chapter 18 technology

Intermediate Microeconomics

Chapter 18 Technology

Gaoji Hu

Shanghai University of Finance and Economics

Note: This lecture is based on Chapter 18 of Varian, H. R. (2010).

Intermediate Microeconomics: A Modern Approach: 8th Edition. WW Norton & Company.

All figures in the notes with blue background are from the book.

Gaoji Hu Intermediate Microeconomics Market Demand 1 / 17

Summary

1. Consumer theory and producer theory.

2. Input and output, technology, isoquant and production set.

3. Production function and fixed/flexible factors.

4. Examples of technologies.

5. Properties of technologies: monotonicity and convexity.

6. Marginal product, TRS, and return to scale.


Consumer Theory versus Producer Theory

Consumer Theory Producer Theory

Starting points: Preference Technology

Perspectives: Subjective Objective

Outcomes: Utility Output

(Only ordering matters) (The magnitude matters)

(Unobservable utility) (Observable Output)


Inputs and Outputs

Inputs to production or factors of production:

land, labor, capital, and raw materials.

In particular, capital goods are themselves produced.

Examples of capital goods (not a classification):

Financial capital : the money used to start up or maintain a business.

Physical capital : produced factors of production.


Technological Constraints and Production Set

Let x ∈ R+ represent the input and y ∈ R+ the output.

Technological constraints on firms: y ≤ f(x),∀x ∈ R+,

where f(·) is called the production function.

A production function describes the maximum possible output

that you can get from a given amount of input.

The production set shows the possible technological choices.

Production set = {(x, y) ∈ R2+ : y ≤ f(x)}.

What if we have two inputs?Gaoji Hu Intermediate Microeconomics Market Demand 5 / 17

Isoquant

An isoquant is the set of all possible combinations of inputs 1 and 2 that are

just sufficient to produce a given amount of output.

Isoquants are similar to indifference curves.

Difference:

Utility level does not matter (ranking matters).

Output level has economic meaning!

What is the relationship between production function and isoquants?

[Like utility function and indifference curves.]


Production Function

f(x1, x2, . . . , xn), where (x1, x2, . . . , xn) ∈ Rn+.

Unlike in the consumer theory, two inputs are usually not enough.

We often work with one or two goods only because of simplicity!

Interpretations of f(x) or f(x1, x2):

Other inputs are not important: free and don’t affect decision.

Other inputs are fixed at some level. f(x1, x2, x3, . . . , x4).


Fixed Factors and Flexible Factors

Suppose there are two factors x1 and x2.

One factor fixed No factor fixed

Production function g(x1) = f(x1, x2) f(x1, x2)

Figures

Interpretations: Short run Long run

Warning: the “time” here is just an interpretation;

we don’t have a dynamic model/timeline here!


Examples: Fixed Proportions/Perfect Complements

f(x1, x2) = min{x1, x2}, where (x1, x2) ∈ R2+.

[From now on, when we don’t specify the domain, it is Rn+.]


Examples: Perfect Substitutes

f(x1, x2) = x1 + x2.


Examples: Cobb-Douglas

f(x1, x2) = Axa1x

b2.

More concrete example: y = f(K,L) = AKaLb.

A measures the scale of production:

how much output we would get if we used one unit of each input.


Properties of Technology

(Weak) monotonicity: f(x1, x2) ≥ f(x′1, x

′2) if (x1, x2) ≥ (x′

1, x′2).

This is also referred to as the property of free disposal:

if the firm can costlessly dispose of any inputs, having more can’t hurt.

Convexity: for each t ∈ [0.1],

tf(x1, x2) + (1− t)f(x′1, x

′2) ≤ f(tx1 + tx′

1, (1− t)x2 + (1− t)x′2).

Convexity is just a property, which may or may not be satisfied.

We will investigate these and other properties of f(·) through:

1. Marginal product: how one factor affects output.

2. The Technical Rate of Substitution: maintaining the same output,

how two factors can be replaced with each other.

3. Returns to Scale: how scaling the amount of all inputs affect output.


1. The Marginal Product

MP1(x1, x2) = ∂f∂x1

(x1, x2).

MP2(x1, x2) = ∂f∂x2

(x1, x2).

Unlike marginal utilities,

the magnitude of marginal product has economic meaning!

And that specific number can in principle be observed!

1. Marginal product is positive if technology is increasing.

2. Marginal product is diminishing if one factor is used more and more.


2. The Technical Rate of Substitution

y = f(x1, x2)

TRS(x1, x2(x1)) =dx2

dx1(x1).

Convex production technology results in diminishing TRS.

[What if we adopt a convex technology?]

Conversely, diminishing TRS justifies convex technology.

[What technology should we use? Why convex technology?]


Diminishing Marginal Product versus Diminishing Technical Rate of

Substitution

<A 3D figure here>


3. Returns to Scale

Constant return to scales: For any t > 0, f(tx1, tx2) = tf(x1, x2).

Example: the firm just replicates what it was doing before!

Increasing return to scales: For any t > 1, f(tx1, tx2) > tf(x1, x2).

Example: Oil pipeline.

If we double the diameter of a pipe, we use twice as much materials,

but the cross section of the pipe goes up by a factor of 4.

Decreasing return to scales: For any t > 1, f(tx1, tx2) < tf(x1, x2).

Something wrong:

The usual reason is that we forgot to account for some input.

Of course, a technology can exhibit different kinds of returns to scale

at different levels of production.


Examples: Datacenters and Intel

Datacenters: racks.

Intel: copy plants exactly.


chapter 18 technology

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