PhD Macroeconomics question

Profit Maximization

Profit (π) = Total Revenue (TR)- Total Cost (TC)

TR = QxPQ (price times quantity sold)

TC = I x PI (cost times input purchased)

There are two ways to find maximum profit:

A. Graphical way

B. Calculus

Graphical approach to profit maximization focuses on maximizing profit by comparing the total revenue (TR) and total cost (TC) of a firm observing behavior of the TR and TC curve (Figure 1).

Figure 1. Profit maximization : where there is maximum distance between TR curve and TC curve

Calculus approach

Firm aims to maximize its profit:

Π = TR – TC

Where π = profit, TR = total revenue and TC = total cost

Clearly, TR = f(Q) and TC = f(Q), given the price P.

a. The first-order condition for the maximization of function is that its first derivative (with respect to Q in our case) be equal to zero.

=

The term ∂TR/∂Q is the slope of the TR, that is, Marginal revenue (MR) and ∂TC/∂Q is the slope of the Total Cost curve, that is, Marginal Cost (MC). Thus, first-order condition for profit maximization is

MR = MC

Given that MC> 0, MR must also be positive at equilibrium. Since MR = P, the f.o.c. may be written as MC = P.

b. The second-order condition for maximization requires that the second derivative of the function be negative (less than zero).

- < 0

Which yields the condition

<

(slope of MR)< (slope of MC)

Thus, the MC must have a steeper slope than the MR curve or the MC must cut the MR curve from below. In pure competition the slope of the MR curve is zero.

Here is one numerical example of profit maximization using calculus.

A competitive poultry firm estimated both revenue and cost functions of a business as: Total Revenue(TR)= 2000Q – 10Q2 and Total Cost (TC)= 2000 + 500Q.

Step 1: Set a profit function (Profit= Revenue- Cost).

π = 2000Q – 10Q2 – (2000 + 500Q)

π = -10Q2 + 1500Q – 2000

Step 2: Find the first order derivative of the profit function/equation and set the function equal to ZERO.

-20Q + 1500 = 0

Q = 75 (this is the answer to how many units are to be produced for π maximization)

For example, the profit equation -10x2 + 1500x – 2000 becomes -20x + 1500.

Step 3: Calculate the  maximum profit using the number of units produced calculated in step 2. Inserting Q = 75 into the profit function

π = -10Q2 + 1500Q – 2000

π = -10(75)2 + 1500(75) – 2000

π= 54,250

Check s.o.c. for the profit maximization i.e. less than zero.

21