Question

2. A firm has two variable factors and a production function f(11, 12) = 211 + 4.12. (a) On a graph, draw production isoquants corresponding to an ouput of 3 and to an output of 4. (b) If the price of the output good is 4, the price of factor 1 is 2, and the price of factor 2 is 3, find the amount of factor 1, the amount of factor 2 and the amount of output that maximizes the profit.

Answer 1

A)T production isoquant can be plotted by simply plotting the coordinates on the x1-x2 plane.

dues a) f(x4, de) = 524+4 42 we Ham to plost the isoquants for output levels and 4. 0 52x4+ urz : 3 " Jerut ane : 4 2x2 + 442 = 9 2x4+ 442 = 16 au lo alla 2 1018 as lulo ular lo __24+ 412 : 16 2x414x₂ = 9 - 1 2 3 4 5 6 7 8

B) Given is a linear production function, it has perfect substitutability between the factors , therefore logically that factor is used which is cheaper for the production process as compared to the productivity of that factor. This can be proved :

MRTS = MPx1/ MPx2 = 2/4 = =0.5

Given, Price Ratio = Px1/Px2 = 2/3 = 0.67

This implies x1 s half as productive as x2 but costs more than half that of x2, therefore, using x2 is a better deal.

- optimal Point . Wil line - 24+ " 2x4+4+3=9 Ul2 = 16

From the isoquant and cost line, it can be seen that given the cost line , the highest isoquant that can be achieved is as shown (2x1 + 4x2 = 16) and it can be produced only using x2.

Hence , amount of factor1 is 0,

Amount of factor 2 is :

Hence, MC= MR gives , 6Q/4 = 4 which gives Q=8/3

MR= p (price at 34 Output) Guinen a, 4=0 f=5204 + 4 =Q 442 = Q² *2=02 Total cost = 342 = 30² me= 60/4

Hence amount of factor 2 is (8/3)×(8/3) /4=16/9