Costs and Supply
© Allen C. Goodman, 2014
1
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?
2
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?
3
Another Way to Look at it
Let’s let:
+ +
?
+(+ or -)
Q = f (L, K, X, E)
L = Labor
K = Capital
X = Other materials and supplies
E = 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.
4
Fisher Distinguishes between
What goes into
utility ftn.
Direct Outputs and Consumption
Service
Fire Protection
Inputs
Firefighters,
Inspectors,
Stations,
Trucks,
Equipment,
Water Supply
Direct Outputs
Stations/sq.mi.,
FF/station,
Trucks/station
Hydrants/sq. mi.
Consumption
Fires suppressed,
Property damage
prevented,
Deaths prevented
VERY
HARD to MEASURE
5
Two Types of Pictures
Output Q
• Typically, all
else equal, more
inputs  more
output, but at a
decreasing rate.
• What does this
imply about
marginal
product?
ΔQ
Δ
X
Much
smaller
ΔQ
Δ
X
Average
Product
Input X 6
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.
7
Like we
Putting them Togetherdid with
C1 = wL + rK utility,K/L1
MP/$ is
We have talked about
Q1
equal for
consumption
all inputs
indifference curves. K
Let’s do production
C2 < C1
indifference curves,
K1*
sometimes called
isoquants.
Pick two inputs
C2 = wL + rK
L1*
L
8
So … when people talk about cutting
expenditures … and saving …
C1 = wL + rK
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/L1
Q1
K
K1*
C2 = wL + rK
L1*
L
9
Elasticity of substitution, .
K/L1
•  = the
%
change in the
factor input
ratio, brought
about by a 1%
change in the
factor price
ratio.
K
K/L2
L
10
Elasticity of substitution, .
K/L1
•  = the
%
change in the
factor input
ratio, brought
about by a 1%
change in the
factor price
ratio.
K
K/L2
Elastic
 big
change
L
11
Elasticity of substitution, .
K/L1
•  = the
%
change in the
factor input
ratio, brought
about by a 1%
change in the
factor price
ratio.
K/L3
K
K/L2
Inelastic
 small
change
L
12
Some Production Functions
Several different types of production functions. The
typical Cobb-Douglas production function for capital
and labor can be written as:
or
Q = A L K
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 / rK
1% x 1%
s = (w/r)
(L/K)
Increase in (w/r) means that (L/K) should fall. With
matching 1% changes, shares stay constant.
13
Production Functions – CD
Consider Cobb-Douglas production function with capital and labor.
Q = A La Kb
If profits are:
 = pQ - rK - wL, when we substitute in for the quantity
relationship, we get:
k  K / L,
Differentiating with respect to L and K, we get:
Define:
a-1
b
  / L = aAL K - w= 0
  w/ r
a
b-1
  / K = bAL K - r= 0
Simplifying, we get:
[(a/b] (K/L) = w/r
 dk 
(a/b) k
= ψ  ψ/k = a/b
  dk 
k
(a/b) dk = dψ  dk/dψ = b/a
Elas    
Elas = (dk/dψ)(ψ/k) = (b/a)*(a/b)= 1 !
 d  d k
 


14
For 6520
Production Functions
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
15
[(1-)/] k1- =  For 6520
[(1-)/] k- = /k
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?
16
What if workers negotiate a wage
K/L
hike?
2
Why does line rotate
C1 = w1L + rK
inward?
K
What must occur?
Either reduce
quantity produced
or
Increase costs!
What if capital is a
good substitute for
C'1 = w2L + rK
labor?
What if it isn’t?
To get back to original production?
K/L1
What
Happened?
L
17
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?
18
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?
19
Progressive
Two Sectors
Wage
Traditional
Wage
DP
Wages
are the same
in each sector
W1P
SP
DT
ST
W1T
L1P
L1T
Labor
Labor
20
Progressive
Two Sectors
Wage
Traditional
Wage
Productivity ↑
DP
Wages ↑
But so did
productivity
DT
Wages ↑
but w/o ↑
in productivity
W2P
W1P
W1T
L1P
L2P
Labor
L1T
Labor
21
Two Sectors
Wage
Traditional
Wage
Why is this demand curve
Productivity ↑
so steep?
Wages ↑
Answer – Elasticity ofBut so did
DP
productivity
substitution
is very small
(relate
to isoquants).
P
DT
Wages ↑
but w/o ↑
in productivity
W2
P
WWhat
1
happens to wage
bill?
Answer – Probably
increases because
L2P is
elasticity ofL1Pdemand
Labor
very small.
W1T
New
wage bill
Initial
Wage bill
Labor
22
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
23
does).
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
24