Question

1- Suppose that a firm producing commodity with the following production function: Y = 20X,X2 Then, assume that the maximum amount the firm can spend on these two inputs is $100 and price of commodities are as follow: Xi = 4, X2 = 5 a. Use Lagrange Multiplier to determine the optimal production level at this firm. b. What is the meaning of shadow price? How you can interpret it using the solution of part a? 2- Assume the following utility function for cloth (C) and food (F) U = 200 + CF - 5F2 - 6 C2 and the consume wants to maximize her utility U, but has a limited budget of 24 unit. The price of cloth is 1 unit and the price of food is 2 unit. on a- Use equal marginal principle approach to determine the optimal consumption point b- Approximately show the optimal consumption point in a figure

Answer 1

Insuur - Production function = 20x, xz Manimum amount of spend = $100 brice of input x = 4$ brinog Input 12 = 54 spending function » 100 = 4x + 5X2 Applying Langa ragian Rose multiplier me é z = 20 X, X, - al 100-ux, 5x2) first evedere condition for manimi sation sutbust 02 - 20X2 - 0(-4)=0 Jai 2002 = -04 - 22 20x, -d1-5) =0 20x₂= -05 - ə X2 22 - 100 - ux, - SX ₂ =0 - 100 = ud, +5X2=

By Eq 0 a 20 82 .208, 43 S X₂ = 4x, Put aboue value of xz into Ea - 3 100 = 4x + 5 X₂ 100 = uxit uxos 5 1072 иҮ, Тих, о 100 = 8X X = 100 - 12.5 _ ī Putting about the function ^2=442 4x12.5 = 10

- The optimal broduction of 2 X, & X2 are x = 12.5 & X2 = 10 here. maximum Production is. Y=20x1x2 -20 (12.5) (10) = 20 X125 = 2500 manimun droduction is 2500 cose output leul shadow Price : brice Indicating the true value of factors de broduct in the sense of Equilibrium beice. It is estimate buire of good or factor for which no marpet brice Enist. In aboue Pareta lase the shadow brice is bor x = 12.5 &e for U₂ =10 it show that the at these production level & the production is maximise at the gine Production function,

Ans 2 mareginal Principal Approach marginal whitity off - brice off mereginal utility oc brice of mof=20 U= 200-cf-52-662 tes ao :-C-1012 df moci do oc -f - 126 marginal Approach brice of cloth = 1 brine of food = 2

MOF med muc - -10F 2 -f - 12c | tCt0f)2 + (F +120) | +10f= 라 +2uC 10 -20 - 21C-C gf = 23C f= 23C constraint =24 24= 2f +IC 124 = ( c) 왜품

- 24= 23 ctc au= 23C IUC 4. 24. u= 23c+UC a6 = 270 c = 96 = 3.555 27 F-230 the optimal level of Consumption of food is 10:2235 unit & for cloth 3.555 unit. f = 23 . 3.555 e = 10.2235