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.
j)  Suppose x2 is fixed at some level x2. Graph the solution to the profit maximization problem where profit is maximized by choosing x1. [Note that this profit maximization problem is the short-run version of the profit maximization problem

(k) Suppose that our hypothetical firm operates in a perfectly competitive industry with free entry and exit. How much profit can the firm expect to make in the long run? Why? Explain your answer.

Answer 1

The perfectly competitive firm 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) It is given that x2 is fixed at some level x2.

Hence, The production function becomes a function of x1 only in the short-run i.e.

y = f(x1,x2), where x2 = constant

Hence, the Total Revenue from production is

TR = p×y = p.f(x1,x2).........(1)

Also the Total Cost of production is

TC = w1.x1+w2.x2.............(2)

Hence, the Total Profit is

π = TR - TC

or, π(x1,x2) = p.f(x1,x2) - (w1.x1+w2.x2).........(3)

Now in the profit function, x2 is constant. Hence π is a function of x1 only. Hence we will maximize the profit with respect to x1 only. This is the short-run profit maximization.

We will differentiate the profit function with respect to x1 and will set it equal to 0 for maximizing π at the optimim input level x1*.

Hence, dπ/dx1 = 0

or, p.fx1(x1,x2) - w1 = 0........(4)

Where, f​​​​​​x1​​​​(x1,x2) = d{f(x1,x2)}/dx1

From equation 4, we will be able to solve the value of x1* for which the profit is maximized as function of w1 and p in the short run.

Hence, the solution would be

x1* = x1*(w1,p)

The solution is graphed in the following diagram.

From the above diagram we can see, the short-run profit is maximized at point E.

(b) In the long run, the x2 is no longer fixed at x2. Hence, the profit function in equation (3) must have to be differentiated with respect to x2 also. Hence the first order conditions or FOCs are as following,

dπ/dx1 = 0

or, p.fx1(x1,x2) - w1 =0..........(5) and

dπ/dx2 = 0

or, p.fx2(x1,x2) - w2 = 0.........(6)

Now we will solve these two equations (5) and (6) to get the optimal values of x1* and x2* for which the long run profit is maximized.

Hence, if we solve these equations, we will get x1* and x2 * as functions of w1, w2 and p.

Hence, x1* = x1*(w1,w2,p) and

x2* = x2*(w1,w2,p)

Hence, these are the profit maximizing inputs. If we put these input values in the profit funcrion, we will get the maximum profit in the long run.

Hence, long run maximum profit is

π*=p.f{x1*(w1,w2,p),x2*(w1,w2,p)} - [w1.x1*(w1,w2,p) + w2.x2*(w1,w2,p)]

or, π* = π*(w1,w2,p)...........(7)

This is the maximum profit the firm can make in the long run. The reason is that, in the long run both the inputs are variable and they are optimized from our above calculations. Hence this is the long run maximum profit for the optimal long run input combination (x1*,x2*).

Hope the solution is clear to you my friend.