Question

Consider a perfectly competitive firm that produces output using a wellbehaved production function y = f(x1,x2). Let the price of output be denoted by p, the price of input 1 be denoted by w1, and the price of input 2 be denoted by w2.
(a) State the firm’s cost-minimization problem, and draw the solution to the cost minimization problem on a graph.
(b) What are the solutions to the cost-minimization problem? What are these solutions called? Which variables are these solutions a function of?
(c) Which 2 conditions must be satisfied for an input bundle to be considered cost minimizing, assuming an interior solution? Why? Explain your answer.
(d) Derive the firm’s cost function. What role do your answers in part (b) play in deriving the cost function?

Answer 1

Acceoding to the given informations, the perfectly competitive firm that produces output using a well behaved production function y = f(x1,x2). Let the price of output be denoted by p, the price of input 1 be denoted by w1, and the price of input 2 be denoted by w2.

(a) The cost of producing x1 units of input 1 is w1.x1 and the cost of producing x2 units of input 2 is w2.x2.

Hence, total cost of production is

C = w1.x1+w2.x2.......(1)

and to minimize the cost subject to the production function, we need to set the production at a fixed level and the minimize cost of producing that level of y.

Let's say y=f(x1,x2) is fixed at y°.

Hence, the cost minimization problem is

Minimize C = (w1.x1+w2.x2) subject to

f(x1,x2) y°.

This is the cost minimization problem.

The solution to the problem is shown below in the diagram.

From the above diagram we can see the solutions are labelled in the graph.

(b) Hence, we need to write down the Lagrange's equation to solve the problem above.

L=w1.x1+w2.x2 + a.{y° - f(x1,x2)}, where a>0.

Hence, the first order conditions are,

dL/dx1 = 0........(1) and

dL/dx2 = 0........(2).

The solutions to the xost minimization problem would be

x1* = x1*(w1,w2,y°) and

x2* = x2*(w1,w2,y°)

These are the solutions of the cost minimization problem.

These solutions are called "Conditional Input Demand Functions".

These solutions are functions of w1, w2, y°.

(c) The conditions, which must be satisfied for an input bundle to be considered cost minimizing, assuming an interior solution are written below.

The above 2 conditions (3) and (4) must be satified for an input bundle to be cost minimizing. The reason is, conditions (3) and (4) are the First Order Conditions or FOCs for minimizing the Cost of Production. If these conditions are satisfied, the inputs must be cost minimizing.

(d) Now we will put our solutions which are

x1*=x1*(w1,w2,y°) and x2*=x2*(w1,w2,y°), in the Cost Function. Hence,

C* = w1.x1*(w1,w2,y°)+w2.x2*(w1,w2,y°)

or, C* = C*(w1,w2,y°)

This is the cost function.

The answer in part (b) helps to use the values of optimal input bundle in the equation of Cost Function to determine the Cost Function. Here this is the minimum cost as function of w1, w2 and y°.

Hope the solution is clear to you my friend.