Production with Two Variable Inputs

• Most production processes use many variable inputs: labor, capital, materials, and land

• Capital inputs include assets such as physical plant, machinery, and vehicles

• Consider a firm that uses two inputs in the long run:– Labor (L) and capital (K)– Each of these inputs is homogeneous– Firm’s production function is Q = F(L,K)

7-1

Production with Two Variable Inputs

• When a firm has more than one variable input it can produce a given amount of output with many different combinations of inputs– E.g., by substituting K for L

• Productive Inputs Principle: Increasing the amounts of all inputs strictly increases the amount of output the firm can produce

7-2

Isoquants

• An isoquant identifies all input combinations that efficiently produce a given level of output– Note the close parallel to indifference curves– Can think of isoquants as contour lines for the

“hill” created by the production function

• Firm’s family of isoquants consists of the isoquants for all of its possible output levels

7-3

Isoquant Example

7-4

Properties of Isoquants

• Isoquants are thin• Do not slope upward• The boundary between input

combinations that produce more and less than a given amount of output

• Isoquants from the same technology do not cross

• Higher-level isoquants lie farther from the origin

7-5

Substitution Between Inputs

• Rate that one input can be substituted for another is an important factor for managers in choosing best mix of inputs

• Shape of isoquant captures information about input substitution– Points on an isoquant have same output but different input mix– Rate of substitution for labor with capital is equal to negative the

slope

• Marginal Rate of Technical Substitution for input X with input Y: the rate as which a firm must replace units of X with units of Y to keep output unchanged starting at a given input combination

7-6

MRTS and Marginal Product

• Recall the relationship between MRS and marginal utility

• Parallel relationship exists between MRTS and marginal product

• The more productive labor is relative to capital, the more capital we must add to make up for any reduction in labor; the larger the MRTS

K

LLK MP

MPMRTS

7-7

Declining MRTS

• Often assume declining MRTS

• Here MRTS declines as we move along the isoquant, increasing input X and decreasing input Y

7-8

Isocost curves

Various combinations of inputs that a firm can buy with the same level of expenditure

PLL + PKK = M

where M is a given money outlay.

Labor

Capital

0

M/PK

M/PL

Slope = -PK /PL

Maximization of output for given cost

Labor

Capital

0100

200300

R

MPL/PL = MPK/PK

Labor

Capital

0100

200300

R

EXPANSION PATH

Returns to scale

If the firm increases the amount of all inputs by the same proportion:

• Increasing returns means that output increases by a larger proportion

• Decreasing returns means that output increases by a smaller proportion

• Constant returns means that output increases by the same proportion

Output elasticity

The percentage change in output resulting from 1 percent increase in all inputs.

> 1 ==> increasing returns < 1 ==> decreasing returns = 1 ==> constant returns

Returns to Scale

Types of Returns to ScaleProportional change in ALL

inputs yields…What happens when all

inputs are doubled?

ConstantSame proportional change in

outputOutput doubles

IncreasingGreater than proportional

change in outputOutput more than doubles

DecreasingLess than proportional

change in outputOutput less than doubles

7-16

Returns to Scale

7-17