Download Monopoly2 Manual

Monopoly2 Manual
Purpose of the Module
This module is designed to allow the user to practice running a firm in a monopoly
market. You will have to use information about the demand for the product and the firm’s
costs to maximize long run profits for the firm. In order to do a good job you will have to
figure out how the firm can minimize its costs for any given level of output, how its costs
change as its level of production changes, how its revenues change as its production
changes, and how profits are affected by cost and revenue changes.
There are not many monopoly markets, especially ones with no government control or
regulation, but the principles governing successful profit maximization that apply in other
kinds of markets apply in this one. The most important thing to remember is that you
need to use marginal analysis. You do not start out by thinking about whether you should
do everything or nothing. That question may come up eventually, but the more important
questions are “Should I use more labor relative to fuel, or less,” and “Should I produce
more, or less?”
The first part of this module involves setting the firm’s technology and then, given that
technology, discovering the least cost combinations of resources. Once that is
accomplished and you know how to minimize the firm’s costs at any level of output, the
second part of the module allows you to determine the firm’s profit maximizing level of
output, given that the firm sells all its output at a single price. The third part of the
module offers a comparison of the firm’s profitability with and without the ability to sell
parts of its production at different prices (price discrimination).
The Model
This module concerns a firm that produces electricity and whose managers want to
maximize profits. The first part of maximizing profit is to minimize their costs. The
production process of the firm allows for a given amount of product to be produced using
infinitely many combinations of three resources. The three resources discussed in the
module are labor, fuel, and capital. It also assumes that each resource in the production
process faces diminishing returns when more of the given resource is used with fixed
amounts of the other resources. The technology of the firm can be set by the user and can
involve increasing, constant, or decreasing returns to scale. (The last may be a
mathematical fiction or an indication that there is a fourth, fixed, resource not being
accounted for in the module.)
Isoquants
With the assumptions just described, the production process can be represented
graphically by a set of “isoquants” as in Figure 1. Each isoquant is the set of all possible
combinations of labor and fuel assuming a fixed amount of capital that permit the
production of a given level of output. In Figure 1 the combinations for 10 and 20 units
1
of output are shown. There are other such curves for each possible level of production.
Due to the assumption of diminishing returns the slopes of these isoquants get flatter and
flatter as the proportion of labor to fuel increases.
Figure 1 Production Isoquants
Fuel
Q = 20
Q = 10
Labor
The higher the isoquant, the higher the level of production. The reason can be seen by
noting that you can get from one isoquant to a higher one by holding an input (e.g., labor)
constant while raising the amount of the other (e.g., fuel). Even with diminishing returns,
raising the amount of one input while others are constant will increase production
(negative marginal returns are not possible in this module).
There are similar isoquants for combinations of capital and labor (holding fuel constant)
and for capital and fuel (holding labor constant). Output combinations with all three
inputs free to vary can be shown as 3-dimensional surfaces in a 3-D graph. However,
since 3-D graphs are hard to read, it is common to use two sets of isoquants for three
variable problems. One set shows combinations of labor and fuel as in Figure 1. The
second set shows combinations of labor and capital for fixed amounts of fuel.
The Marginal Rate of Technical Substitution
The slopes of the isoquants in Figure 1 can be described by the term Marginal Rate of
Technical Substitution (MRTS). The MRTS is obtained by using measures of the extra
output that can be obtained from an additional amount of each individual resource. The
extra output of an extra unit of labor (holding fuel and capital constant) is defined as the
“Marginal Product of Labor” or MPL. The extra output from an extra unit of fuel
(holding labor and capital constant) is defined as the “Marginal Product of Fuel” or MPF.
The marginal rate of technical substitution between labor and fuel is then defined as:
MRTSLF= MPL/MPF
Looking at the definition it should be clear how the assumption of diminishing returns
affects the MRTSLF. If the amount of labor was increased, while the amount of fuel was
2
decreased just enough to keep production constant, MPL would fall and MPF would rise.
This combination of effects would make the slope of the isoquant (MRTSLF) fall, so the
curve gets flatter and flatter.
Isocost Lines
In order to solve the problem of cost minimization two things are needed, a description of
the production process -- the isoquants discussed above -- and a description of the costs
of using resources. This module assumes that firms can acquire additional amounts of all
resources at fixed prices (the prices neither rise nor fall as the firms buys/hires more of
them). As a result, for a fixed level of cost, the combinations of resources that the firm
can get can be described (in the two dimensional case) as a straight line, an isocost line,
such as the ones in Figure 2. In the three dimensional case combinations that incur the
same cost would be a flat plane.
Figure 2 Isocost Lines
Fuel
TC2
TC1
Labor
In Figure 2 the line TC 1 represents one level of cost (e.g., $1000) and TC 2 represents a
higher level of cost (e.g., $2000). The slopes of TC1 and TC2 are the same, given the
constant prices of the inputs.
Cost Minimization
The difficulty of the problem of minimizing cost depends on how many inputs the firm
can control. In the very short run when the firm can control only a single input (labor in
this module) there really is no “problem” just a set of results. These results can be seen
for the default technology in the module in the table below. (The default technology is
the one you get if you do not choose to set parameters of the production function -described later in this manual -- for yourself.)
3
Table 1 Short Run Costs (Capital = 46, Fuel = 80)
Output
4139.092
7868.867
10890.556
13609.788
16147.125
18558.697
20876.229
23119.789
25303.065
27435.899
29525.663
31578.052
Labor
20
60
100
140
180
220
260
300
340
380
420
460
Total Cost Average Cost
$ 1216
$ 0.294
1536
0.195
1856
0.170
2176
0.160
2496
0.155
2816
0.152
3136
0.150
3456
0.149
3776
0.149
4096
0.149
4416
0.150
4736
0.150
Marginal Cost*
0.070
0.098
0.113
0.122
0.130
0.136
0.140
0.145
0.148
0.152
0.155
0.157
* The marginal cost is calculated using the extra cost of producing more output using
additional labor since the other two inputs are fixed. Marginal cost of adding output using
more of one of the other inputs could be very different.
When more than one input can be varied the problem is to find the combination of inputs
that offers the lowest cost of producing a given output, or that offers the largest output for
a given total cost. How difficult that is depends, in this module, on whether the firm
controls two inputs or three (the more inputs the harder the problem, but the more the
firm can minimize its costs).
In the two input case the solution to the problem of cost minimization can be found by
comparing labor and fuel costs within the firm (“internal”) and costs in outside markets
(“external”). Suppose the firm currently can hire more labor for $10 an hour, and can get
fuel for $20 a pound in the “outside”, or labor and capital markets overall. In other words
the input price ratio, the measure of external opportunity costs, is $10/$20 = 1/2. Thus
the “external” opportunity cost of another pound of fuel is two hours of labor.
Suppose the current method the firm is using to produce (the combination of labor, fuel,
and capital) creates an MRTS of one -- one hour of labor can substitute for one pound of
fuel with production constant. If the firm hires another unit of labor for $10, it could use
one less pound of fuel -- costing $20, and still get the same output. The “internal”
opportunity cost of labor is less than the “external” opportunity cost of labor and that
means the firm was not minimizing its cost. The firm could get the same production
while spending $10 less, giving it a lower cost per unit.
4
If the “internal” and “external” opportunity costs are different that means the firm can
change its current practices and reduce its costs. If you recall the basic concept of
comparative advantage, in which differences in opportunity costs mean that there can be
“gains from trade,” you can see this as just another application of that concept. Where
the firm’s opportunity costs differ from those of others in the market (as reflected in the
relative prices of the inputs), the firm can gain by trading with others in the market.
The only circumstance in which the firm could not reduce its cost for a given level of
production is when the “internal” opportunity costs match the “external” opportunity
costs, in other words, when the MRTSLF is equal to the ratio of prices for the resources
(PL/PF). If a firm can find the combination of inputs that achieves that equality, there are
no more potential gains from trade and the cost minimization problem has been solved.
The logic of the three input case is the same. However, if the firm controls three inputs it
must compare the internal and external costs of inputs in more ways -- labor versus fuel,
labor versus capital, and capital versus fuel. (One of these is redundant -- if you find the
best proportion of labor to fuel and labor to capital you already have the best proportion
of capital to fuel.)
Figure 3 Cost Minimization with Two Variable Inputs
Fuel
B
A
Q = 20
Q = 10
TC1
Labor
In Figure 3, Point B is a combination of inputs that allows the firm to produce 10 units
of output at a cost of TC1. However, at that combination the firms “internal” opportunity
costs, represented by the slope of the isoquant for 10 units, does not equal the “external”
opportunity costs, represented by the slope of TC1. The firm can produce more, up to 20
units, by moving to Point A -- where the tangency between TC1 and the isoquant for 20
units indicates that the “internal” and “external” opportunity costs are the same.
A firm that has three variable inputs is looking for the combination that matches
“internal” and “external” opportunity costs in more respects. In the previous example the
best point was one where the ratio between labor and fuel prices matched the ratio
between the marginal products of those two inputs. If the firm also controlled how much
capital it used, it would want this equality to hold, but it would also want the ratio
5
between the price of labor and the price of capital to equal the ratio of the marginal
products of those two inputs (MRTSLK).
In Figure 4a, the firm is at the tangency for labor and fuel, just as in Point A in Figure 3.
However it is not at the similar tangency A’ in Figure 4b, so it is not minimizing all the
costs under its control. Instead, since it is not minimizing its costs, the firm is a point like
Point B, producing less at the current cost than the amount it would get at A’, and it does
not have the lowest possible unit cost of production. The choice of Point B in the graph
has the marginal product of capital too low compared to the marginal product of labor –
too much capital is being used relative to labor. It is equally true that Point C in the
graph shows a “wrong” combination, too little capital and too much labor. Only at A’ is
there no way to reduce the cost of producing the given output by varying the proportions
of capital and labor.
Figure 4
Cost Minimization with 3 Variable Inputs
(a)
(b)
Fuel
Capital
A
B
A’
C
Labor
Labor
Cost Curves
In the long run a firm’s marginal and average costs can be expressed in terms of long run
cost curves. If a firm has constant returns to scale its lowest possible average and
marginal costs do not change as the level of production change. In other words the firm
has constant (unit) costs. This is shown by the horizontal LRAC, LRMC line in Figure
5. A firm that controls all of its costs (in other words one that has no fixed costs because
it has no fixed inputs) could produce any amount at a cost per unit shown by that line.
Constant returns to scale is one of the possible cases in this module and you can
determine whether the firm has increasing, constant, or decreasing returns when you run
the module. The default case features increasing returns to scale, common in monopolies.
If the short run is defined as a period in which the firm has any fixed inputs, in the short
run the firm has average and marginal cost curves whose positions are determined by the
amount of the fixed input that they have. In this module choosing to control only two
inputs puts you into this short run, with capital as the fixed input that you cannot change.
For one particular value of capital the firm would have as potential least average and
6
marginal cost levels those indicated by the SRATC and SRMC curves. A smaller value
for capital would create similar curves, but they would be more to the left. More capital
would create similar curves to the right.
It is vital to realize that input choices that do not minimize either long or short run cost
would create costs that are not on any of these cost curves. In Figure 5 a firm would be
able to produce output Q1 at an average cost of A if it could minimize long run costs.
(Figure 5 assumes that the technology chosen has constant returns to scale, that is why
the long run cost curves are horizontal, for simplicity the figure also omits the Average
Variable Cost curve often shown in such diagrams.) It could produce at B if it could only
minimize short run cost. If it used a combination that did neither it would create a
cost/output point such as Point C.
Figure 5 Short and Long Run Costs (Constant Returns to Scale Case)
$
B
A
C
SRMC
SRATC
●
LRAC=LRMC
Q1
Quantity
Profit Maximization
This module features a firm in a monopoly market. To understand the firm’s problems
once it has learned to minimize costs you need to understand how this kind of market
works. First, some definitions of terms that are important in understanding the model are
provided. Next, an explanation of the monopoly market and how it functions is given.
Price (P)
At a given moment in time the market price is the dollar amount that will be paid in the
market for one unit of the product. Every unit sold will be sold at this price (in part two
of the module the firm is not a discriminating monopoly).
Marginal Revenue (MR)
In a monopoly the (non-discriminating) firm must lower the price to sell additional units
of the product. When the firm sells another unit it does not get extra revenue equal to the
7
price of the last unit sold, it gets less. The firm gets the revenue from the price of the last
unit, but loses revenue on each of the previous units that would have been sold at higher
prices. As a result the marginal revenue from selling another unit is less than the price of
the product. Note that when very small quantities are being sold the loss from selling
previous units is small (since there are few such units) and MR is very close to price.
Average Total Cost (ATC)
The average total cost of the product is the firm’s total cost (all costs regardless of type)
divided by the total number of units of the product produced. This is also referred to as
per unit cost. If the monopoly firm is acting in the long run, all costs are variable and
there are no fixed costs, so there is no distinction between average total cost (ATC) and
average variable cost (AVC). If you choose to control only two inputs there is a
distinction between total cost and total variable cost and therefore between average total
and average variable cost.
Marginal Cost (MC)
The firm’s marginal cost is the additional cost of producing an additional unit of the same
product. In this module if you control all three inputs there are three measures of
marginal cost, each representing the cost of adding output using one of the three inputs. If
you choose to control only two inputs there are two measures of marginal cost. Note that
if the marginal cost of increasing output using more of one input does not equal the
marginal cost of increasing output using more of some other input that proves the firm is
not currently using an input combination that is cost minimizing.
Total (Economic) Profit
The firm’s total profit is its total revenue (TR) (price per unit multiplied by the total
quantity produced) minus its total cost (TC). In economics, the total costs include all of
the firm’s opportunity costs. The profit that remains after subtracting TC from TR is the
total economic profit. This economic profit is what the firm is earning over and above
what it could get using the same resources in some other way.
The Monopoly Market
A pure monopoly is a firm that is the only one in a given market. It produces a product
that has no close substitutes. The particular case used in this module is a firm that sells
electricity to final users (not distributors or wholesalers) in a particular geographical area.
Electricity has substitutes. You can cook on an electric stove, or a gas stove, or even a
wood stove. You can use gas for lighting, or candles. The assumption of this module is
that the alternatives are not close substitutes. (Technically, that means that the crossprice elasticities with these alternatives are fairly small.) As a result, the demand for the
firm’s product is the same as the total market demand for the product.
8
Since in second part of the module the monopolist does not discriminate, the more units
the firm wants to sell, the lower it has to set its price. It cannot have a high price and a
large quantity sold, it can’t make people buy the product -- they must be willing to buy at
the firm’s price, or it will have a bunch of leftover, unsold, products.
The firm wants to set the production level, and the price, at its most profitable level. In
Figure 6, at output QA the firm gets additional revenue, MRA, from producing and selling
an additional unit of the product. (In Figure 6 and Figure 7 costs are drawn as if the firm
had increasing returns to scale, so the cost lines slope downward.) The way this
additional, or marginal, revenue is determined is this: the firm gains revenue equal to the
price it is paid when it sells that additional unit. However, that revenue is less than the
revenue it received from selling the previous unit since it must cut price to sell another
unit. For example: If the firm was selling 100 units at $10 each, to increase the number
of units sold to 101, it must cut its price to $9.95. The firm gains $9.95 in revenue from
selling the 101st unit. The firm loses $0.05 on each of the 100 units it was already selling
for $10 each since its only getting $9.95 each now. It loses 100 x $0.05 or $5 in revenue
this way. Overall, it gets extra revenue of $4.95 from selling one more unit.
Figure 6
Monopoly Price and Output Determination
Price
PE
MRA
MCA
MR
0
QA Q E
D
AC
MC
Quantity
At output QA marginal revenue is higher than marginal cost (MCA). That means that the
extra revenue the firm receives from selling this unit is larger than the extra cost of
producing it -- so selling this unit adds to total profit. The firm should continue to
increase sales until it equates MR with MC at QE. At QE the firm maximizes total profit.
In Figure 7, at output QB the firm gets less additional revenue (MRB) from the last unit it
sells than the additional cost of producing it (MCB). It is bad idea to produce that last
unit because making and selling it reduces the firm’s total profit. At QB it would be
better if the firm cut production. The best place for the firm to produce is at Q E, where
MR equals MC and total profits are maximized.
9
Figure 7
Monopoly Price and Output Determination
Price
PE
MCB
AC
MRB
0
Price Discrimination
MR
QE
QB
MC
D
e
Quantity
m
a
n
d
In the third part of the module the firm can use price discrimination. In fact, two levels of
price discrimination are available. The firm can practice perfect price discrimination,
often called third order discrimination. In perfect discrimination each unit of the product
sold is bought at the highest price that will be paid for that unit. Near the vertical
intercept of the demand function only one unit would be sold at that very high price. With
perfect discrimination that unit will be sold at that price, and then the next unit will be
sold at a lower price -- the highest at which that next unit could be sold. This continues-up to whatever output the firm decides to produce--with each unit being sold at a lower
price than the preceding unit. The consequence is that the separation of price and
marginal revenue found in the second part of the module does not occur. For each added
unit sold, the marginal revenue received by the firm is the same as the price at which the
unit is sold.
The firm can also discriminate in pricing between two markets (groups of customers)
with different demand characteristics. In this type of imperfect discrimination the
monopolist will have two prices for the same product, one in each market.
Perfect Discrimination and Profit Maximization
Since marginal revenue and price are the same with perfect price discrimination the profit
maximizing point, where marginal revenue equals marginal cost, is also where price
equals marginal cost. That means the monopolist will sell the same number of units as a
perfectly competitive industry would sell in long run equilibrium but will receive more
revenue. The perfectly competitive industry would receive revenue equal to price times
quantity, with all units sold at the same price. A perfectly discriminating monopolist
receives the same revenue for the last unit as the competitive industry would, but more
revenue (higher prices) for the preceding units.
10
If the competitive industry and monopoly would have the same cost (this is not possible
in the increasing or decreasing returns to scale cases) that guarantees the monopoly has
higher profits than the competitive industry would have. (In the increasing returns to
scale case a competitive industry cannot exist in the long run, so there is no proper
comparison.) The profit maximizing solution for perfect discrimination (constant returns
case), compared to the basic monopoly solution, is shown in Figure 8.
Figure 8
Perfect Price Discrimination
MRND indicates marginal revenue in the absence of discrimination, QND indicates the
profit maximizing output if there is no discrimination, while PND indicates the nodiscrimination price paid by all buyers. QD indicates the output of the perfectly
discriminating monopolist, while PLast indicates the price paid by buyer of the last unit
(the lowest price paid by anyone). The blue rectangle in the figure shows the nondiscriminatory profit amount, while the profit made by perfect discrimination is the sum
of the blue rectangle and the two green triangles.
Imperfect Discrimination
There are various possible degrees of discrimination that fall short of perfection. In this
module the alternative to perfect discrimination and no discrimination is the ability to
separate two markets and charge a different price in each. Discrimination in these
circumstances requires the firm to: (a) find the output at which the marginal revenue
from the two markets combined equals marginal cost -- this tells the firm how much to
produce in total; and (b) find the division of that output that makes marginal revenue in
the two markets the same. Condition (a) is no different than the non-discrimination case.
Condition (b) is now required for profit maximization, however.
The reason for (b) is most easily seen by looking at what would happen if the two values
for marginal revenue were different. If the MR in Market 1 was $2 and the MR in
Market 2 was $1 then the firm could increase profit simply by moving a unit of the
product from Market 2 to Market 1. (This assumes there are no special costs to selling
units in one market versus the other, e.g., higher transportation costs to get the goods to
11
the other market.) Moving the unit would lose the firm $1 in revenue from Market 2, but
increase its revenue by $2 in Market 1. Since there is no change in the firm’s costs, the
net increase in revenue of $1 for the firm is also an increase in profit of $1. The
opportunity to increase profits in this way disappears only when the two marginal
revenues are the same, so giving up $1 in revenue in one market produces only $1 in
extra revenue in the other market.
It should be noted that discrimination -- charging different prices in the two markets -will only be profitable if the price elasticities of demand in the two markets are different
where the marginal revenues equal each other and equal marginal cost. This can be seen
if you recall that in the two markets prices are related to marginal revenues:
MR1 = P1 * (1 + (1/e1 ))
where e1 is the price elasticity of demand in Market 1
MR2 = P2 * (1 + (1/e2 ))
where e2 is the price elasticity of demand in Market 2
(The above equations are derived from algebraic manipulations of the definition of
marginal revenue as the change in total revenue due to a change in output.)
If, in order to maximize profit, output is divided so that MR1 = MR2 then:
P1 * (1 + (1/e1 )) = P2 * (1 + (1/e2 ))
If the two elasticities are equal, then the two prices must also be equal.
Figure 9
Discrimination Between Two Markets
Market 1
Market 2
Combined Markets
P1
P2
D2
Q1
MR1
Q2
MR2
D=D1 + D2
QT
MRtotal
In Figure 9 the demand lines for Markets 1 and 2 are assumed to have the same slope,
but different intercepts. That is sufficient to ensure that the price elasticities of demand
for the two will be different at the profit maximizing points. The market demand line has
twice the slope of the individual demand lines up to the price level at which market 2’s
quantity demanded drops to zero. Above that price the total demand slope (indeed, the
total demand itself) will match the demand in Market 1. In Figure 9 QT = Q1 + Q2 and
P1 is not equal to P2.
12
Running the Module
When you begin this module you need to make some choices that will set up the costminimizing problem for you. First, you need to decide whether you will control two
inputs or three inputs (the long run). Once you have determined that, you can have the
module use the default values for the parameters (“Use Assigned Technology”), or set the
values of the parameters (“a” through “e”) of the production function yourself.
If you set the parameter values yourself you will find that there are limits to the values
these parameters can have. The values of “a”, “b”, and “c” have to lie strictly between
zero and one (greater than zero and less than one) and they must add up to one (a + b + c
= 1). If, for example, you increase the value of “a” and leave the value of “b” the same
you will find that the program automatically lowers the value of “c” to maintain the sum
equal to one. This is required because these parameters are, in effect, setting the relative
“weight” each input has in the production process.
Once you have clicked on “Set Technology” you will find that your cursor is in the box
for parameter “a”. Set the value, then either use the tab key to move to the box for “b”.
Once you have set “b” use the tab key and your cursor will move to the box for parameter
“d”. (Having set “a” and you cannot set the value for “c” independently.)
The value of “d”, which determines the degree of substitutability between the inputs,
must lie strictly between 0.1 and 1 in this module. That means the inputs are overall
substitutes. Having set the value of “d”, use the tab key to move your cursor to the box
for parameter “e”. This parameter determines the returns to scale for the firm. A value
less than one means the firm has decreasing returns to scale, if “e” is equal to one that
means constant returns to scale, greater than one means increasing returns to scale. In the
module the value of “e” must lie between 1.1 and 0.9 (no greater than 1.1 no less than
0.9). Any attempt to enter a number that does not meet the restrictions for a given
parameter will get you a warning message -- and will not work.
Once you have decided on the parameters and on whether you wish to solve the two or
three variable input problem, you get to decide on how much of each (variable) input you
will have the firm hire. Once you have entered values for either two or three inputs,
depending on your choice of problems, you can get the numerical results. Note: If you
chose the two input problem the amount of capital your firm uses will be automatically
set at a given value of about 46000 hours (expressed as 46 since everything is measured
in thousands). If you enter 1 as your labor input that would be 1,000 labor hours, and
entering 1 as your fuel input would be 1,000 pounds.
Depending on how many variable inputs you chose to have you will get the marginal
products of two or three inputs, one or two MRTS values, total costs, unit costs, and
marginal cost by two or three different measures (MC of increasing output if you hire
more labor, or if you buy more fuel, or -- in the three input case -- if you lease more
capital). You also get reminded of the resource prices. When you click on “graph” you
will see the isocost - isoquant graph, as well as information about how much this output
13
would have cost you if you had found the best solution, and how much you could have
produced for the cost you actually expended. Determining the firm’s inputs determines
output, and market demand determines the price, given the output value. Consequently
you also receive the price, marginal revenue, and profit information for the firm.
A sample result for the three input case, using the assigned technology with the default
inputs of Capital = 46; Labor =300, Fuel = 80, is:
Output =
23119.789
Total Cost =
$ 3546.000
Average Cost =
$ 0.149
Price =
$ 0.243
Marginal Revenue = $ 0.236
Profit =
$ 2136.590
MCL =
$ 0.145
MCK =
$0.425
MCF =
$0.054
MTRS(L,F) = 2.71
MRTS(L,K) = 0.60
MRTS(K,F) = 4.55
MPL =
55.288
MPK =
37.606
MPF =
74.494
Price Ratios
$ 8/4
$ 8/16
$16/4
Additional Definitions
Total cost = cost of capital plus the cost of labor plus the cost of fuel
= (PK * quantity of capital) + (PL * amount of labor) + (PM *amount of fuel)
Unit (average) cost is total cost divided by the number of units produced,
AC = Total Cost/Q
Marginal cost is the EXTRA cost paid out to cover the production of the last unit
produced.
MC = ∆Total cost/∆Electricity
When there are multiple variable inputs how cost changes depends on which way output
is increased (which input was changed).
Marginal Cost is determined by the largest ratio of input price to input marginal product.
If you have found the best (least cost) combination then:
MC
= PL /∆Electricity/∆Labor = PL/MPL
= PM /∆Electricity/∆Fuel = PM /MPM
= PK /∆Electricity/∆Capital = PL /MPK
If you have not found the least cost combination then these three measures of marginal
cost will differ. The best way of increasing output would be to increase the use of the
input that offers the lowest marginal cost. To cut costs while keeping output fixed,
increase use of the input with the lowest MC and reduce use of the one with the highest
MC. (If the marginal costs differ it can be very difficult to tell whether output overall
14
should be increased or decreased. If all the marginal cost measures are less than marginal
revenue then it is safe to conclude output should be increased. If they are all greater than
MR it is safe to conclude output should decrease. If some are greater than MR and some
are less, there is no safe conclusion.)
Only after you have found the optimum proportions of the three inputs (e.g., 1 of capital
to 2 of labor to 1.5 of fuel -- like a cooking recipe) then it is possible, by changing input
amounts without changing their proportions, to increase or reduce output in a cost
minimizing way. Comparing, under these circumstances, marginal revenue with
marginal cost (all three MC measures being the same) tells you whether to increase or
reduce output in order to increase profit to its maximum value.
Price Discrimination
Once you have determined the optimum output and optimum input combinations to
maximize profit when there is no discrimination, you can use the last part of the module
to see the impact of price discrimination on optimal output and pricing. Selecting “Price
discrimination” gives you the results (given the production function you chose) for the
firm if it practices either of the two levels of discrimination allowed by the module. Of
course, you can try out price discrimination without finding the cost minimizing method
of production or the no-discrimination profit maximizing output but you will not see the
full benefit of discrimination compared to what can be achieved without discrimination.
On the results page showing output, price, etc. look for Discrimination on the tool bar and
click on it. You will now be faced with the choice of degree of discrimination -- perfect
or imperfect (two market). If you select Perfect Price discrimination the next screen you
see will show you the results your firm would get, keeping the level of output where you
previously set it, with and without discrimination. You had two or three measures of
marginal cost on the previous results screen but only the lowest of these will be seen at
this point.
You can now compare the profit you were making without discrimination with the profit
you could make (no change in output) with discrimination. Note that even if you found
the profit maximizing output (no discrimination) the profit you now get from
discrimination is still not the highest possible profit you could get – because the best
output when you do not use perfect price discrimination is not the best output when you
are using perfect price discrimination. To see why, compare the marginal revenue figure
in the column labeled Before Discrimination with the value for marginal revenue in the
column labeled After Discrimination. At this point you can choose to see a graph
showing the output level you chose and the demand, marginal revenue (no
discrimination), average cost and marginal cost lines.
If you chose imperfect price discrimination (or if you use the “Change Discrimination”
button on the tool bar on the Perfect Discrimination graph screen) you have to make
another decision. You must decide what proportion of your current output you want to
15
sell in market 1 (the rest will be sold in market 2). Once you set that proportion (or leave
it at the default value of 0.50 (50%) click on “Discrimination Results” on the tool bar.
The next screen reminds you of the results you got with no discrimination (again with
only one value -- the lowest -- for marginal cost). It also shows you the results you are
not getting in the two markets. The values for your costs are the same regardless of
whether you discriminate or not, or how you divide your output between the two markets.
The key information for you is the comparison of marginal revenue in the column for
market 1 with that for market 2. If they are not the same you are not making as much
profit out of discriminating (given the current output) as you could make.
If you select “See Graph” on this screen you will see your output, the cost curves, and
demand and marginal revenue for the two markets combined. To see how the current
decision affects the two markets (division of output, prices) and the separate demand and
marginal revenue lines for the two, select “Two Markets” on the tool bar.
The Mathematical Model
The production function in this module is a Constant Elasticity of Substitution function:
Electricity = 100 [a (capital d) + b (labor d) + c (fuel d)]e/d
Each input amount is raise to the "d" power, while the combination in the square brackets
is raised to the (e/d) power. The parameters “a”, “b”, and “c”, involve the importance of
the inputs in the production process, and in costs. The value of “d” determines the
degree to which the various inputs are substitutes for each other. The “e” in the exponent
for the square brackets determines the returns to scale, that is, if “e” is greater than one
the firm has increasing returns to scale (common in monopolies), is “e” is equal to one
the firm has constant returns to scale, and if “e” is less than one, the firm has decreasing
returns to scale.
The solution to the cost minimization problem depends on the MRTS values. Looking at
the two variable input problem, you need the marginal products of labor and fuels.
These can be obtained by using the “partial derivatives” of electricity with respect to each
input. A partial derivative is the change in the dependant variable (electricity) when one
independent variable (e.g., labor) changes, while the others are kept constant (same
definition as the marginal product).
MPL = ∆Electricity/∆labor = e* b(labor d-1)100[a (capital d) + b (labor d) + c (fuel d)](e/d)-1
MPM=∆Electricity/∆fuel =e*c(fueld-1)100[a (capital d)+b (labor d)+c (fuel d)](e/d)-1
MPK=∆Electricity/∆capital= e* a(capitald-1)100[a (capital d)+b (labor d)+c (fuel d)](e/d)-1
There is a MRTS between labor and fuels (this is the MRTS relevant to the two variable
input problem). This is:
16
MRTSLF = MPL/MPM = (b/c)(labor/fuel) d-1
The solution to the two variable input problem is found when (if P L is the price of labor
and PF is the price of fuel):
MRTSLF = MPL/MPM = (b/c)(labor/fuel) d-1 = PL / PF
If you know the values of the prices and of the three parameters, “b”, “c”, and “d”, then
this relationship will tell you the ratio of labor to fuel that will minimize the costs of
production. The actual amounts of these resources will determine how much of the
product will be produced, but it is the ratio that determines the cost per unit.
The only difference between the solution for the two variable input problem and the three
variable input problem is that in the latter the internal opportunity cost and external
opportunity cost of labor versus capital must also be equal. (It is equally valid to state
the comparison in terms of fuels versus capital; the final results will be the same in any
case.) This second condition is:
MRTSLK = MPL/MPK = (b/a)(labor/capital) d-1 = PL / PK
Adding this condition makes it possible to determine the lowest cost ratio of labor to
capital that the firm can use. (If you know the best labor to fuel ratio and the best labor to
capital ratio, then you already know the best fuel to capital ratio; no further conditions are
needed to determine that.)
If both conditions are satisfied the firm will be producing it output at a unit cost that
matches the one found on the long run average cost curve. If only the first condition is
satisfied the firm is on a short cost curve, though not the usual short run curve for which
there is only one variable input.
To maximize profit the firm must produce where MC = MR. To determine marginal
revenue it is necessary to specify the demand curve. In the module (no discrimination)
the demand curve is assumed to be:
Price = 0.25 - (0.0000003 * electricity)
Multiplying this by the quantity of electricity to get total revenue, and then finding the
change in revenue when output changes yields:
MR = 0.25 - (0.0000006 * electricity)
The function for marginal cost in the constant returns to scale case where “e” equals one
(assuming that the optimal input proportions are being used) is:
MC=((100*b)*(((a*(((32/16)*(b/a))^(d/(d-1))))+b+(c*(((8/16)*(b/c))^(d/(d-1)))))^((1-d)/d)))
17
(Note: the values of 32, 16, and 8 refer to the input prices. The 100 is a constant from
the production function that you have not been allowed to re-set.) If you know all the
parameter values, this tells you the value for marginal cost and therefore the value for
marginal revenue. Using that in the MR function would allow you to solve for the
optimal output level.
If “e” is not equal to one, marginal cost is a non-linear function of output and in general
there is no way of solving the MR = MC condition for the optimum value of output using
the usual algebraic methods. The cost curves are:
Defining certain combinations of constants:
Ccost=((32*(((b/a)*(32/16))^(1/(d-1)))+16+(8*(((b/c)*(8/16))^(1/(d-1))))
Cq=100*(((a*(((b/a)*(32/16))^(d/(d-1))))+b+(c*(((b/c)*(8/16))^(d/(d-1))))^(e/d))
Cmc=e*b*((a*(((b/a)*(32/16))^(d/(1-d))))+b+(c*(((b/c)*(8/16))^(d/(1-d)))))^((e/d)-1)
AC=(electricity^((1-e)/e))*Ccost*(Cq^(-1/e))
MC=((electricity/Cq)^((1-e)/e)*16/(100*Cmc)
The optimum value can be found using a trial and error approach. For example, enter
input values that give you an output, see if MR is greater than MC, if it is, enter larger
inputs in the same proportions, if MR is less than MC enter smaller inputs in the same
proportions, continue until MR and MC are as close to equal as you feel is necessary.
The formal method of solution is quite similar to this, using a technique from a branch of
mathematics called Numerical Methods. (Of course, when used by a computer this
technique is a great deal quicker than you are likely to be in finding a value very close to
the best output.)
Numerical methods are used for the general case (“e” not equal to 1) in order to find the
output levels where MR = MC for both perfect and imperfect competition. The
difference between the two is that for perfect discrimination the demand line is also the
MR line. In the imperfect discrimination case the optimum requires an extra step. After
you determine how to minimize cost (best input proportions) and the output for which
MR = MC for the combined markets, you must then vary proportions sold in the two
markets until MR in market 1 equals that in market 2.
Since the two demand curves are:
Market 1
Price1 = 0.24 - (0.0000006 * Q1)
MR1 = 0.24 - (0.0000012 * Q1)
18
Market 2
Price2 = 0.26 - (0.0000006 * Q2)
MR2 = 0.26 - (0.0000012 * Q2)
Q1 + Q2 = the total output you produced.
Given the total output you produced and the condition that the two amounts must add up
to that total, it is possible to solve for the profit maximizing amounts. It is easiest to do
so using a number you would already have -- marginal revenue for the combined
markets. Since to maximize profit it is necessary that:
MR = MR1 = MR2
If you have a number for MR that will give you the values for Q1 and Q2 (if they do not
add up to total Q you have made a mistake somewhere).
19