We have shown that any given level of output can be produced by many combinations of inputs—as illustrated by isoquants. When a manager wishes to produce a given level of output at the lowest possible total cost, the manager chooses the combination on the desired isoquant that costs the least. This is a constrained minimization problem that a manager can solve by following the rule for constrained optimization set forth in Chapter 3.
F I G U R E 9.3
Shift in an Isocost Curve
1 2
1 2
Capital (K)
10 8 6 4 2
0 2 6 10 14
Labor (L)
4 8 12 16 18 20
12
K = 10 – L
K = 8 – L
Now try Technical Problem 2.
Although managers whose goal is profit maximization are generally and primar- ily concerned with searching for the least-cost combination of inputs to produce a given (profit-maximizing) output, managers of nonprofit organizations may face an alternative situation. In a nonprofit situation, a manager may have a budget or fixed amount of money available for production and wish to maximize the amount of output that can be produced. As we have shown using isocost curves, there are many different input combinations that can be purchased for a given (or fixed) amount of expenditure on inputs. When a manager wishes to maximize output for a given level of total cost, the manager must choose the input combination on the isocost curve that lies on the highest isoquant. This is a constrained maximization problem, and the rule for solving it was set forth in Chapter 3.
Whether the manager is searching for the input combination that minimizes cost for a given level of production or maximizes total production for a given level of expenditure on resources, the optimal combination of inputs to employ is found by using the same rule. We first illustrate the fundamental principles of cost mini- mization with an output constraint; then we will turn to the case of output maxi- mization given a cost constraint.
Production of a Given Output at Minimum Cost
The principle of minimizing the total cost of producing a given level of output is illustrated in Figure 9.4. The manager wants to produce 10,000 units of output
Labor (L) 0
40 60 100 120 140
210 180
150 90
60
Capital (K)
C A
E K
K
L L L Q1 = 10,000 B
A B
100
90
60 66 K''
K'
L' L'' 134
201 K
F I G U R E 9.4
Optimal Input Combina- tion to Minimize Cost for a Given Output
at the lowest possible total cost. All combinations of labor and capital capable of producing this level of output are shown by isoquant Q1. The price of labor (w) is
$40 per unit, and the price of capital (r) is $60 per unit.
Consider the combination of inputs 60L and 100K, represented by point A on isoquant Q1. At point A, 10,000 units can be produced at a total cost of $8,400, where the total cost is calculated by adding the total expenditure on labor and the total expenditure on capital:2
C 5 wL 1 rK 5 ($40 3 60) 1 ($60 3 100) 5 $8,400
The manager can lower the total cost of producing 10,000 units by moving down along the isoquant and purchasing input combination B, because this combination of labor and capital lies on a lower isocost curve (K0L0) than input combination A, which lies on K9L9. The blowup in Figure 9.4 shows that combination B uses 66L and 90K. Combination B costs $8,040 [5 ($40 3 66) 1 ($60 3 90)]. Thus the manager can decrease the total cost of producing 10,000 units by $360 (5 $8,400 2 $8,040) by moving from input combination A to input combination B on isoquant Q1.
Since the manager’s objective is to choose the combination of labor and capital on the 10,000-unit isoquant that can be purchased at the lowest possible cost, the manager will continue to move downward along the isoquant until the lowest possible isocost curve is reached. Examining Figure 9.4 reveals that the lowest cost of producing 10,000 units of output is attained at point E by using 90 units of la- bor and 60 units of capital on isocost curve K'''L''', which shows all input combi- nations that can be purchased for $7,200. Note that at this cost-minimizing input combination
C 5 wL 1 rK 5 ($40 3 90) 1 ($60 3 60) 5 $7,200
No input combination on an isocost curve below the one going through point E is capable of producing 10,000 units of output. The total cost associated with input combination E is the lowest possible total cost for producing 10,000 units when w 5 $40 and r 5 $60.
Suppose the manager chooses to produce using 40 units of capital and 150 units of labor—point C on the isoquant. The manager could now increase capital and reduce labor along isoquant Q1, keeping output constant and moving to lower and lower isocost curves, and hence lower costs, until point E is reached. Regardless of whether a manager starts with too much capital and too little labor (such as point A) or too little capital and too much labor (such as point C), the manager can move to the optimal input combination by moving along the isoquant to lower and lower isocost curves until input combination E is reached.
At point E, the isoquant is tangent to the isocost curve. Recall that the slope (in absolute value) of the isoquant is the MRTS, and the slope of the isocost curve
2Alternatively, you can calculate the cost associated with an isocost curve as the maximum amount of labor that could be hired at $40 per unit if no capital is used. For K9L9, 210 units of labor could be hired (if K 5 0) for a cost of $8,400. Or 140 units of capital can be hired at $60 (if L 5 0) for a cost of $8,400.
(in absolute value) is equal to the relative input price ratio, w/r. Thus, at point E, MRTS equals the ratio of input prices. At the cost-minimizing input combination,
MRTS 5 __ w r
To minimize the cost of producing a given level of output, the manager employs the input combination for which MRTS 5 w/r.
The Marginal Product Approach to Cost Minimization
Finding the optimal levels of two activities A and B in a constrained optimiza- tion problem involved equating the marginal benefit per dollar spent on each of the activities (MB/P). A manager compares the marginal benefit per dollar spent on each activity to determine which activity is the “better deal”: that is, which activity gives the higher marginal benefit per dollar spent. At their optimal levels, both activities are equally good deals (MBA/PA 5 MBB/PB) and the constraint is met.
The tangency condition for cost minimization, MRTS 5 w/r, is equivalent to the condition of equal marginal benefit per dollar spent. Recall that MRTS 5 MPL/ MPK; thus the cost-minimizing condition can be expressed in terms of marginal products
MRTS 5 MPL
_____
MPK 5 w __ r
After a bit of algebraic manipulation, the optimization condition may be expressed as
MPL
____ w 5 MP_____ K r
The marginal benefits of hiring extra units of labor and capital are the marginal products of labor and capital. Dividing each marginal product by its respective input price tells the manager the additional output that will be forthcoming if one more dollar is spent on that input. Thus, at point E in Figure 9.4, the marginal product per dollar spent on labor is equal to the marginal product per dollar spent on capital, and the constraint is met (Q 5 10,000 units).
To illustrate how a manager uses information about marginal products and input prices to find the least-cost input combination, we return to point A in Figure 9.4, where MRTS is greater than w/r. Assume that at point A, MPL 5 160 and MPK 5 80; thus MRTS 5 2 (5 MPL/MPK 5 160/80). Because the slope of the isocost curve is 2/3 (5 w/r 5 40/60), MRTS is greater than w/r, and
MPL
____ w 5 160 ____ 40 5 4 . 1.33 5 ___ 80 60 5 MPK
_____ r
The firm should substitute labor, which has the higher marginal product per dollar, for capital, which has the lower marginal product per dollar. For example, an additional unit of labor would increase output by 160 units while increasing labor cost by $40. To keep output constant, 2 units of capital must be released,
Now try Technical Problem 3.
causing output to fall 160 units (the marginal product of each unit of capital released is 80), but the cost of capital would fall by $120, which is $60 for each of the 2 units of capital released. Output remains constant at 10,000 because the higher output from 1 more unit of labor is just offset by the lower output from two fewer units of capital. However, because labor cost rises by only $40 while capital cost falls by $120, the total cost of producing 10,000 units of output falls by $80 (5 $120 2 $40).
This example shows that when MPL/w is greater than MPK/r, the manager can reduce cost by increasing labor usage while decreasing capital usage just enough to keep output constant. Because MPL/w . MPK/r for every input combination along Q1 from point A to point E, the firm should continue to substitute labor for capital until it reaches point E. As more labor is used, MPL falls because of diminishing marginal product. As less capital is used, MPK rises for the same reason. As the manager substitutes labor for capital, MRTS falls until equilibrium is reached.
Now consider point C, where MRTS is less than w/r, and consequently MPL/w is less than MPK/r. The marginal product per dollar spent on the last unit of labor is less than the marginal product per dollar spent on the last unit of capital. In this case, the manager can reduce cost by increasing capital usage and decreas- ing labor usage in such a way as to keep output constant. To see this, assume that at point C, MPL 5 40 and MPK 5 240, and thus MRTS 5 40/240 5 1/6, which is less than w/r (5 2/3). If the manager uses one more unit of capital and 6 fewer units of labor, output stays constant while total cost falls by $180. (You should verify this yourself.) The manager can continue moving upward along isoquant Q1, keeping output constant but reducing cost until point E is reached. As capital is increased and labor decreased, MPL rises and MPK falls until, at point E, MPL/w equals MPK/r. We have now derived the following:
Principle To produce a given level of output at the lowest possible cost when two inputs (L and K) are variable and the prices of the inputs are, respectively, w and r, a manager chooses the combination of inputs for which
MRTS 5 ___MPL MP_ K 5 __ w r which implies that
MPL ____ w 5 ____ MPK
r
The isoquant associated with the desired level of output (the slope of which is the MRTS) is tangent to the isocost curve (the slope of which is w/r) at the optimal combination of inputs. This optimization condition also means that the marginal product per dollar spent on the last unit of each input is the same.
Now try Technical Problem 4.
Production of Maximum Output with a Given Level of Cost
As discussed earlier, there may be times when managers can spend only a fixed amount on production and wish to attain the highest level of production consistent
with that amount of expenditure. This is a constrained maximization problem the optimization condition for constrained maximization is the same as that for con- strained minimization. In other words, the input combination that maximizes the level of output for a given level of total cost of inputs is that combination for which
MRTS 5 w __ r or MPL
____ w 5 MPK
_____ r
This is the same condition that must be satisfied by the input combination that minimizes the total cost of producing a given output level.
This situation is illustrated in Figure 9.5. The isocost line KL shows all possible combinations of the two inputs that can be purchased for the level of total cost (and input prices) associated with this isocost curve. Suppose the manager chooses point R on the isocost curve and is thus meeting the cost constraint. Although 500 units of output are produced using LR units of labor and KR units of capital, the manager could produce more output at no additional cost by using less labor and more capital.
This can be accomplished, for example, by moving up the isocost curve to point S. Point S and point R lie on the same isocost curve and consequently cost the same amount. Point S lies on a higher isoquant, Q2, allowing the manager to produce 1,000 units without spending any more than the given amount on inputs (represented by isocost curve KL). The highest level of output attainable with the
Q3 = 1,700
Labor (L) 0
KE
LE
Capital (K)
E K
L
Q2 = 1,000 Q1 = 500 KS
KR
LS LR S
R F I G U R E 9.5
Output Maximization for a Given Level of Cost
given level of cost is 1,700 units (point E), which is produced by using LE labor and KE capital. At point E, the highest attainable isoquant, isoquant Q3, is just tangent to the given isocost, and MRTS 5 w/r or MPL/w 5 MPK/r, the same conditions that must be met to minimize the cost of producing a given output level.
To see why MPL/w must equal MPK/r to maximize output for a given level of expenditures on inputs, suppose that this optimizing condition does not hold.
Specifically, assume that w 5 $2, r 5 $3, MPL 5 6, and MPK 5 12, so that MPL
____ w 5 6 __ 2 5 3 , 4 5 ___ 12 3 5 MPK
_____ r
The last unit of labor adds 3 units of output per dollar spent; the last unit of capital adds 4 units of output per dollar. If the firm wants to produce the maximum output possible with a given level of cost, it could spend $1 less on labor, thereby reducing labor by half a unit and hence output by 3 units.
It could spend this dollar on capital, thereby increasing output by 4 units.
Cost would be unchanged, and total output would rise by 1 unit. And the firm would continue taking dollars out of labor and adding them to capital as long as the inequality holds. But as labor is reduced, its marginal product will increase, and as capital is increased, its marginal product will decline.
Eventually the marginal product per dollar spent on each input will be equal.
We have established the following:
Principle In the case of two variable inputs, labor and capital, the manager of a firm maximizes output for a given level of cost by using the amounts of labor and capital such that the marginal rate of technical substitution (MRTS) equals the input price ratio (w/r). In terms of a graph, this condition is equivalent to choosing the input combination where the slope of the given isocost curve equals the slope of the highest attainable isoquant. This output-maximizing condition implies that the marginal product per dollar spent on the last unit of each input is the same.
We have now established that economic efficiency in production occurs when managers choose variable input combinations for which the marginal product per dollar spent on the last unit of each input is the same for all inputs. While we have developed this important principle for the analysis of long-run production, we must mention for completeness that this principle also applies in the short run when two or more inputs are variable.