Transcript Outline 1. Locating a shopping mall 2. A model of the firm

Outline
1. Locating a shopping mall
2. A model of the firm
3. Profit maximization
4. Marginal analysis
5. Sensitivity analysis
Locating a shopping mall in a
coastal area
•Villages are located East to West
along the coast (Ocean to the North)
•Problem for the developer is to locate
the mall at a place which minimizes
total travel miles (TTM).
Number of Customers per Week (Thousands)
West
15
10
A
B
3.0
10 10
xC
3.5
D
2.0
2.5
5
20
10
E
F
G
4.5
2.0
Distance between Towns (Miles)
15
H
4.5
East
Minimizing TTM by enumeration
•The developer selects one site at a time, computes the
TTM, and selects the site with the lowest TTM.
•The TTM is found by multiplying the distance to the
mall by the number of trips for each town (beginning
with town A and ending with town H).
•For example, the TTM for site X (a mile west of town
C) is calculated as follow:
(5.5)(15) + (2.5)(10) + (1.0)(10) + (3.0)(10) + (5.5)(5) +
(10.0)(20) + (12.0)(10) + (16.5)(15) = 742.5
Marginal analysis is more effective
Enumeration takes lots of
computation. We can find the
optimal location for the mall
easier using marginal analysis—
that is, by assessing whether
small changes at the margin will
improve the objective (reduce
TTM, in other words).
Illustrating the power of marginal analysis
1. Let’s arbitrarily select a location—say, point X. We
know that TTM at point X is equal to 742.5—but we
don’t need to compute TTM first.
2. Now let’s move in one direction or another (We will
move East, but you could move West).
3. Let’s move from location X to town C. The key
question: what is the change in TTM as the result of
the move?
4. Notice that the move reduces travel by one mile for
everyone living in town C or further east.
5. Notice also that the move increases travel by one
mile for everyone living at or to the west of point X..
Computing the change in TTM
To compute the change in total travel miles (TTM)
by moving from point X to C:
TTM = (-1)(70) + (1)(25) = - 45
Reduction in TTM for
those residing in and
to the East of town C
Increase in TTM for
those residing at or to
the west of point X.
Conclusion: The move to town C unambiguously
decreases TTM—so keep moving East so long as
TTM is decreasing.
Rule of Thumb
Make a “small” move to a nearby alternative if, and
only if, the move will improve one’s objective
(minimization of TTM, in this case). Keep moving,
always in the direction of an improved objective, and
stop when no further move will help.
•
Check to see if moving from town C to town D
will improve the objective.
•
Check to see if moving from town E to town F
will improve the objective.
Model of the firm
Assumptions:
1. A firm produces a single good or service
for a single market with the objective of
maximizing profit.
2. The task for the firm is to establish price
and output at levels which achieve the
objective.
3. Firms can predict the revenue and cost
consequences of its price and output
decisions with certainty.
A microchip manufacturer
A microchip is a piece of semiconducting material that
contains a large number of integrated circuits.
•
The problem for the microchip manufacturer
is to determine the quantity of chips to
manufacture, as well as their price.
•
The objective of management is to maximize
profits—the difference between revenue and
cost.
•
In algebraic terms, we have:
=R–C
where  is profit; R is revenue, and C is cost.
Definitions
•Demand: The quantities of a good or service (or factor of
production) buyers are willing and able to buy at various prices,
other things being equal.
•Quantity demanded: The quantities of a good or service (or
factor of production) buyers are willing and able to buy at a
specific price, other things being equal.
•Law of demand: Other things being equal, price and quantity
demanded of a good or service (or factor of production) are
inversely related.
The demand for microchips
Price (Thousands of Dollars)
200
The firm uses the
demand curve to
predict the revenue
consequences of
alternative pricing
and output policies
150
A
B
100
C
50
0
2
4
6
8
10
Quantity (Lots)
Algebraic representation of demand
•The demand curve for microchips is
given by:
Q = 8.5 - .05P,
[2.1]
Where Q is the quantity of lots
demanded per week, and P denotes the
price per lot (in thousands of dollars).
•We see, for example, that if the P = 50,
then according to [2.1] Q = 6. This
corresponds to point C on our demand
curve.
Inverse demand and the revenue function (R)
By rearranging equation [2.1], we obtain the following
inverse demand equation for microchips:
P = 170 – 20Q
[2.2]
Note that revenue from the sale of microchips (R) is given by
price (P) times quantity sold (Q) or:
R=PQ
Substituting [2.2] into this equation yields the revenue
function (R);
R = P  Q = (170 – 20Q)Q = 170Q – 20Q2 [2.3]
The revenue function (R)
Total Revenue (Thousands of Dollars)
400
300
200
Quantity
(Lots)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
8.5
Price Revenue
($000s)
($000s)
170
0
150
150
130
260
110
330
90
360
70
350
50
300
30
210
10
80
0
0
100
0
2
4
6
8
10
Quantity (Lots)
Check Station 1
The inverse demand function is given by:
P = 340 - .8Q
Find the revenue function:
Thus the revenue function is given by:
R = P  Q = (340 - .8Q)Q = 340Q - .8Q2
The cost function (C)
To produce microchips, the firm must have a plant,
equipment, and labor.
•The firm estimates that for each chip produced, the cost
of labor, materials, power, and other inputs is $38. This
converts to a variable cost of $38,000 per lot.
•In addition, there are $100,000 in cost the firm could not
avoid even if it shut down—that is, fixed cost = $100,000.
•Thus, the cost function is given by:
C = 100 + 38Q
[2.4]
C = 100 + 38Q
Total Cost (Thousands of Dollars)
450
400
350
T otal cost
300
250
200
Quantity
(Lots)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
150
100
50
0
2
4
6
8
10
Quantity (Lots)
Cost
(000s)
100
138
176
214
252
290
328
366
The profit function ()
Given a revenue function (R) and a cost function (C), we
can derive a profit function ():
=R–C
= (170Q – 20Q2) – (100 + 38Q)
= -100 + 132Q – 20Q2
[2.5]
Profit from microchips
Total Profit (Thousands of Dollars)
150
100
50
Total prof it
Quantity Profit Revenue Cost
(Lots) ($000s) ($000s) (000s)
0.0
-100
0
100
1.0
12
150
138
2.0
84
260
176
3.0
116
330
214
4.0
108
360
252
5.0
60
350
290
6.0
-28
300
328
7.0
-156
210
366
0
-50
-100
-150
0
1
2
3
4
5
6
7
8
Check Station 2
Suppose the demand function is
P = 340 - .8Q
And the cost function is:
C = 120 + 100Q
Write the profit() function:
  120  240Q  .8Q
2
Marginal Analysis--Again
OK, we have a profit equation.
Now we want to find the profit
maximizing quantity (Q). One
method is enumeration—that is,
we substitute different values
for Q into [2.5] until we find the
Q that gives the highest profit
(). But this is too cumbersome.
Marginal analysis is better.
The marginal profit (M) function
Marginal profit (M) is the change in profit
resulting from a small change in a managerial
decision variable, such as output (Q).
The algebraic expression for marginal profit is
Marginal profit =
Change in profit
Change in output
  1   0


Q Q1  Q 0
where the term “” stands for “change in,” Q0 is the
initial level of output (0 is the corresponding level of
profit) and Q1 is the new level of output.
Marginal profit
Quantity
(Lots)
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Profit Marginal Profit
(dollars)
Per Lot
105000
108000
30000
110600
26000
112800
22000
114600
18000
116000
14000
117000
10000
117600
6000
117800
2000
117600
-2000
117000
-6000
116000
-10000
114600
-14000
To compute M when
Q increases from 2.5 to
2.6 lots:
M 
 $3000

 $30,000
Q
.1
Marginal profit (M) is equal to the slope of a line tangent to
the profit function
Prof it (Thousands of Dollars)
119
•Slope at point A = M = 8,000
•Slope at point B = M = 0
B
118
A
117
Notice that profit
() is maximized
when the slope of
the  function is
equal to zero
116
3.0
3.1
3.2
3.3
3.4
3.5
Quantity (Lots)
Maximum profit is attained at the output level at
which marginal profit is zero.
Again our profit function is given by
 = -100 + 132Q – 20Q2
[2.6]
Marginal profit (the slope of the profit function) can
be found by taking the first derivative of the profit
function with respect to output:
M 
d
 132  40Q
dQ
[2.7]
Set M = 0 and solve for Q
We know that profit is
maximized when the slope of
the profit function is equal to
zero. So set the first derivative
of the function equal to zero to
find the optimal output
M = 132 – 40Q = 0
Solving for Q yields:
Q = 132/40 = 3.3 lots
Check Station 4
Suppose the demand function is
P = 340 - .8Q
And the cost function is:
C = 120 + 100Q
Write the marginal profit (M) function. Set M = 0 to
find the optimal output:
d
M 
 240  1.6Q
dQ
240  1.6Q  0  Q  240 / 1.6  150
Marginal revenue
•Marginal revenue is the additional revenue that comes from a
unit change in output and sales. The marginal revenue (MR) of
an increase in sales from Q0 to Q1 is given by:
Change in revenue
Marginal revenue =
Change in output
R R1  R 0


Q Q1  Q 0
Marginal cost
•Marginal cost is the additional cost of producing an extra unit
of output. The marginal revenue (MC) of an increase in output
from Q0 to Q1 is given by:
Marginal cost =
Change in cost
Change in output
C C1  C 0


Q Q1  Q 0
Profit maximization revisited
We know that  = R – C. It follows that:
M  = MR – MC
[2.9]
We also know that profit is maximized when M  = 0. Another
way is say this is that profit is maximized when MR – MC = 0.
This leads to profit maximizing rule of thumb:
The firm’s profit-maximizing level of output
occurs when the additional revenue from
selling an extra unit just equals the extra cost
of producing it, that is, when MR = MC
Equating MR and MC
(a) Total Revenue, Cost, and Profit (Thousands of Dollars)
400
•MR is given by the
slope of a line tangent to
the revenue function
300
Total cost
200
Revenue
•Profit is maximized
when the slopes of the
revenue and cost
functions are equal
100
Profit
0
–100
0
2
•MC is given by the
slope of a line tangent to
the cost function.
4
6
8
Microchip example--again
We know that MR = 170 – 40Q. We know also that MC = 38.
To solve for the profit maximizing output set MR = MC and
solve for Q:
170 – 40Q = 38
Thus:
40Q = 132, therefore Q = 3.3 lots
Check station 5
Suppose the inverse demand and cost functions are given by:
P = 340 - .8Q
C = 120 + 100Q
Solve for the profit maximizing level of output using the MR = MC approach.
Step 1 is to obtain the revenue function (R):
R = P · Q = (340 - .8Q)Q = 340Q - .8Q2
Now find MR by taking the first derivative of R with respect to Q:
MR = dR/dQ = 340 - 1.6Q
Now find MC by taking the first derivative of C with respect to Q:
MC = dC/dQ = 100
Now set MR = MC and solve for Q:
340 – 1.6Q = 100
1.6Q = 240. Thus Q = 240/1.6 = 150
Sensitivity analysis
In light of changes in the
economic facts of a given
problem, how should the
decision maker alter his or her
course of action? Marginal
analysis is a big help.
For any change in economic conditions, we can trace the
impact (if any) on the firm’s marginal revenue or marginal
cost. Once we have identified this impact, we can appeal to
the MR = MC rule to determine the new, optimal decision.
Changing economic facts
Using sensitivity analysis, we can determine the change in
the optimal output resulting from the following:
•A change in overhead (fixed) costs;
•A change in materials (variable) cost; and
•A change in demand
Initial optimum output of microchips
(a) Marginal Revenue and Cost (Thousands of Dollars)
150
MR = 170 - 40Q
MR = MC when
Q = 3.3 lots
100
50
MC = 38
3.3
Quantity (Lots)
Overhead increases by $12,000
(a) Marginal Revenue and Cost (Thousands of Dollars)
150
MR = 170 - 40Q
MR = MC when
Q = 3.3 lots
100
50
MC = 38
3.3
Notice that the optimal output does not
change, since MC is unaffected by a change
in overhead cost. Profit, however, decreases
by $12,000—whatever the level of output
Quantity (Lots)
Silicon prices rise from $38,000
to $46,000 per lot
(b) Marginal Revenue and Cost (Thousands of Dollars)
150
MR = 170 - 40Q
100
Note this will shift the MC
function up the vertical axis
MC = 46
50
MC = 38
3.1 3.3
Setting MR = MC we obtain:
170 – 40Q = 46
Q = 124/40 = 3.1 lots
Quantity (Lots)
Increased demand for microchips
(c) Marginal Revenue and Cost (Thousands of Dollars)
MR = 190 - 40Q
150
100
Note the increase in
demand is manifest in a
shift to the right of the MR
function
50
MC = 38
3.3 3.8
Setting MR = MC, we obtain:
190 – 40Q = 38
Q = 152/40 = 3.8 lots
Quantity (Lots)