For any production function y = Ax1b with positive values for A and b, output, y, increases without limit as x1 increases. However, if b falls between zero and one, that is, 0 < b < 1, it is still possible to find the quantity of the input x1 that maximizes the profit for the firm.
First, assume that there is a constant output price called p and a constant input price called v, and that these prices vary neither with the quantity of input used nor the quantity of output produced. The assumption of fixed input and output prices is a key assumption of the model of pure competition, in which firms are price takers in both the input and output markets. To illustrate, an individual farmer can produce and sell as much or as little corn as desired without having the amount of corn that he or she produces affect the price for the corn. From a calculus perspective, the market price of corn, p, is a constant.
Further, we assume also that this same farmer could purchase as much or as little nitrogen fertilizer as needed without affecting the market price of nitrogen, so v, the price of the fertilizer, is also a fixed constant. If the price of corn and the price of nitrogen fertilizer are both constants, we say there is perfect competition in both the factor (resource) and product markets.
Assuming a constant product price p, then total revenue to the corn producer is py. In finding the profit‐maximizing level of input use, Total Revenue is often called the Total Value of the Product, and abbreviated as TVP. TVP is the number of bushels of corn produced multiplied by the constant price of corn. Mathematically TVP = p y, the price of corn times the quantity of corn produced. But y = Ax1b, so py = TVP= pAx1b. This is the firm’s production function multiplied by the price of the product, y, or in this case the price of corn.
What about cost? The Total Factor Cost, TFC, is the total cost of the input. Since the price of the input, v, is a constant, TFC = vx1, the price of the input multiplied by the quantity of input used.
Profit, sometimes represented as (the Greek letter pi) is the difference between revenue and costs. We can write
= revenue ‐ costs.
= Total Value of the Product ‐ Total Factor Cost.
= TVP ‐ TFC.
= py ‐ vx.
= pAx1b ‐ vx.
Table 5.3 illustrates these data assuming the price of corn is $7 per bushel and the price of nitrogen fertilizer is $1.50 per pound. From Table 5.3, the profit function appears to be nearly flat for nitrogen application levels between 150 and 170 pounds per acre, generating profits over the cost of the nitrogen of $711.33 at 150 pounds of nitrogen, 711.56 at 160 pounds of nitrogen and
$711.10 at 170 pounds of applied nitrogen (x1). We will calculate the exact amount on nitrogen (x1) that needs to be applied to maximize profits, and the exact profit level that occurs.
Table 5.3 TPP, TVP, TFC and Profit for a Corn Producer, p = $7.00, v = $1.50
Nitrogen TPP TVP TFC Profit
0 0 0.00 0 0.00
2 45.5 318.17 $3.00 $315.17
4 54.1 378.38 $6.00 $372.38
6 59.8 418.74 $9.00 $409.74
8 64.3 449.97 $12.00 $437.97
10 68.0 475.78 $15.00 $460.78
20 80.8 565.80 $30.00 $535.80
30 89.5 626.16 $45.00 $581.16
40 96.1 672.86 $60.00 $612.86
50 101.6 711.46 $75.00 $636.46
60 106.4 744.64 $90.00 $654.64
70 110.6 773.90 $105.00 $668.90
80 114.3 800.17 $120.00 $680.17
90 117.7 824.08 $135.00 $689.08
100 120.9 846.07 $150.00 $696.07
110 123.8 866.48 $165.00 $701.48
120 126.5 885.53 $180.00 $705.53
130 129.1 903.43 $195.00 $708.43
140 131.5 920.32 $210.00 $710.32
150 133.8 936.33 $225.00 $711.33
160 135.9 951.56 $240.00 $711.56
170 138.0 966.10 $255.00 $711.10
180 140.0 980.00 $270.00 $710.00
190 141.9 993.34 $285.00 $708.34
200 143.7 1,006.16 $300.00 $706.16
The data in the Table 5.3 might not be exactly correct, since nitrogen use is in 10 pound increments. Let us use calculus to find out for certain if these numbers are correct.
The profit function is = pAx1b ‐ vx.
The first order conditions for profit maximization require the first derivative of the profit function treating input and output prices (v and p) as fixed constants (see Chapter 10). Then find the point where the derivative is equal to zero. This should correspond with the exact point where the profit function is maximum, or flat. So
= pAx1b ‐ vx.
d/dx1 = bpAx1b‐1 ‐ v = 0, or, find the value for x1 where bpAx1b‐1 = v.
The left‐hand term in this expression, bpAx1b‐1, is the derivative of TVP with respect to x1, or dTVP/dx1. It is the equation for the slope of the TVP function. Economists refer to it as the Value of the Marginal Product function, or VMP for short. Note particularly that it is the Marginal Physical Product function, MPP, multiplied times the price of the product, corn. So VMP = pMPP = bpAx1b‐1. The right‐hand term of this expression, sometimes called the Marginal Factor Cost, or MFC for short, is the derivative of the Total Factor Cost function with respect to x1. So, dTFC/dx1 = dvx1/dx1 = v, assuming that the input price is a constant and equal to v. A graph of a constant is a horizontal line drawn at the value of the constant. If v = 1.75, then the horizontal line is drawn at
The so‐called second order conditions require that dVMP/dx1 is less than dMFC/dx1. But, since MFC is a constant when the input price is a constant when the input price is fixed, its derivative with respect to x1 will be zero.
dVMP/dx1 will be negative so long as b falls between zero and 1, an assumption necessary for the law of diminishing marginal returns to hold!
The profit‐maximizing amount of x1 is the quantity of x1 that solves the equation VMP = MFC for x1. Find the value of x1 where VMP, or bpAx1b‐1. is equal to MFC, or v.
bpAx1b‐1 = v.
x1b‐1 = v/bpA.
x1 = (v/bpA)1/(b‐1).
Now, insert A = 38.222 v = $1.50, p = $7 b = 0.25, and B ‐1 = ‐ 0.75 .
x1 = ($1.75/(0.25 × $7 × 38.222)(1/‐0.75), in steps 0.25 $7 38.222 = 66.888 .
1.50/68.888 = 0.0224256 . 1/‐0.75 = ‐1.3333 .
The number 0.0224256‐1.3333 rounded to two decimals is 158.13 pounds of nitrogen needed to maximize profits. Verify that the profits at this input level are $711.58, slightly larger than at either 150 or 160 pounds of nitrogen. Table 5.4 provides the data for VMP and MFC. Also VMP = MFC should equal 1.55 at x1 = 158.13 pounds of nitrogen fertilizer.
At the profit‐maximizing level of input use, x1 = 158.13, and the slope of the TFC curve (the constant, v) is the same as the slope of the TVP curve at that point. Further, the slope of the profit function will be zero at x1 = 158.13, since profits are maximum at x1 = 158.13 (Figure 5.3).
Figure 5.4 illustrates the VMP and MFC functions. These are the derivatives of the TVP and TFC functions illustrated in Figure 5.3. Note that VMP and MFC intersect (VMP= MFC = 1.75 at the same 128.75 pounds of nitrogen.
VMP is pMPP or, alternately, dTVP/dx1. MPP does not stay constant as the input (x1) use increases, and will decline with increasing input levels if the law of diminishing marginal returns holds. MFC is equal to dTFC/dx1. However, if the input price for x1, v, does not vary with the amount of x1 used, the MFC will be equal to the input price, v.
At the profit‐maximizing level of input use, VMP = MFC, or with a constant input price, VMP =v.
Table 5.4 VMP and MFC for our Production Function Ax1b = 38.222x10.25
Nitrogen TPP TVP TFC Profit VMP = pMPP MFC = v
0 0 0.00 0 0.00 undefined 1.50
2 45.5 318.17 $3.00 $315.17 $39.77 1.50
4 54.1 378.38 $6.00 $372.38 $23.65 1.50
6 59.8 418.74 $9.00 $409.74 $17.45 1.50
8 64.3 449.97 $12.00 $437.97 $14.06 1.50
10 68.0 475.78 $15.00 $460.78 $11.89 1.50
20 80.8 565.80 $30.00 $535.80 $7.07 1.50
30 89.5 626.16 $45.00 $581.16 $5.22 1.50
40 96.1 672.86 $60.00 $612.86 $4.21 1.50
50 101.6 711.46 $75.00 $636.46 $3.56 1.50
60 106.4 744.64 $90.00 $654.64 $3.10 1.50
70 110.6 773.90 $105.00 $668.90 $2.76 1.50
80 114.3 800.17 $120.00 $680.17 $2.50 1.50
90 117.7 824.08 $135.00 $689.08 $2.29 1.50
100 120.9 846.07 $150.00 $696.07 $2.12 1.50
110 123.8 866.48 $165.00 $701.48 $1.97 1.50
120 126.5 885.53 $180.00 $705.53 $1.84 1.50
130 129.1 903.43 $195.00 $708.43 $1.74 1.50
140 131.5 920.32 $210.00 $710.32 $1.64 1.50
150 133.8 936.33 $225.00 $711.33 $1.56 1.50
160 135.9 951.56 $240.00 $711.56 $1.49 1.50
170 138.0 966.10 $255.00 $711.10 $1.42 1.50
180 140.0 980.00 $270.00 $710.00 $1.36 1.50
190 141.9 993.34 $285.00 $708.34 $1.31 1.50
200 143.7 1,006.16 $300.00 $706.16 $1.26 1.50
Note that Profit, , is the difference between revenue and cost, or, in this case, = TVP‐TFC. At the point where the difference between TVP and TFC is maximum, VMP and MFC will be equal.
0.00 200.00 400.00 600.00 800.00 1,000.00 1,200.00
0 20 40 60 80 100 120 140 160 180 200
$
Nitrogen x1
Figure 5.3 TVP, TFC and Profit Functions for y = 38.222x
10.25TVP
Profit
TFC
158.13
$711.58
Profit is maximum using 158.13 units of x1, for a profit of
$711.58.