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Costs and Supply

© Allen C. Goodman, 2015

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Production Functions

• Thus far we’ve talked about demand. Let’s start looking at supply!

• We wish to relate outputs to some measure of inputs.

• Consider the police, for example.– What are the outputs?– What are the inputs?

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Production functions

Let:

Q = f (L, K, X)

L = Labor

K = Capital

X = Other materials and supplies

Presumably, as L, K, or X ↑, what would happen to Q?

Why?

+ + +

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Another Way to Look at it

Let’s let:

Q = f (L, K, X, E)L = LaborK = CapitalX = Other materials and suppliesE = Economic environment, including type of population

Maybe some people volunteer in schools, maybe individuals patrol their neighborhoods. Maybe some students are easier to teach than others.

All of these may have additional impacts on output.

+ + +(+ or -)?

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Fisher Distinguishes between Direct Outputs and Consumption

Fire Protection

Service Inputs Direct Outputs Consumption

Firefighters,Inspectors,Stations,Trucks,Equipment,Water Supply

Stations/sq.mi.,FF/station,Trucks/stationHydrants/sq. mi.

Fires suppressed,Property damage prevented,Deaths prevented

VERYHARD to MEASURE

What goes intoutility ftn.

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Two Types of Pictures

• Typically, all else equal, more inputs more output, but at a decreasing rate.

• What does this imply about marginal product?

Input X

Output Q

ΔQ

Δ X

Δ X

ΔQ

Muchsmaller

AverageProduct

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Expenditures

• To get output, we must spend money on factors of production, or inputs.

• Cost of output 1 is:– Cost = wL1 + rK1 + pX1

– w, r, and p might refer to wage rates (cost of labor), rental fees (cost of capital), and other materials prices.

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Putting them Together

We have talked about consumption indifference curves.

Let’s do production indifference curves, sometimes called isoquants.

Pick two inputs

K

L

K/L1

Q1

C1 = wL + rK

C2 = wL + rK

L1*

K1*

If we want C2 < C1

Like we did with utility, MP/$ is equal for all inputs

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So … when people talk about cutting expenditures … and saving …

1. They are implying that current production is inefficient. What exactly does “efficient” mean?

2. They are saying that they want lower levels of inputs into public services.

K

L

K/L1

Q1

C1 = wL + rK

C2 = wL + rK

L1*

K1*

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Elasticity of substitution, .

= the % change in the factor input ratio, brought about by a 1% change in the factor price ratio.

K

L

K/L1

K/L2

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Elasticity of substitution, .

= the % change in the factor input ratio, brought about by a 1% change in the factor price ratio.

K

L

K/L1

K/L2Elastic big change

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Elasticity of substitution, .

= the % change in the factor input ratio, brought about by a 1% change in the factor price ratio.

K

L

K/L1

K/L2

Inelastic small change

K/L3

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Some Production FunctionsSeveral different types of production functions. The

typical Cobb-Douglas production function for capital and labor can be written as:Q = A L K or ln Q = ln A + α ln L + β ln K

It turns out that there is a property of the Cobb-Douglas function that

= 1. What does this mean? This gives an interesting result that factor shares stay constant. Why?s = wL / rKs = (w/r) x (L/K)

Increase in (w/r) means that (L/K) should fall. With matching 1% changes, shares stay constant.

1% 1%

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Consider Cobb-Douglas production function with capital and labor. Q = A La Kb

If profits are: = pQ - rK - wL,

Differentiating with respect to L and K, we get: / L = aALa-1 Kb - w= 0 / K = bALaKb-1 - r= 0 Simplifying, we get:

[(a/b] (K/L) = w/r (a/b) k = ψ ψ/k = a/b

(a/b) dk = dψ dk/dψ = b/aElas = (dk/dψ)(ψ/k) = (b/a)*(a/b)= 1 !

Production Functions – CD

Define:/ ,

/

k K L

w r

dkdkk

Elasd kd

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Consider C.E.S. production function with capital and labor. Q = A [K + (1-) L] R/.If profits are: = pQ - rK - wL, when we substitute in for the quantity relationship, we get:

Differentiating with respect to L and K, we get: / L = A(R/) (1-) L-1[K + (1-) L] (R/)-1 - w= 0 / K = A(R/) K-1 [K + (1-) L] (R/)-1 - r= 0

Simplifying, we get:[(1-)/] (K/L)1- = w/r

Production FunctionsFor 6520

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For 6520

Production Functions

Redefine k = K/L, and = w/r, so:

[(1-)/] k1- = Now, differentiate fully. We get:

[(1-)/] (1-) k- dk = d, or:

dk/d = [/(1-)] [1/(1-)] k. Multiplying by /k, we get the elasticity of substitution, or:

= 1/(1-).

What does a Cobb-Douglas function look like? What do others look like?

[(1-)/] k1- = [(1-)/] k- = /k

[(1-)/] k1- = [(1-)/] k- = /k

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What if workers negotiate a wage hike?

Why does line rotate inward?

What must occur?

Either reduce quantity produced or

Increase costs!

What if capital is a good substitute for labor?

What if it isn’t?

K

L

K/L1

C1 = w1L + rK

C'1 = w2L + rK

K/L2

WhatHappened?

To get back to original production?

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Do Local Governments Minimize Costs?

• Model above showed how either output could be maximized, or costs minimized.

• In a competitive model, competition will (in theory) lead to minimum cost production.

• Will this happen among localities?

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Baumol’s Cost Hypothesis

• Consider two sectors. He calls them– Progressive – subject to productivity

improvements.– Traditional – Generally more labor

intensive and not subject to productivity improvements.

• What happens?

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Two Sectors

Labor

Wage

Labor

Wage

Progressive Traditional

W1P W1

T

Wagesare the samein each sectorDP DT SP ST

L1P L1

T

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Two Sectors

Labor

Wage

Labor

Wage

Progressive Traditional

W1P W1

T

DP DT

Productivity ↑

W2P

L1P L2

P

Wages ↑But so did productivity

Wages ↑but w/o ↑

in productivity

L1T

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Two Sectors

Labor

Wage

Labor

Wage

Traditional

W1P W1

T

DP DT

Productivity ↑

W2P

L1P L2

P

Wages ↑But so did productivity

Wages ↑but w/o ↑

in productivity

Why is this demand curve so steep?Answer – Elasticity of substitution is very small (relate to isoquants).

What happens to wage bill?Answer – Probably increases because elasticity of demand is very small.

InitialWage bill

New wage bill

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Does this apply?

• In some cases yes; in others, no.• If you’re doing a woodwind quintet, it’s hard

to do much substitution. On the other hand, rock bands can do so much more now with synthesizers than they ever did!

• Look at what happened with the DSO!• Bill Clinton thought it applied to health care.

I was never sure that it did (or does).

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Fisher (P. 154-5) – Good summary

• Costs of state-local goods seem to have gone up relative to private sector over the last 25 years.

• Fiscal pressure on states and localities was somewhat hidden in 1990s because the overall national economy grew quickly and provided lots of revenues.

• Real estate values also ↑, providing revenues.• With national recession in 2001, slow growth since then,

and “Great Recession” of 2008-2010 we have seen increasing costs for state-local sector and increasing fiscal pressure.

• Possible solutions?– Use of new technology– Substitute private production for public production