Question

(a) Consider a one input production technology utilizing labor that is represented by the function, x = Aln(L) with A>0. Is

the producer’s choice set convex? Show your working.

(b) Find the total cost function associated with this production technology.

Note: Show all necessary working and steps so that understanding the solution is easy.

Answer 1

a).

Consider the given problem here the production function is given by, => X = A*ln(L), where A > 0.

=> So, the 1st order differentiation is, dX/dL = A/L > 0 for all “L>0”.

=> The 2nd order differentiation is, d^2X/dL^2 = (-1)*A/L^2 < 0 for all “L>0”. So, here the production function is upward sloping concave in nature. Consider the following fig.

Here OA represent the production function and the shaded region is the production choice set of a producer. Now, if we take any two point in the producer choice set and connect the two point with a dotted line, then the entire line will also belong to the same production set. So, the production choice set convex set.

b).

Here the production function is, => Y = A*ln(L), => Y/A = ln(L), => e^(Y/A) = L, => L = e^(Y/A).

So, the cost function is given by.

=> C = W*L + R*K, “K=capital is fixed”.

=> C = R*K + W*[e^(Y/A)].