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Intermediate Microeconomic Theory

Technology

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Inputs

In order to produce output, firms must employ inputs (or factors of production)

Sometimes divided up into categories:

Labor

Capital

Land

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The Production Function

To produce any given amount of a good a firm can only use certain combinations of inputs.

Production Function – a function that characterizes how output depends on how many of each input are used.

q = f(x1, x2, …, xn)

units of output units of input 1 units of input 2…units of input n

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Examples of Production Functions

What might be candidate production functions for the following? Vodka Distillary – can be made from either potatoes or corn.

Lawn mowing service – requires both Labor and “Capital”, though not necessarily in fixed proportions.

So what are Production functions analogous to? How are they different?

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Production Functions vs. Utility Functions Unlike in utility theory, the output that gets produced has

cardinal properties, not just ordinal properties.

For example, consider the following two production functions: f(x1,x2) = x1

0.5x20.5

f(x1,x2) = x12x2

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Isoquants

Isoquant – set of all possible input bundles that are sufficient to produce a given amount of output.

Isoquants for Vodka?

Isoquants for acres of lawn mowed?

Isoquants for Axes (i.e. each axe requires one blade and one handle)?

So what are Isoquants somewhat analogous to? How do they differ?

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Isoquants

Again, like with demand theory, we are most interested in understanding trade-offs.

What aspect of Isoquants tells us about trade-offs in the production process?

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Marginal Product of an Input

Consider how much output changes due to a small change in one input (holding all other inputs constant), or

Now consider the change in output associated with a “very small” change in the input.

Marginal Product (of an input) – the rate-of-change in output associated with increasing one input (holding all other inputs constant), or

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Marginal Product of an Input

Example: Suppose you run a car wash business governed by the production

function q = f(L, K) = L0.5K0.5

(q = cars washed, L = Labor hrs, K = machine hrs.)

What will Isoquants look like?

What will be the Marginal Product of Labor at the input bundle {L=4, K= 9}?

What will be the Marginal Product of Labor at the input bundle {L=9, K= 9}?

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Substitution between Inputs

Marginal Product is interesting on its own

MP also helpful for considering how to evaluate trade-offs in the production process. Consider again the following thought

exercise: Suppose firm produces using some

input combination (x1’,x2’).

If it used a little bit more x1, how much less of x2 would it have to use to keep output constant?

Δx1

Δx2

x2

x1

x1’ x1”

x2’

x2”

f(x1’,x2’)

f(x1”,x2’)

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Technical Rate of Substitution (TRS)

Technical Rate of Substitution (TRS): 1. TRS = Slope of Isoquant

2.

Also referred to as Marginal Rate of Technical Substitution (MRTS) or Marginal Rate of Transformation (MRT)

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Substitution between Inputs (cont.)

We are often interested in production technologies that exhibit Diminishing Technical Rate of Substitution (TRS).

So what would be the expression for the TRS for a generalized Cobb-Douglas Production function F(x1,x2) = x1

ax2b?

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Diminishing TRS

1 4 worker hrs (L)

machine hrs (K)

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4

4 cars

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Economies of Scale

What do we mean by the term “economies-of-scale.”

Increasing Returns-to-Scale

Decreasing Returns-to-Scale

Constant Returns-to-Scale

What kind of assumption would these be with respect to a generic production function q = f(x1,x2)?

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Returns-to-scale graphically

1 4 8 worker hrs (L)

machine hrs (K)

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16

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4 cars

? cars