1. Cost Minimizing Input Combinations
Micro:
Econ: 36 72
Module
Cost Minimizing Input Combinations
KRUGMAN'S
MICROECONOMICS for AP*
Margaret Ray and David Anderson 1

2. What you will learn in this Module:
How do firms determine the optimal input mix?
What is the cost-minimizing rule for hiring inputs?
The purpose of this module is to show how firms decide the optimal combination of factors for producing the desired level of output. 2

3. Small group discussion
What use is this?
If there are multiple ways to produce some output (like digging a ditch) then how does the firm know which one to choose?
The firm will choose the combination of inputs that can produce the output at the lowest cost. This combination is called the least-cost combination of inputs.
A firm that wants to minimize the cost of hiring inputs (labor and capital) to produce as much output as possible. Employing labor and capital requires paying factor prices, the wage and rental rate, and each provides output to the firm, the marginal product of labor and capital. The firm will find that any level of output is produced at the lowest cost when labor and capital are hired such that:
MPL/w = MPK/r
Anytime the marginal product per dollar is not equal, the firm can reshuffle employment of labor and capital to increase output while keeping costs unchanged. Once the firm has found the least-cost combination of labor and capital, the firm has found the combination that produces that output at the lowest possible cost. 3

4. Alternative Input Combinations
Substitutes
Note: sometimes inputs can be substitutes for themselves – globalized labor
Complements
We have studied how single inputs are hired to maximize profit. We saw that firms must employ an input up to the point where the marginal revenue product is equal to the marginal factor cost of that input. But what does a firm do when there are different ways in which to produce the same output?
Firms combine inputs, like labor, capital and land, to produce output.
Example: construction is a great example. Carpenters use tools to build houses, but there are different combinations of labor and capital that will get the same house built. One man with a nail gun could be more productive than several men with hammers and nails and the firm must decide if that more expensive, but more productive, nail gun is a better choice than several men with inexpensive hammers.
A. Substitutes and Complements in Factor Markets
Substitutes: two factors that can do essentially the same work.
Examples?
An ATM machine dispenses cash, accepts deposits and allows you to transfer money. The ability to perform these banking tasks makes the machine a substitute for a bank teller.
A Caterpillar backhoe with one driver can dig holes and ditches. A team of men with shovels can also dig holes and ditches. These are also substitutes in production.
Two types of labor can also be substitutable. An American computer programmer and a Korean programmer could do the same work. A group of union autoworkers in Indiana could be considered substitutable with a group of non-union workers in Tennessee.
Complements: two factors that must be combined to produce output. The presence of one factor increases the marginal product of the other.
The Caterpillar backhoe and the driver are complements. Not much digging gets done if they aren’t combined in production.
A team of pilots and a 747 passenger jet are complements.
An 18-wheeler and a truck driver are complements. 4

5. Determining the Optimal Input Mix
So, if there are many ways of a bank exchanging money with their customers, how do they choose?
Least-cost combination of inputs
Cost-minimization rule
MPL/w = MPK/r
If there are multiple ways to produce some output (like digging a ditch) then how does the firm know which one to choose?
The firm will choose the combination of inputs that can produce the output at the lowest cost. This combination is called the least-cost combination of inputs.
A firm that wants to minimize the cost of hiring inputs (labor and capital) to produce as much output as possible. Employing labor and capital requires paying factor prices, the wage and rental rate, and each provides output to the firm, the marginal product of labor and capital. The firm will find that any level of output is produced at the lowest cost when labor and capital are hired such that:
MPL/w = MPK/r
Anytime the marginal product per dollar is not equal, the firm can reshuffle employment of labor and capital to increase output while keeping costs unchanged. Once the firm has found the least-cost combination of labor and capital, the firm has found the combination that produces that output at the lowest possible cost. 5

6. What use is this?
What if the marginal productivity of one factor of production has increased relative to another?
What if the price of one factor has gone significantly up or down?
If one side is unequal, they can move around the proportion of factors of production to stay at about the same level of production without increasing costs.
If there are multiple ways to produce some output (like digging a ditch) then how does the firm know which one to choose?
The firm will choose the combination of inputs that can produce the output at the lowest cost. This combination is called the least-cost combination of inputs.
A firm that wants to minimize the cost of hiring inputs (labor and capital) to produce as much output as possible. Employing labor and capital requires paying factor prices, the wage and rental rate, and each provides output to the firm, the marginal product of labor and capital. The firm will find that any level of output is produced at the lowest cost when labor and capital are hired such that:
MPL/w = MPK/r
Anytime the marginal product per dollar is not equal, the firm can reshuffle employment of labor and capital to increase output while keeping costs unchanged. Once the firm has found the least-cost combination of labor and capital, the firm has found the combination that produces that output at the lowest possible cost. 6

7. What use is this?
A firm uses two inputs, capital and labor, to produce output. Its production function exhibits a diminishing marginal rate of technical substitution.
a) If the price of capital and labor services both increase by the same percentage amount (e.g., 20 percent), what will happen to the cost-minimizing input quantities for a given output level?
b) If the price of capital increases by 20 percent while the price of labor increases by 10 percent, what will happen to the cost-minimizing input quantities for a given output level?
If there are multiple ways to produce some output (like digging a ditch) then how does the firm know which one to choose?
The firm will choose the combination of inputs that can produce the output at the lowest cost. This combination is called the least-cost combination of inputs.
A firm that wants to minimize the cost of hiring inputs (labor and capital) to produce as much output as possible. Employing labor and capital requires paying factor prices, the wage and rental rate, and each provides output to the firm, the marginal product of labor and capital. The firm will find that any level of output is produced at the lowest cost when labor and capital are hired such that:
MPL/w = MPK/r
Anytime the marginal product per dollar is not equal, the firm can reshuffle employment of labor and capital to increase output while keeping costs unchanged. Once the firm has found the least-cost combination of labor and capital, the firm has found the combination that produces that output at the lowest possible cost. 7