# UCLA ECON 11 - Solution for Q123_ex6 (9 pages)

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## Solution for Q123_ex6

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- School:
- University of California, Los Angeles
- Course:
- Econ 11 - Microeconomic Theory

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Economics 11 Byeong Hyeon Jeong Answer for the last extra exercises 1 a x2 y 2 y 1 x1 b Decreasing returns to scale as 1 4 1 2 f pkx1 kx2 q pkx1 q1 4 pkx2 q1 2 k3 4 x1 x2 kf px1 x2 q c 1 4 1 2 max px1 x2 w1 x1 w2 x2 x1 x2 d 1 3 4 1 2 x x2 4 1 1 1 4 1 2 M P2 x1 x2 2 M P1 e T RS December 5 2016 1 x2 M P1 M P2 2 x1 1 Economics 11 Byeong Hyeon Jeong f We can use pM Pi wi condition Solving the following equations simultaneously 1 3 4 1 2 p x1 x2 w1 4 1 1 4 1 2 p x1 x2 w2 2 we can get p4 64w12 w22 p4 x2 32w1 w23 x1 g min w1 x1 w2 x2 x1 x2 1 4 1 2 subject to y x1 x2 h 1 4 1 2 1 We can use T RS w w2 and y x1 x2 1 x2 w1 2 x1 w2 yields x2 1 4 1 2 Plugging this into y x1 x2 2w1 x1 w2 yields y 2w 1 2 1 w2 1 4 1 2 x1 x1 2w 1 2 1 w2 3 4 x1 Solving for x1 x1 x2 December 5 2016 w 2 3 2 y 4 3 2w1 2w 1 3 2w1 1 y 4 3 x1 w2 w2 2 Economics 11 Byeong Hyeon Jeong i 2w 1 3 w 2 3 1 3 2 3 1 2 y 4 3 w2 y 4 3 2 2 3 21 3 w1 w2 y 4 3 cpw1 w2 yq w1 2w1 w2 1 3 2 3 ACpw1 w2 yq 2 2 3 21 3 w1 w2 y 1 3 4 1 3 2 3 M Cpw1 w2 yq 2 2 3 21 3 w1 w2 y 1 3 3 2 a Variable costs y 2 2y fixed costs 1 AVC y 2 AFC 1 y AC y 2 1 y MC 2y 2 MC AC AV C AF C b Firm produces optimally at p M C as long as p M C AV C p 2y 2 y December 5 2016 p 2 2 3 Economics 11 Byeong Hyeon Jeong Also note that M C AV C from a c y 8 2 3 2 Producer s surplus is Total Revenue Variable Costs 8 3 32 2 3 9 Alternatively one can compute the area of triangle MC 8 2 3 3 a x2 y 3 y 1 x1 December 5 2016 4 Economics 11 Byeong Hyeon Jeong b Step 1 Solving cost minimization problem to find conditional factor demands 2 3 1 3 x2 1 3 2 3 p1 3qx1 x2 1 3 1 3 y x1 x2 p1 3qx1 10 10 Solving simultaneously x1 x2 y 3 2 Step 2 Plugging conditional factor demands to w1 x1 w2 x2 cpyq 10y 3 2 10y 3 2 20y 3 2 Average costs ACpyq cpyq y 20y 1 2 Marginal costs M Cpyq c1 pyq 30y 1 2 p MC AC y c Conditional factor demand for x1 1 3 y x1 1001 3 x1 y3 100 Cost function cS pyq w1 x1 w2 x2 December 5 2016 y3 1000 10 5 Economics 11 Byeong Hyeon Jeong AC y 2 10 1000 y AVC y 2 10 AFC 1000 y 2 MC 3y 10 AC p AV C MC AF C y d Note that M C AC for all y Therefore M C curve is the long run supply curve e Note that M C AV C for all y Therefore M C curve is the short run supply curve f Long run cpyq 20y 3 4 ACpyq 20y 1 4 M Cpyq 15 1 4 p AC MC y December 5 2016 6 Economics 11 Byeong Hyeon Jeong Short run y 3 2 1000 10 y 1 2 1000 ACpyq 10 y 1000 y 1 2 AF Cpyq AV Cpyq 10 y 3y 1 2 M Cpyq 20 cS pyq p MC AC AV C AF C y g a Iso quants will look like indifference curves for perfect complements preference b Solving cost minimization problem p2x1 q2 3 p5x2 q2 3 y x1 0 5y 3 2 x2 0 2y 3 2 Long run cost function cpyq 5y 3 2 2y 3 2 7y 3 2 ACpyq 7y 1 2 21 M Cpyq y 1 2 2 c Short run y mint2x1 500u2 3 x1 0 5y 3 2 y 5002 3 December 5 2016 7 Economics 11 Byeong Hyeon Jeong Short run cost function cS pyq 5y 3 2 1000 ACpyq 5y 1 2 1000 y AV Cpyq 5y 1 2 AF Cpyq 1000 y 15 M Cpyq y 1 2 2 In both d and e the marginal cost curves are supply curves as M C AC in long run and M C AV C in short run h a Iso quants will look like indifference curves for perfect substitutes preference b In order to solve cost minimization problem notice that the firm will only use x2 as it is more efficient x1 0 y 2x2 x2 y 2 Long run cost function cpyq 5y ACpyq M Cpyq 5 c Short run y x1 200 x1 y 200 y 200 If the firm is to produce less than 200 the firm only has to pay the fixed cost 1000 Short run cost function cS pyq 10py 200q 1000 1000 10py 200q 1000 y y ACpyq 1000 y 10py 200q if y y AV Cpyq 0 if y if y 200 if y 200 if y 200 if y 200 200 200 1000 AF Cpyq y 0 if y 200 M Cpyq 10 if y 200 December 5 2016 8 Economics 11 Byeong Hyeon Jeong d In long run the firm will produce infinite amount if p M C 5 and produce 0 if p M C 5 If p M C 5 the firm will choose any amount e In short run the firm will produce infinite amount if p M C 10 and produce 0 if p M C 10 In case p M C 10 the firm will choose any amount as long as AV C 10 Notice that for any amount y AV C 10 December 5 2016 2000 10 y 9

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