x%). g) Suppose the company doubles labor. Without using the equation for C(q) what is the impact on total variable cost? Average variable cost? Average cost Answer: Total variable cost TVC(q) is: PL L. If the company doubles labor, then TVC(q) must also double. Now: AVC(q) = TVC(q)/q. As the company doubles inputs because it has decreasing returns it less than doubles output. Thus, when the company doubles inputs the ratio TVC(q)/q rises because TVC(q) is doubling and q is less than doubling. On the other hand: AC(q) = C(q)/q AC(q) = {TFC + TVC(q)}/q 5 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain AC(q) = TFC/q + TVC(q)/q AC(q) = AFC(q) + AVC(q) With decreasing returns, as labor is doubled, AVC(q) rises but AFC(q) declines with q. Thus as q increases, AC(q) can rise, remain constant, or fall depending on which effect is dominant. In particular, it possible to generate a "U" shaped AC(q) curve (see lecture 9 slide on case of < 1). h) Suppose the company doubles labor. Using the equation for C(q) what is the impact on total variable cost? Average variable cost? Average cost Answer: The cost equation is: C(q) = PK k + PL L = PK k + (PL /k/)q1/ where because of decreasing returns < 1. Now: as L doubles from q = L k q less than doubles. Thus, TVC(q) doubles. Now, AVC(q) = TVC(q)/q = (PL /k/)q1/ /q = (PL/k/) q(1 )/. In this question < 1 so that (1 )/ > 0 which implies that as q increases AVC(q) will also increase. Now AC(q) = C(q)/q = (PK k)/q + (PL/k/) q(1 )/. Here: (PK k)/q decreases with quantity while (PL/k/) q(1 )/ rises with quantity. Thus, AC(q) can rise, remain constant, or fall with quantity. Question 2 Ajax Inc. produces cranberry juice and has the short run cost function C(q) = 5 + Qc. Don Damiano Inc. produces grape juice and has the short run cost function C(q) = 5 + Qg. If Ajax and Don Daminao merge the short run cost function for both cranberry and grape juice is: C(q) = 5 + Qc + Qg. Are there economies of scope if Ajax and Don Damiano merge? Answer: To see if there economies of scope, we simply need to check if cost of cranberry and grape juices is lower than the sum of cranberry plus grape juice. Now: 6 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain C(Qc ,Qg) = 5 + Qc + Qg Whereas C(Qc) + C(Qg) = 5 + Qc +5 + Qg.= 10 + Qc + Qg Observe: C(Qc ,Qg) < C(Qc) + C(Qg) From lecture 8, this shows that it is cheaper to produce the juices together than separately. Therefore, there are economies of scope. 7 ...
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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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