Profit Maximization:The First-order Condition

ATC, MC

MC

ATC

0 Q1 Q2 Q3 Q4

Q5 Q

FIGURE 2.14 The relationship between average total cost and marginal cost.

At output level Q4 the slopes of the ray and tangent are identical (ATC = MC).Thus,at Q4 ATC is neither rising nor falling (i.e., dATC/dQ = 0). After Q4 the slope of the tangent not only becomes greater than the slope of the ray,but the slope of the ray changes direction and starts to increase. Thus,we see that at output level Q5, MC > ATC and ATC are rising.These relationships are illustrated in Figure 2.14.

The situation depicted in Figure 2.14 illustrates a U-shaped average total cost curve in which the MC intersects ATC from below.The reader should visually verify that when MC < ATC,even when MC is rising, ATC is falling. Moreover,when MC > ATC,then ATC is rising.Finally,when MC = ATC, then ATC is neither rising nor falling (i.e., ATC is minimized).In some cases, the average curve is shaped not like U but like a hill:that is,the marginal curve intersects the average curve from above at its maximum point.An example of this would be the relationship between the average and marginal physical products of labor,which will be discussed in detail in Chapter 5.

PROFIT MAXIMIZATION: THE FIRST-ORDER CONDITION

We are now in a position to use the rules for taking first derivatives to find the level of output Q that maximizes p,as illustrated in Table 2.3.Consider again the total revenue and total cost functions introduced earlier:

(2.67)

TRQ PQP ( ) = =; $18 TCQ Q Q Q ( ) = + - + 6 33 9 2 3 p= - = - + - +( ) TR TC Q Q Q Q 18 6 33 9 2 3 p=- - + 6 15 9 2 3Q Q Q

It should be noted in Table 2.3 and Figure 2.11 that profit is maximized (p= 19) at Q = 5.What is more,it should be immediately apparent that if a smooth curve is fitted to Figure 2.11,the value of the slope at Q = 5 is zero:that is,the profit function is neither upward sloping nor downward sloping.Alternatively,at Q = 5,then dp/dQ = 0.These observations imply that the value of a function will be optimized (maximized or minimized) where the slope of the function is equal to zero.In the present context,the first-order condition for profit maximization is dp/dQ = 0,thus

dp dQ Q Q =- + - = 3 18 15 0 2 (2.68)

This equation is of the general form:

ax bx c 2 + + = 0 (2.69)

where a =-8, b = 10 and c =-15.Quadratic equations generally admit to two solutions,which may be determined using the quadratic formula.The quadratic formula is given by the expression:

x12 , b b ac 2 4- ± 2a (2.70)

After substituting the values of Equation (2.68) into Equation (2.70) we get

Q1 = 2 18 18 4 3 15 2 3 18 324 180 4 18 12 -6 30 6

- + ( ) - -( ) ( ) 5

Q2 = 18 12- + -6 66= 1

Referring again to Figure 2.11,we see that the value of p reaches a minimum and a maximum at output levels of Q = 1 and Q = 5,respectively. Substituting these values back into the Equation (2.67) yields values of p=-13 (at Q = 1) and p= 19 (at Q = 5).In this example,therefore,the entrepreneur of the firm would maximize his profits at Q = 5.As this example illustrates,simply setting the first derivative of the function equal to zero is not sufficient to ensure that we will achieve a maximum,since a zero slope is also required for a minimum value as well.Thus,we need to specify the second-order conditions for a maximum or a minimum value to be achieved.