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

2. What you will learn in this Module:
How firms determine the optimal input mix.
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.

3. Alternative Input Combinations
Substitutes
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. Ditch Digging – 100 feet Combination Labor Cost Capital Cost
Total Cost 1
Driver
$500
Backhoe $2,500 2
10 diggers
$1,000
10 shovels
$ 250
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. Backhoe 300 ft., Diggers 100 ft. Combination Labor Cost Capital Cost
Total Cost per 100 ft. 1
Driver
$500
Backhoe $2,500 2
10 diggers
$1,000
10 shovels
$ 250
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. Least Cost Rule MPL/w = MPK/r 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.

7. Profit Maximizing Rule
Firm produces at MR=MC=P
MPRL = MPL x MR
MRPL = MCL (wage) or
MRPL/w = 1
Profit maximizing rule:
MRPL/w = MRPK/r=1
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.

8. Summary Factors of production can be complements or substitutes.
If substitutes, the key economic question is how much of each input to use in production.
Determine the marginal product per dollar for each input. MP of factor / cost per unit of factor
Cost minimizing input combination is where 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.