Question

Consider a firm producing one output using two inputs, capital and labor. If the weak axiom of revealed profit maximization holds, which of the conditions below describes the constraint implied by profit maximizing behavior across any two periods?

1. delta(p)delta(q) >= delta(w)delta(L) - delta(r)delta(K)

2. delta(p)delta(q) <= delta(w)delta(L) + delta(r)delta(K)

3. delta(p)delta(q) >= delta(w)delta(L) + delta(r)delta(K)

4. delta(p)delta(q) <= - delta(w)delta(L) + delta(r)delta(K)

Answer 1

Let us suppose a firm uses two inputs Labor i.e. L and capital i.e. K and the labor costs w and capital rent is r.

The firm produces output i.e. y and sells at a price p.

Now, suppose there are two periods.

• In period 1 the firm uses input combination (L1,K1). The input price combination is (w1,r1). Output is q1 and output price is p1.

• In period 2 the firm uses input combination (L2,K2). The input price combination is (w2,r2). Output is q2 and output price is p2.

Now, if the WARM or Weak Axiom of Revealed Profit Maximization holds and the production function has not changed between two periods, then we can write the following conditions,

✓ p1.q1-w1.L1-r1.K1 p1.q2-w1.L2-r1.K2......(1)

And

✓ p2.q2-w2.L2-r2.K2 p2.q1-w2.L1-r2.K1.......(2)

If any of the above inequality is violated, the firm will not reamin a profit maximizing firm.

Now, multiplying (-1) with equation 2 gives

-p2.q1+w2.L1+r2.K1 -p2.q2+w2.L2+r2.K2.....(3)

Now, adding equations (1) and (3), we get

(p1-p2).q1-(w1-w2).L1-(r2-r2).K1 q2.(p1-p2) -L2(w1-w2)-K2.(r1-r2)

or,

(p1-p2).(q1-q2)-(w1-w2).(L1-L2)-(r1-r2).(K1-K2)0

or, ∆p.∆q - ∆w.∆L - ∆r.∆K 0

Hence,

∆p.∆q ∆w.∆L+∆r.∆K

Hence, option (3) is the correct answer.

Hope the solution is clear to you my friend.