Study Session 4
quantity of capital for a given quantity of labor will at first result in increasing marginal returns to capital. At some point, however, employing additional units of capital will result in smaller increases in output. This is the point of diminishing marginal returns to capital.
Professor’s Note: To convince yourself that there is a point of diminishing marginal returns for both labor and capital, consider the result if this were not the case. If the marginal product of labor continued to increase, all the output ever required could be produced (given enough workers) with a given amount of capital (one factory). If the marginal product of capital continued to increase, one group of
laborers could (with enough capital) produce all the output ever required. Neither seems likely.
Profit-Maximizing Utilization of an Input
The marginal product of a resource is measured in output units and is sometimes called the marginal physical product of the resource. In order for profit to be at a maximum, a firm must use the mix of inputs that minimizes the cost of producing any given quantity of output. For a firm with productive inputs, cost minimization requires that:
MP^ _ MP^ _ MPn
This equation tells us that the additional output per dollar spent to employ one additional unit of each input must be the same. This result is best understood by examining the case where this does not hold. Consider a production process that uses only two inputs, capital and labor (A'and L), with the following input prices and values for M Pland M I f :
PL = $75 PK = $600 MPl = 5 units MPk = 30 units
Additional output from employing one more unit of labor costs 75/5 = $15 per output unit, and output per additional dollar spent on labor is 1/15 unit.
Additional output from employing one more unit of capital costs 600/30 = $20 per unit of output, and output per additional dollar spent on capital is 1/20 unit.
In this situation, production costs can be reduced by employing less capital and more labor, so this cannot be the optimal (cost-minimizing) combination of inputs. Reducing capital by one unit would reduce output by 30 units and reduce costs by $600. Spending
$450 on 6 additional units of labor would increase output by 6 x 5 = 30 units, restoring output to its previous level. By employing more labor and less capital, we have decreased the cost of production by $150.
If we can produce greater output at the same cost by using more labor and less
diminishing marginal productivity in each input, using more labor and less capital will decrease the MPL and increase the MPK, reducing the difference between MPL / PL and MPk / PK. From this example, we can see that we can reduce production costs by substituting labor for capital until we reach the input quantities for which MPL / PL = MPk / PK, which is the necessary condition for cost minimization.
Although the condition for cost minimization is necessary for costs to be at a minimum, it does not tell us how much of either input to use to maximize profit. That is, we could be minimizing the cost of producing an output quantity either greater or less than the profit-maximizing quantity.
To determine the quantity of each input that should be used to maximize profit, we need to introduce the concept of marginal revenue product (MRP). The MRP is the monetary value of the marginal product of an input. It is calculated by multiplying a production factor’s marginal product by the marginal revenue of the additional output.
MRP is the increase in the firm’s total revenue from selling the additional output from employing one more unit of the factor.
The profit-maximizing quantity of an input i is that quantity for which MRP; = P;. A firm can increase profits by employing another unit of the input as long as MRPj > P;
because employing another unit of the input increases revenue by more that it increases costs. Conversely, if MRP; < P;, the firm could increase profits by reducing the quantity of the input employed. The decrease in total revenue is less than the cost savings from using one less unit of the input.
Recall that under perfect competition, marginal revenue is equal to price, and the MRP of a factor is its marginal product multiplied by price. For a firm that faces a downward-sloping demand curve, marginal revenue is less than price. In either case, we can multiply each factor’s marginal product in the cost minimization condition, by the marginal revenue value of additional output, to get an equivalent relation necessary for cost minimization:
Mfi x MRoutput MP2 x MRoutput MPNxMRoutput
Pi “ P2 Pn
or
MRP, MRP2 MRPn
Pi Pn
Based on the condition for the profit-maximizing utilization of each factor, MRPf = Pf, we can state that for cost minimization and profit maximization, a firm must employ inputs in quantities such that:
Study Session 4
Example: Profit maximizing level of a productive input
Consider the following data for Centerline Industries. The firm’s inputs can be categorized as high technology equipment, unskilled labor, and highly trained workers. The MR, MP, and cost per day of the various inputs are summarized in the following table. Assume that the inputs can substitute for each other in the production process.
Resource Resource MP
(units) Output MR ($) Resource $ Price/
High technology equipment 30 30 Unit800
Unskilled labor 5 30 160
Highly trained workers 15 30 450
1. Is the firm operating at the cost-minimizing levels for its inputs?
2. Assuming diminishing marginal factor returns, what adjustments to its input mix, if any, should the firm make to increase profits?
Answer:
1. Comparing marginal product per dollar of each resource we have:
MPhightech/P hightech= 30 / 800 = 0.03750 MPunskilled/P unskilied = 5 / 160 = 0.03125 MPhighskill/P highskill = 15 /450 = 0.03333
Because these are not equal, the condition for cost minimization is not met.
2. Comparing the MRP for each resource to its price, we have:
M R P hightech = 3 0 x 3 0 = 9 0 0 P .high tech = 8 0 0 M R P unskilled = 5 x 3 0 = 1 5 0 p unskilled = 16 0. M R P high skill = 15 x 3 0 = 4 5 0 ^high skill = 4 5 0
The condition for the profit-maximizing quantity of each resource, MRP = P is met only for highly skilled labor. For high technology equipment, the MRP (900) is greater than the unit cost (800), so the firm should employ more high technology equipment. For unskilled workers, the MRP (150) is less than the unit cost (160), so the firm should employ fewer unskilled workers.
K ey C oncepts