Chapter 7
Production Theory and Estimation
7.1 THE PRODUCTION FUNCTION
Production refers to the transformation of inputs or resources into outputs of goods and services. Inputs can be broadly classified into labor (including entrepreneurial talent), capital, and land or natural resources. Fixed inputs are those that cannot be readily changed during the time period under consideration. Variable inputs are those that can be varied easily and on short notice. The time period during which at least one input is fixed is called the short run. If all inputs are variable, we are in the long run.
A production function is an equation, table, or graph showing the maximum output of a commodity that a firm can produce per period of time with each set of inputs. Inputs and outputs are usually measured in physical rather than monetary units. Technology is assumed to remain constant during the period of the analysis. The general equation of the production function of a firm using labor (L) and capital (K) to produce a good or service (Q) can be written as
Q = f (L, K) (7-1)
EXAMPLE 1. Table 7.1 gives a hypothetical production function, which shows the outputs (the Q’s) that the firm can produce with various combinations of labor (L) and capital (K). The table shows that by using one unit of labor (1L) and one unit of capital (1K), the firm would produce 1 unit of output (1Q). With 2L and 1K, output is 4Q. with 3L and 1K, output is 9Q; with 3L and 2K, output is 15Q; with 4L and 2K, output is 16Q; and so on.
EXAMPLE 2. The production function of Table 7.1 is shown in the top panel of Fig. 7-1, where the height of the bars refers to the maximum output that the firm can produce with each combination of labor and capital shown on the axes. If we assume that inputs and outputs are continuously or infinitesimally divisible (rather than being measured in discrete units), we would have the continuous production surface shown in the bottom panel of Fig. 7-1. This indicates that by increasing L with K1 of capital the firm produces the output shown by the height of cross section K1AB (with base parallel to the labor axis). Increasing L with K2, we have cross section K2CD. Increasing K with L1. we have cross section L1EF (with base parallel to the capital axis).
Fig. 7-1
Table 7.1
Capital (K) / Output (Q)K
/ 6 / 6 / 14 / 16 / 20 / 24 / 255 / 9 / 15 / 20 / 24 / 26 / 24
4 / 9 / 15 / 20 / 24 / 24 / 20
3 / 7 / 16 / 18 / 20 / 20 / 18
2 / 4 / 12 / 15 / 16 / 15 / 12
1 / 1 / 4 / 9 / 12 / 12 / 9
L
/ 1 / 2 / 3 / 4 / 5 / 6Labor (L)
7.2 PRODUCTION WITH ONE VARIABLE INPUT
By changing the quantity used of one input while holding constant the quantity used of another (so that we are in the short run), we generate the total product (TP) of the variable input. We then use TP to derive the marginal and average products. The marginal product (MP) is the change in total product per unit change in the variable input used. The average product (AP) equals total product divided by the quantity used of the variable input. Output elasticity measures the percentage change in output, or total product, divided by the percentage change in the variable input used. If the variable input is labor, we have
(7-2)
(7-3)
(7-4)
Rewriting equation (7-4) in a more explicit form and rearranging, we get
(7-5)
In the short run, we have the law of diminishing returns. It postulates that, after a point, the marginal product of a variable input declines. We can also define the stages of production. Stage I covers the range of increasing average product of the variable input. Stage II covers the range from the point of maximum average product of the variable input to the point at which the marginal product of the input is zero. Stage III covers the range of negative marginal product of the variable input. (See Example 4.)
EXAMPLE 3. With capital held constant at K = 1 and labor increasing from L = 0 to L = 6, we have the total product given in the last row in Table 7.1, which is reproduced in column (2) of Table 7.2. Note that adding the sixth unit of labor leads to a decline in TP as workers start getting in each other’s way. From the TP schedule, we derive the MPL and APL schedules [columns (3) and (4)1 and, from them, the EL schedule [column (5)].
EXAMPLE 4. The TP, MPL, and APL schedules of Table 7.2 are plotted in Fig.
7-2. Note that each value of MPL is plotted halfway between the quantities of labor used. Past 2.5L, the MPL curve declines (i.e., the law of diminishing returns begins to operate). The MPL curve intercepts the APL curve at its highest point. The APL curve rises when the MPL is above it, and falls when the MPL curve is below it. The bottom panel of Figure 7-2 also shows the stages of production for labor. A rational producer (firm) would produce only in stage II, where MPL is positive but declining. [See Problem 7.5(b).1
Table 7.2 Total, Marginal, and Average Product of Labor, and Output Elasticity
L
/ TP / MPL = DTP / DL / APL = TP / L / EL = MPL / APL( 1 ) / ( 2 ) / ( 3 ) / ( 4 ) / ( 5 )
0 / 0 / –– / –– / ––
1 / 1 / 1 / 1 / 1
2 / 4 / 3 / 2 / 1.5
3 / 9 / 5 / 3 / 1.67
4 / 12 / 3 / 3 / 1
5 / 12 / 0 / 2.4 / 0
6 / 9 / –3 / 1.5 / –2
Fig. 7-2
7.3 OPTIMAL USE OF THE VARIABLE INPUT
The marginal revenue product (MRP) of a variable input equals the marginal product (MP) of the input times the marginal revenue (MR) received from the sale of the extra output produced. If the variable input is labor (L) and the commodity price (P) is constant (so that MR = P), we have
MRPL = (MPL) (MR) = (MPL) (P) (7-6)
The marginal resource cost (MRC) of a variable input, is equal to the increase in total costs that results from hiring an additional unit of the variable input. If the variable input is labor and the wage rate (w) is constant, we have
(7-7)
As long as MRP exceeds MRC, it pays for the firm to expand the use of the variable input because by doing so it adds more to its total revenue than to its total costs (so that the firm’s total profits rise). The firm should not hire those units of the variable input for which MRP falls short of MRC. Thus, the optimal use of the variable input (i.e., the quantity at which the firm maximizes profits) is at MRP = MRC. If the variable input is labor and the wage rate is constant, the firm should hire labor until
MRPL = w (7-8)
EXAMPLE 5. Column (2) in Table 7.3 gives the marginal product of labor as read off from the MPL curve in stage II, in the bottom panel of Figure 7-2. The fractional units of labor are based on the assumption that the firm can hire labor for half a day at a time. Column (3) gives P = MR = $10. Column (4) gives MRPL, which is equal to (MPL) (MR). Column (5) gives MRCL = w = $30 for each half day of work. In order to maximize profits the firm should hire 3.5L, at which MRPL = MRCL = w. This is shown at point H* in Figure 7-3. Note that MRPL = DL represents the firm’s demand curve for labor.
Fig. 7-3
Table 7.3
L
/ MPL / P = MR / MRPL = (MPL) (MR) / MRCL = DTC / DL = w( 1 ) / ( 2 ) / ( 3 ) / ( 4 ) / ( 5 )
2.5 / 5 / $10 / $50 / $30
3.0 / 4.5 / 10 / 45 / 30
3.5 / 3 / 10 / 30 / 30
4.0 / 1.5 / 10 / 15 / 30
4.5 / 0 / 10 / 0 / 30
7.4 PRODUCTION WITH TWO VARIABLE INPUTS
The production function with two variable inputs can be depicted graphically with isoquants. An isoquant shows the various combinations of two inputs (say, labor and capital) that a firm can use to produce a specific level of output. A higher isoquant refers to a larger output, while a lower isoquant refers to a smaller output. A firm would never operate on the positively sloped portion of an isoquant because the firm could produce the same output with less labor and capital (see Example 6). Ridge lines separate the relevant (i.e., negatively sloped) from the irrelevant (or positively sloped) portions of the isoquants. The absolute value of the slope of the isoquant is called the marginal rate of technical substitution (MRTS). MRTS = MPL / MPK and diminishes as we move down an isoquant, so that the isoquant is convex to the origin. The smaller the degree of curvature of an isoquant, the greater is the degree of substitutability of inputs in production.
EXAMPLE 6. In Table 7.1, we saw that 9 units of output can be produced with 3L and 1K, 6L and 1K, 1L and 4K, or 1L and 5K. These are shown by isoquant 9Q in Fig. 7-4. The figure also shows the isoquants for 15Q, 20Q, and 24Q. The firm would never produce 9Q with 6L and 1K (point T) because it could produce 9Q with 3L and 1K (point S). Similarly, the firm would not produce 9Q with 1L and 5K (point M) because it could produce 9Q with 1L and 4K (point N). Thus, only the negatively sloped portion of the isoquants between the ridge lines is relevant. The firm can move from point N to point R on isoquant 9Q by substituting 0.5L for 2K. Thus, the absolute slope or MRTS of isoquant 9Q between points N and R is 2K/0.5L = 4. The MRTS at point R is 4/3 (the absolute value of the slope of the tangent to the isoquant at point R). Note that as we move down an isoquant its absolute slope or MRTS diminishes so that the isoquant is convex to the origin.
Fig. 7-4
7.5 OPTIMAL COMBINATION OF INPUTS
To determine the optimal combination of labor (L) and capital (K) for the firm to use, we also need an isocost line. This shows the various combinations of L and K that the firm can hire or rent at given input prices (w and r) and total cost (C). That is
C = wL + rK (7-9)
By subtracting wL from both sides of equation (7-9) and then dividing by r, we get
(7-10)
where C/r is the vertical intercept of the isocost line and –w / r is its slope.
The optimal combination of inputs used to minimize costs or maximize output is given at the tangency point of an isoquant and an isocost. At the tangency point, the (absolute) slope of the isoquant (MRTS = MPL / MPK) is equal to the (absolute) slope of the isocost line (w / r). That is
(7-11)
By cross multiplying. we get
(7-12)
That is, to minimize costs or maximize output, the marginal product per dollar spent on labor should be equal to the marginal product per dollar spent on capital.
Finally, to maximize profits a firm should produce the profit-maximizing level of output with the optimal (least-cost) input combination. This occurs when the firm employs each input until the marginal revenue product of the input equals the marginal resource cost of hiring the input. With constant input prices, this condition becomes
MRPL = (MPL) (MR) = w (7-13)
MRPK = (MPK) (MR) = r (7-14)
EXAMPLE 7. If C = $12, w = $4 and r = $3, the firm could either hire 3L or rent 4K, or any combination of L and K shown on isocost line AB in Fig. 7-5. The equation of the isocost line is then
Fig. 7-5
and the firm must give up 4/3K for each additional unit of L it wants to hire. The lowest cost of producing 9Q is given by point R, where isoquant 9Q is tangent to isocost line AB and
Thus, the firm should use 1.5L and 2K at a total cost of C = $12. If w declined from
w = $4 to w = $2, while r increased from r = $3 to r = $6 and C remained at $12, the new isocost line would be A"B' (see Fig. 7-5). The firm would then substitute L for K in production until it reached the new optimal input combination at point S. By changing total costs or expenditures while holding input prices constant, the firm can define different but parallel isocost lines. By joining tangency points of isoquants with isocosts the firm defines its expansion path [see Problem 7.17(a)]. By hiring labor and renting capital until
MRPL = w = $4 and MRPK = r = $3
the firm will produce the profit-maximizing level of output with the optimal input combination [see Problem 7.20(b)].
7.6 RETURNS TO SCALE AND EMPIRICAL PRODUCTION FUNCTIONS
If the quantity of all inputs used in production is increased by a given proportion (so that we are in the long run), we have constant returns to scale if output increases in the same proportion as the increase in inputs, increasing returns to scale if output increases by a greater proportion than the increase in inputs, and decreasing returns to scale if output increases by a smaller proportion than the increase in inputs (see Example 8). Increasing returns to scale arise because, as the scale of operation increases, a greater division of labor and specialization can take place and more specialized and productive machinery can be used. Decreasing returns to scale arise primarily because, as the scale of operation increases, it becomes more difficult to manage the firm effectively and to coordinate its various operations and divisions.
The most commonly used production function is the Cobb-Douglas of the form
Q = AKa Lb (7-15)
where Q, K, and L refer, respectively, to the quantities of output, capital, and labor, and A, a, and b are the parameters to be estimated. Parameters a and b represent, respectively, the output elasticities of capital and labor (EK and EL). If a + b = 1 we have constant returns to scale, if a + b > 1 we have increasing returns to scale, and if a + b < 1 we have decreasing returns to scale. To estimate the Cobb-Douglas production function by regression analysis, we must first transform it:
ln Q = ln A + a ln K + b ln L (7-16)
EXAMPLE 8. The left panel of Fig. 7-6 shows that doubling inputs from 3L and 3K to 6L and 6K doubles output from 100 (point A) to 200 (point B). Thus, OA = AB along ray OE, and we have constant returns to scale. The middle panel shows that output triples by doubling inputs. Thus, OA < AC (i.e., the isoquants come closer together) and we have increasing returns to scale. The right panel shows decreasing returns to scale. Here, output changes proportionately less than labor and capital, and OA > AD. Empirical estimates of the Cobb-Douglas production function indicate that most industries exhibit near-constant returns to scale (see Problem 7.23).