Question

Hello below is the question , Many thanks for your help :)

A competitive firm uses two inputs, capital (?) and labour (?), to produce one output, (?). The price of capital, ??, is $1 per unit and the price of labor, ??, is $1 per unit. The firm operates in competitive markets for outputs and inputs, so takes the prices as given. The production
function is ?(?,?)=3?0.25?0.25. The maximum amount of output produced for a given amount of inputs is ?=?(?,?) units.

a) Use the method of Lagrange to find the conditional factor demands for cost minimization.


e) Draw the firm’s total cost function, average cost function, and marginal cost function on a diagram. Clearly label the axes, the curves, and any key points on the graph (eg., axis intercepts, curve intersections, and minimums) with the numbers specifying the exact prices and quantities at these points. What are the coordinates of the points where the average cost curve and marginal cost curve intersect with the total cost curve? Show the firm’s long-run supply function on your diagram and write a supply function for the firm

h) Using your supply function, find the profit maximising quantity if the price of output ?=4. What price would be needed for the firm to supply 18 units of output?

Answer 1

.k Com function: Cryp) = 12 Indelicon fumtion: y = {(1,9 - on y = f(HU) - 320.25 0.25 form & the Langrangian probleem is of the Min C8.0. y. fus) : 21+k + 2 (4-380.250.25) Firar Onder Condition: ƏZ = OI - 320 de 10.75 > = 0.75 3 20.25 az = 0 -30 10.25 = 0 or 12 = k0.75 1 320.25 22 - 0 7 yo y = 38025,10.35 =0) - ( 02 From Ol @) 10.75 0.75 10.25 = k0.75 310.26 310.26 Setting in C; y-310.5 = 0 lor, 310.5 = y Binilanly, 16 = 4² which are the conditional factor demains.

6) Zetting in total cost [(y) y funtion, we get en or, kly) - 24², which is the total copt (TC) function is twice Cleanly, Mc of Al INC T, AC, MC the polur. AC cure T at олон АС = TC or 2y = 2 - AL Similarly, MC cuts TC at the point where ML = TC - все A Supply function = MC (t in the long-run) Ar P=A, opply A = 4y or, y = 9, ce, the profit maximising quantity for P=4 ip y = 9 Ar og = 18; P= AX 18 = p = 8), cide, the price needs to Supply 18 unipo is