Production

Dr. Andrew McGee

Simon Fraser University

Productions funtions

• Production functions: a recipe. For given levels of inputs, how much output you get.

– P.f. characterize the production technology used by the firm

– The technology describes what a firm can do in terms of converting input into output

Inputs Production process(Technology)

Output

Production functions

Marginal product

Marginal product

• MP is evaluated holding other inputs constant (fixed)

• Imagine the MPL of the 51st farm worker with and without a tractor:– Without a tractor:

• 50L → 1000 bushels of wheat

• 51L → 1002 bushels of wheat

– With a tractor:• 51L → 1050 bushels of wheat

• Notice how different your calculation of the MPLof the 51st worker would be if you did not hold K (tractor) fixed

Diminishing Marginal Product

• Assume diminishing MP → Holding other inputs constant, the extra output from using more of an input goes down as you use more of the input after a given usage level of the input

Workers Output MPL

1 100 100 (100-0)

2 300 200 (300-200)

3 550 250 (550-300)

4 600 50 (600-550)

Diminishing Marginal Product• As the usage of an input increases, there are

benefits from specialization resulting in a region of increasing marginal product

• After a point, it becomes inefficient to increase the usage of an input holding constant the other inputs as the marginal product declines

Gains fromspecialization

DiminishingMP

L (input)

Ouputper unit input (MPL)

MPL

L’

Diminishing Marginal Product

L’

L’

L

L

Totaloutput

Output per unit input (MPL)

L

fLL

Q=f(L)

MPL

0

L’’

L’’ L’

Marginal Product & Malthus

• Thomas Malthus (1766-1834) argued that Britain’s population would be constrained by its ability to feed people. If the population got too large, famine and disease would check an underfed population

• Problem: Britain’s population today is many times larger than when Malthus wrote. What was his mistake?

• He never took ECON201/301!

Marginal Product & Malthus

Land

Q(food)

Q=f(L,K)

All the land in Britain

Max food production?

Malthus argued that because the quantity of arable land in Britain was fixed, its potential food output was capped at Q’. This would act as an effective constraint on population growth.

Q=f(L,K’) or Q=g(L,K)

Malthus failed to appreciate his own ceteris paribus assumptions. In particular, he assumed that both (a) the capital stock would remain fixed and (b) the technology used to produce food would remain the same. In the end, neither proved to be true.

Definitions & Terminology

Relationship between Average & Marginal Products

L

Output per unit input (L)

MPL

APL

Relationship between MPL & APL:(1) MPL>APL → APL increasing(2) MPL<APL → APL decreasing(3) MPL=APL → APL at its maximum value

This is a purely mathematical property having nothing to do with economics. Quiz example.

Example

Isoquants

• Given a production technology Q=f(K,L), an isoquant depicts all combinations of inputs K & L that yield the same level of output

K

L

Isoquant Q=Q0

While isoquants might remind you of indifference curves, note that there is absolutely no connection between the two. The similarity arises because we adopt the same technique for collapsing a 3-dimensional function into 2-dimensions for the sake of depicting it.

Isoquant Map

Direction of increasing quantity produced

K

L

Q=Q0

Q=Q1

Q=Q2

Q=Q3

Slope of an Isoquant

Marginal Rate of Technical Substitution

Why are isoquants negatively sloped?

Diminishing MRTS

MRTS

K K K

L L L

Diminishing MRTS Increasing MRTSConstant MRTS

Example: Diminishing MRTS

• Imagine a bakery that employs labor (bakers but also security guards, accountants, etc.) and capital (whisks, rolling pins, industrial-sized mixers, ovens, etc.)

• You can replace an electric mixer with a person with a wooden spoon, you could even take away the wooden spoon and mix with your bare hands, but how do you replace the oven with labor?

• Similarly, you can automate all of the mixing, pouring into pans, and baking, but who monitors this automated process?

What Isoquants Tell Us about Technology

• The shape of an isoquant tells us both how easily capital is substituted for labor and vice versa as well as how important an input is to the production process:

K

L

K

L

Q=Q0

Q=Q0

Capital (machine) intensive technology Labor-intensive technology

Relationship between MRTS and MP• Does diminishing MRTS follow from assumption of

diminishing MPL and diminishing MPK? No.• Diminishing MPL and diminishing MPK speak to what

happens to output when you increase usage of one input holding the other input constant. Moving along an isoquant necessarily changes both inputs at the same time.

• It turns out that with an additional assumption, diminishing MPL and diminishing MPK will guarantee diminishing MRTS: if fKL>0, then diminishing MPL and diminishing MPKguarantee diminishing MRTS.

• If fKL<0, then whether you have diminishing MRTS will depend on how “rapidly” the MP of K & L diminish.

• Note that it is hard to imagine why fKL<0. It is far more reasonable to assume that fKL>0, but this means that the assumption of diminishing MP will indeed guarantee diminishing MRTS.– Important because it means we don’t need to check SOC for

firm’s cost minimization problem

Returns to Scale in Production

• Diminishing MPL and MPK refer to what happens to output when you change one input holding the other constant

• Returns to scale refer to how output changes when you change all inputs by the same proportion (t>0)

Types of Returns to Scale (RTS)

Seeing RTS in an Isoquant MapK

L2 4 8 100

2

4

8

100

Q=2

Q=6

Q=12

Q=16DRS

CRS

IRS

The isoquant map reveals a lot about the production technology. A production function can exhibit all 3 types of returns to scale: IRS, CRS, & DRS.

A conundrum

• Economists are skeptical of the existence of DRS in production technologies.– If one can combine a given level of capital and labor

(or any combination of inputs) to produce a given level of output, why shouldn’t one be able to exactly replicate this production process with exactly the same amounts of the inputs and get exactly the same amount of output? That is, shouldn’t we always expect to see at least constant returns to scale in production?

• Fact: DRS is observed in real-world production data.

• Why the disconnect between theory and reality?

DRS: Myth or Reality?

• Suppose you combine 3 workers, a storefront, a cash register, a fry machine, and a grill and get 500 Happy Meals per day. What would prevent you from simply getting another storefront, hiring 3 more workers, getting a cash register, fry machine and grill and then producing 500 Happy Meals per day at this other location?

DRS: Resolution

• Economists’ explanation for DRS observed in data: there must be some unobserved input not increasing by the same proportion as other inputs– What we see is not really DRS; it only looks like it

• Most likely unobserved input?– Management

• 2 McDonalds require 2 managers

• Economists generally expect CRS– CRS production functions have nice properties—

namely constant marginal costs as we shall see

DRS and diminishing MP

• Diminishing MPL (or MPK) does not imply DRS. Indeed, there is no connection between the two concepts. MP is evaluated holding all other inputs constant, while RTS necessarily speaks to how output changes when all inputs are varied.

RTS with more than 2 inputs

Homothetic Production Functions

K

L

q0

q1

K/L

MRTS is the same at all input combinations where the capital-labor ratio is the same

All CRS production functions are homothetic

Ease of substitution &MRTS

“Ease of substitution” measure

• Why do we need a units-free measure of substitutability and one that reflects how substitutability changes with the capital labor ratio?

K K

L L

dk/dL=-5dK/dL=-1

Is it really any harder to substitute labor for capital in the graph on the left? It might just reflect units of measure. In both cases, you can always substitute a fixed amount of capital for a fixed amount of labor.

q0 q0

“Ease of substitution” measureK

L

MRTS=constant

MRTS=straight line: easy to substitute K for L; can always substitute K for L in some fixed ratio

K

L

q0

q0

MRTS undefined

Impossible to substitute K for L: K and L must always be used in fixed proportions to one another

K

L

q0

MRTS changing

Somewhere between perfectly easy to substitute and impossible to substitute K for L

Elasticity of substitution between inputs

Elasticity of substitution(K/L)

(K’/L’)

∆(K/L)

MRTS1

MRTS2

∆MRTS

Types of Production Functions

K

L

q0/a

q0/b

-b/a

What sort of input mix (K/L) will industries with perfect substitutes in production use?--Never use both inputs. Only use the relatively cheaper input.--Looking ahead to the cost minimization problem firms solve, this type of production technology will lead to corner solutions.

Types of Production Functions

K

L

q0

b/a

Ratio of K to L is fixed at b/a when firms solve cost minimization problem.

Types of Production Functions

K

L

q0

Types of Production Functions

Finding the Elasticity of Substitution

Finding the Elasticity of Substitution