Question

Problems 1 a) A firm has the production function y = 22212 and faces input prices W1 and w2. Derive the conditional input demand functions for both inputs.

please show all work, thanks!

Answer 1

Solution-

Conditional input demand function describes the input demand function in terms of output level and input prices.

In order to determine the conditional input demand, we set a Lagrangian function. Here the objective function is of output subject to the constraint of cost function (C=w1z1 +w2z2)

1a) 1/2 y = 2, 22 Objective function] cost Function (C) = w, 2, + W₂ Zz [Constraint] we have to set a (6) to derive the input demand inputs. lagrangian function conditional a functions for both d = Z, 22 " + x[K- W, 2, - W₂ Zz] ad = 22 ²2 _aw, aa = 22,22 2 - 1 W2 az 2 da = K-W, 2, - W₂ Zz To maximize output and minimize cost, we have to set Lag rangian function to o.

da = zz!__ 1w, = 0 Z2/2 = n wi n = Z212 = I 2, 222_7 W2 =0 n = ₂2, 22 Wz da an =k- w, z, - W₂ Z2 = 0 k=w,2, + W₂ Zz - Equating C & W we get : - 1/2 w 2 = 12, Z₂ / 2 -1/2 2 - W2 - 922 wi - 12 Wz 2.22 Z1 Wz

Putting for value of 2, from equation (2) in output function Y = Z, 22 2 1/2 y= 2 W2 Zz. 2212 ow, 3/2 Z2 = eear 22 Jolor input demand Thus, the conditional function for 2₂ is

wa calls y 23 (W x y la Thus, the conditional input-demand function for 2, is 1216= (a w..

Minimum cost will be (* = W, 2,*+ W₂ Zz * Here, w,= $5 and Wq = $10 output= 27 units] (* = 52,*+ 10 22 * So, 2*= (2. z 1² , 2/3 2. 2015. 27213 413x9 = = 1.587489 - 14.286

NL = [x27x5125 = 277213 10 = 3. 57 16 Thus, (* = 52*+1022* i 5814.286 + 10x3.5716 = 71. 43 + 35. 716 a T* 107.146 107.15 =

b) Thus the minimum cost associated with production of 27 units of output is $107.15. ( Minimum cost is found by putting conditional input demand functions for both inputs in place of z1 and z2 in the cost function)