ccer各种习题作业-2003-秋20次

9,October28,1yx4x1 1y(xx)1/, Solution:(1)Themarginalproductofinput1 11MP 21

.Andwe 1

x

7/4

4x1

Themarginalproductofinput2 3

1/2MP 12

.Andwe 2x

x5/4

4x2

Thusthemarginalproductofbothinputsisdiminishing.Thetechnicalrate-of-substitutionisasfollow:

1x23Soit’sobviousthattheabsolutevalueofthetechnicalrate-of-substitutiondiminishesasffkx,kxkx1/43/4kx1/4kfx,x,theproduction 1212 exhibitsconstantreturn-to-(2)Themarginalproductofinput11

MP

yy

.Andwehave:MP11y1

x

x x 1 1 x x 1 Themarginalproductofinput21

y

.Andwehave:MP21y1

1x x 1 2 2 1 x 2 Thusthemarginalproductofbothinputsherealsoisdiminishing.Thetechnicalrate-of-substitutionisasfollow:2 2 xxdx MP2 x 1Soit’sobviousthattheabsolutevalueofthetechnicalrate-of-substitutiondiminishesas fkx,kxkx

1/kxx1/kfx,x,the functionexhibitsconstantreturn-to-scalef(x,x)x1/2x1/给定生产函 x216时，求使利润最大化的 fxfx,x4x1/2,thentheproblem maxpfx1w1x1w2x2 Foc:2px1/2 22p1Thuswegettheoptimalinput x* 1w1x, x*,当 2都可变时，求使利润最大化的 (2)It’seasytonoticethattheproductionfunctionexhibitsconstantreturn-to-scale.Theumprofitinthissituationwillbeeitherpositivelyinfiniteorzerodependingonsuchvariables Theproblemwenowfacemaxpfx1,x2w1x1w2x2px 22x1px

12x2

x Whichcanbereduced 2 1x1 x1Theumprofit(valuefunction)canthusbedenoted2p,w,wpfx*,x*wx*wx*x*w11/2p2ww052 1 2 1

Ifp2w

0 p,w,w,andx*x* , ,Ifp2w

p,w,w0,andx*w2x* 20 20, 2 Ifp2w1w205,p,w1,w20,andx*x*0 f(x,x)x1/2x1/给定生产函 p4,ww x*, ，求使利润最大化的 max:4x1/2x1/4x