Production Modeling
Accounting Cost - concerned with historical expenditures.
Economic Cost takes into account opportunity cost that has to be taken into
consideration in the decision process (It is more costly for a doctor to mow the lawn on
Monday at 10:00 than a janitor).
Sunk costs - costs that cannot be recovered.
Valuable Alternative use of land => high opportunity cost of present use.
Zoning in many cases reduces opportunity cost.
Total Cost (TC) = Variable Cost (VC) + Fixed Cost (FC)
Marginal Cost (MC) = VC / Q  TC / Q
Average Cost (AC) = TC / Q
Table 7.1 illustrates FC, AC VC MC, AFC, AVC, ATC
In the short run can change labor to increase output. Given the wage w ,
MC  VC / Q  wL / Q  w / MPL
Assuming labor is the only variable input
AVC  wL / Q  w / APL
where APL is the average product of labor.
Figure 7.1 The MC curve goes through the minimum of the ATC and AVC but not the
AFC. As Q  , AFC  0 .
In the long run all inputs are variable.
Isocost line (= production budget line) is locus of points shows different combinations of
inputs such that cost is fixed.
C  wL  rK
solving for K gives
K  (C / r )  (( w / r ) L)
Isocost slope
K / L   ( w / r )
of ratio of wage rate to rental cost of capital.
Want to maximize production for a given total cost TC.
K
A=TC/Pk
A
B=TC/PL
Q
B
L
The MRTS   K / L  MPL / MPK . In equilibrium the slope of the isocost PL / PK =
MRTS
MPL / MPk  PL / PK  w / r
or
MPL / w  MPK / r
Figure 7.5 show effect of an effluent fee raising the relative price of the input water.
Production moves to be less water intensive.
Expansion path shows the amounts of labor and capital used as firm grows (figure 7.6).
This is a long run concept.
K
Long Run
Short Run
L
Economies of scale includes increasing returns to scale as a special case. Increasing
returns to scale requires inputs be used in fixed proportions. Economies of scale allows
the ratio of inputs to change.
Figure 7.9 Long-Run Cost with Constant Returns to scale
Figure 7.10 Long-Run Cost with first Economies then Diseconomies of Scale. The LAC
traces tangency points of SAC curves.
At minimum point of LAC the LMC and SMC intersect.
Economics of scale => MC < AC. Diseconomies of scale => MC > AC. Define
EC
= elasticity of total cost
EC  [ TC / TC] / [ Q / Q]  [ TC / Q] / [TC / Q]  MC / AC
Economics of scale => E c  1, of MC < AC.
Economies of scope are present when joint output of a single firm producing two
products is greater that the output of two firms each producing one product. Define
C ( Qi ) as cost of producing Qi of the ith good. Economies of scope => SC > 0
SC  [ C(Qi )  C(Q1 ,..., Qn )] / C(Q1 ,..., Qn )
i
Diseconomies of scope SC < 0
Learning Curve => Over time labor costs go down as more units are produced. Define L
as labor per unit, A> 0, B>0, 0<  <1 and N = # of units produced. Learning curve
implies
L  A  BN  
As N  , L  A
If N=1 then L=A+B or the amount of labor for first unit. If   0 there is no learning.
B
When   0 then second unit costs A   which converges to A as N increases.
2
Costs can decline because of increasing returns to scale or learning. See figure 7.11
which shows cumulative number of machines on the x axis. Figure 7.12 shows
production rate.
$
Economies of scale => A to B
Learning Curve => A to C
A
B
C
AC1
AC2
Q
In the early life of a product the learning curve is steep. After while the "learning" slows
down. In chip business had a 20% learning curve. => 10% increase in cumulative
production => costs fall 2%. In aircraft industry learning curve rate was 40%.
Estimating a cost curve
Linear Model used if MC is constant at  . MC  
VC  Q
Quadratic Model used if have straight line MC. MC    2Q.
VC  Q  Q2
Cubic cost function used if have U shaped MC curve of form
MC    2Q  3Q2
VC  Q  Q2  Q3
Can use Excel and other software to estimate the MC curve.
Problem 10 page 244. Computer firm cost is AC  10.1(CQ) .3Q where CQ is
cumulative quantity, and Q is quantity per year.
a. There is a learning curve effect since -.1
b. There is decreasing returns to scale?
TC  10Q.1(CQ)(Q) .3Q2
MC  TC / Q  10.1(CQ) .6Q
Since .6Q .3Q => decreasing returns to scale
c. CQt=40,000, Qt = 10,000, Qt+1= 12,000
ACt  10.1(40) .3(10)  9., ACt 1  10.1(50) .3(12)  8.6
Problem 12 page 245.
Assume TC  a  bQ  cQ2  dQ3 , MC  (TC) / Q  b  2cQ  3dQ2
AC  [a / Q]  b  cQ  dQ2 . AC not defined for Q=0. For a U shaped cost
curve we want c<0 and d>0. At minimum point of AC, MC=AC. Set a=0 to normalize y
axis and equate MC=AC. => b  2cQ  3dQ2  b  cQ  dQ2 or c  2dQ . If c=-1 and
d=1 then Q=1/2
Key Question: How do we use this theory to solve a practical problem?
Firm has a production function F ( K , L)  AK  L which could be estimated using Excel
when we note that log[ F ( L, K )]  log( A)   log( K )   log( L) . Want to determine a
general case that shows the optimum K and L given the cost of capital r and the wage w.
The following is a simpler treatment than 246-250. In equilibrium we note that
(1)
[ MPK / MPL ]  r / w
Given the production function
(2)
Q0  AK  L
(3)
[ AK  1 L / AK  L 1 ]  [r / w]
(4)
[L / K ]  r / w
(5)
L  [rK / w]
Substituting (5) into the production function (2) gives
(6)
Q0  [ AK    r  K  /   w ]
(7)
K (   )  (w / r )  Q0 / A
The equilibrium amount of capital is
(8)
K  [(w /  r )  /(   ) ](Q0 / A)1/(   )
If the cost of capital, r, increases or the marginal product of capital,  , falls K will fall.
Direct substitution of (8) into (5) gives
(9)
L  [(  r / w) ( /(   )) ](Q0 / A) (1/(   ))
If the wage rate, w, increases or the marginal product of labor,  , falls then less labor
will be used. The firms cost function is
(10)
C  wL  rK
At equilibrium
(11)
C  w(  /(   )) r ( /(   )) [( /  ) (  /(   ))  ( /  ) (  /(   )) ](Q / A) (1/(   ))
Equation (11) shows how C is related to wages, w, cost of capital, r, and returns to scale
(   ) . MATLAB can be used to calculate the appropriate derivatives
Equation (11) shows how costs are related to the production function of the firm as well
and input costs.
See ch7_1.xls for solution