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) =
Average Cost (AC) =
Table 7.1 illustrates FC, AC VC MC, AFC, AVC, ATC
In the short run can change labor to increase output. Given the wage ,
Assuming labor is the only variable input
where 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 .
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.
solving for K gives
Isocost slope
of ratio of wage rate to rental cost of capital.
Want to maximize production for a given total cost TC.
K
A=TC/Pk
AB=TC/PL
Q
B L
The. In equilibrium the slope of the isocost = MRTS
or
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
= elasticity of total cost
Economics of scale => 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
as cost of producing of the ith good. Economies of scope => SC > 0
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
As
If N=1 then L=A+B or the amount of labor for first unit. If there is no learning. When then second unit costs which converges to A as N increases.
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
CAC1
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 .
Quadratic Model used if have straight line MC.
Cubic cost function used if have U shaped MC curve of form
Can use Excel and other software to estimate the MC curve.
Problem 10 page 244. Computer firm cost is where CQ is cumulative quantity, and Q is quantity per year.
- There is a learning curve effect since -.1
- There is decreasing returns to scale?
Since => decreasing returns to scale
- CQt=40,000, Qt = 10,000, Qt+1= 12,000
Problem 12 page 245.
Assume
. 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. => or . 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 which could be estimated using Excel when we note that . 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)
Given the production function
(2)
(3)
(4)
(5)
Substituting (5) into the production function (2) gives
(6)
(7)
The equilibrium amount of capital is
(8)
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)
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)
At equilibrium
(11)
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