Economics 11 Byeong-Hyeon JeongAnswer for the last extra exercises.1.(a)x1x2y “ 2y “ 1(b)Decreasing returns to scale asf pkx1, kx2q “ pkx1q1{4pkx2q1{2“ k3{4x1{41x1{22ă kf px1, x2q.(c)maxx1,x2px1{41x1{22´ w1x1´ w2x2.(d)MP1“14x´3{41x1{22,MP2“12x1{41x´1{22.(e)T RS “ ´MP1MP2“ ´12¨x2x1.December 5, 2016 1Economics 11 Byeong-Hyeon Jeong(f)We can use pM Pi“ wicondition. Solving the following equations simultaneously,p14x´3{41x1{22“ w1,p12x1{41x´1{22“ w2,we can getx1“p464w21w22,x2“p432w1w32.(g)minx1,x2w1x1` w2x2subject to y “ x1{41x1{22.(h)We can use T RS “ ´w1w2and y “ x1{41x1{22.´12¨x2x1“ ´w1w2yieldsx2“2w1w2x1.Plugging this into y “ x1{41x1{22yieldsy “´2w1w2¯1{2x1{41x1{21“´2w1w2¯1{2x3{41.Solving for x1:x1“´w22w1¯2{3y4{3.x2“2w1w2x1“´2w1w2¯1{3y4{3.December 5, 2016 2Economics 11 Byeong-Hyeon Jeong(i)cpw1, w2, yq “ w1´w22w1¯2{3y4{3` w2´2w1w2¯1{3y4{3“”2´2{3` 21{3ıw1{31w2{32y4{3,ACpw1, w2, yq “”2´2{3` 21{3ıw1{31w2{32y1{3,MCpw1, w2, yq “43¨”2´2{3` 21{3ıw1{31w2{32y1{3.2.(a)Variable costs: y2` 2y,fixed costs: 1,AVC: y ` 2,AFC: 1{y,AC: y ` 2 ` 1{y,MC: 2y ` 2.MCACAV CAF C(b)Firm produces optimally at p “ M C as long as p “ M C ě AV C.p “ 2y ` 2 ùñ y “p ´ 22.December 5, 2016 3Economics 11 Byeong-Hyeon JeongAlso, note that MC ě AV C from (a).(c)y “8 ´ 22“ 3.Producer’s surplus is Total Revenue ´ Variable Costs.8 ˆ 3 ´ 32´ 2 ˆ 3 “ 9.Alternatively, one can compute the area of triangleMC2833.(a)x1x2y “ 3y “ 1December 5, 2016 4Economics 11 Byeong-Hyeon Jeong(b)Step 1. Solving cost minimization problem to find cond itional factor demands:p1{3qx´2{31x1{32p1{3qx1{31x´2{32“1010,y “ x1{31x1{32.Solving simultaneously,x1“ x2“ y3{2.Step 2. Plugging conditional factor demands to w1x1` w2x2:cpyq “ 10y3{2` 10y3{2“ 20y3{2.Average costs: ACpyq “ cpyq{y “ 20y1{2.Marginal costs: MCpyq “ c1pyq “ 30y1{2.ypMCAC(c)Conditional factor demand for x1y “ x1{311001{3ùñ x1“y3100Cost function:cSpyq “ w1x1` w2x2“y310` 1000.December 5, 2016 5Economics 11 Byeong-Hyeon JeongAC: y2{10 ` 1000{y,AVC: y2{10,AFC: 1000{y,MC:3y210.ypAV CACAF CMC(d)Note that MC ě AC for all y. Therefore, M C curve is the long-run supply curve.(e)Note that MC ě AV C for all y. Therefore MC curve is the short-run supply curve.(f)Long-run:cpyq “ 20y3{4, ACpyq “ 20y´1{4, MCpyq “ 15´1{4.ypMCACDecember 5, 2016 6Economics 11 Byeong-Hyeon JeongShort-run :cSpyq “y3{210` 1000,ACpyq “y1{210`1000y,AV Cpyq “y1{210, AF Cpyq “1000y,MCpyq “3y1{220.ypMCACAF CAV C(g)(a): Iso-quants will look like indifference curves for perfect complements preference.(b): Solving cost-minimization problem:p2x1q2{3“ p5x2q2{3“ y ùñ x1“ 0.5y3{2, x2“ 0.2y3{2.Long-run cost function:cpyq “ 5y3{2` 2y3{2“ 7y3{2,ACpyq “ 7y1{2,MCpyq “212y1{2.(c): Short-run.y “ mint2x1, 500u2{3ùñ x1“ 0.5y3{2, @y ď 5002{3.December 5, 2016 7Economics 11 Byeong-Hyeon JeongShort-run cost function:cSpyq “ 5y3{2` 1000,ACpyq “ 5y1{2` 1000{y,AV Cpyq “ 5y1{2, AF Cpyq “ 1000{y,MCpyq “152y1{2.In both (d) and (e), the marginal cost curves are supply curves as M C ě AC in long-run andMC ě 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 x2as it is moreefficient.x1“ 0, y “ 2x2ùñ x2“y2.Long-run cost function:cpyq “ 5y, ACpyq “ M Cpyq “ 5.(c): Short-run.y “ x1` 200 ùñ x1“ y ´ 200, @y ě 200.If the firm is to produce less than 200, the firm only has to pay the fixed cost 1000.Short-run cost function:cSpyq “$&%10py ´ 200q ` 1000 if y ą 200,1000 if y ď 200.ACpyq “$&%10py´200qy`1000yif y ą 200,1000yif y ď 200.AV Cpyq “$&%10py´200qyif y ą 200,0 if y ď 200.AF Cpyq “1000y.MCpyq “$&%0 if y ă 200,10 if y ą 200.December 5, 2016 8Economics 11 Byeong-Hyeon Jeong(d): In long-run, the firm will produce infinite amount if p ą MC “ 5, and pr oduce 0 if p ă M C “5. I f p “ M C “ 5, the firm will choose any amount.(e): In short-run, the firm will produce infinite amount if p ą MC “ 10, and produce 0 if p ăMC “ 10. In case p “ M C “ 10, the firm will choose any amount as long as AV C ă 10. Noticethat for any amount y,AV C “ 10 ´2000yă 10.December 5, 2016