Outline

1
Outline
Optimal Decisions using Marginal Analysis

• Locating a shopping mall
• A model of the firm
• Profit maximization
• Marginal analysis
• Sensitivity analysis

2
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)
15
15
10
10
10
10
20
5
West
East
x
A
B
C
D
E
F
G
H
3.0
3.5
2.5
4.5
4.5
2.0
2.0
Distance between Towns (Miles)
3
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

4
Marginal analysis is more effective
Enumeration takes lots of computation. We can
find the optimal location for the mall easier
using marginal analysisthat is, by assessing
whether small changes at the margin will improve
the objective (reduce TTM, in other words).
5
Illustrating the power of marginal analysis

• Lets arbitrarily select a locationsay, point X.
  We know that TTM at point X is equal to 742.5but
  we dont need to compute TTM first.
• Now lets move in one direction or another (We
  will move East, but you could move West).
• Lets move from location X to town C. The key
  question what is the change in TTM as the result
  of the move?
• Notice that the move reduces travel by one mile
  for everyone living in town C or further east.
• Notice also that the move increases travel by one
  mile for everyone living at or to the west of
  point X..

6
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 TTMso keep moving East so long as TTM
is decreasing.
7
Rule of Thumb
Make a small move to a nearby alternative if,
and only if, the move will improve ones
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.

8
Model of the firm

• Assumptions
• A firm produces a single good or service for a
  single market with the objective of maximizing
  profit.
• The task for the firm is to establish price and
  output at levels which achieve the objective.
• Firms can predict the revenue and cost
  consequences of its price and output decisions
  with certainty.

9
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
  profitsthe difference between revenue and cost.
• In algebraic terms, we have? R Cwhere ?
  is profit R is revenue, and C is cost.

10
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.

11
The demand for microchips
The firm uses the demand curve to predict the
revenue consequences of alternative pricing and
output policies
A
B
C
2
4
6
8
0
12
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.

13
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
14
The revenue function (R)
2
4
6
8
0
15
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
16
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 downthat is,
  fixed cost 100,000.
• Thus, the cost function is given byC 100
  38Q 2.4

17
C 100 38Q
0
2
4
6
8
18
The profit function (?)

• Given a revenue function (R) and a cost function
  (C), we can derive a profit function (?)
• ? R C
  2.5
• (170Q 20Q2) (100 38Q)
• -100 132Q 20Q2

19
Profit from microchips
T
0
0
0
0
0
1
2
3
4
5
6
7
8
20
Check Station 2
Suppose the demand function is P 340 -
.8Q And the cost function is C 120
100Q Write the profit(?) function
21
Marginal Analysis--Again
OK, we have a profit equation. Now we want to
find the profit maximizing quantity (Q). One
method is enumerationthat 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.
22
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
Change in profit
Marginal profit
Change in output
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.
23
Marginal profit
To compute M? when Q increases from 2.5 to 2.6
lots
24
Marginal profit (M?) is equal to the slope of a
line tangent to the profit function

• Slope at point A M? 8,000
• Slope at point B M? 0

B
A
Notice that profit (?) is maximized when the
slope of the ? function is equal to zero
25
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

2.7
26
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
27
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
28
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
29
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

Change in cost
Marginal cost
Change in output
30
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 firms 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
31
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
• MC is given by the slope of a line tangent to the
  cost function.
• Profit is maximized when the slopes of the
  revenue and cost functions are equal

300
200
100
Profit
0
0
2
4
6
8
32
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
33
Check station 5
Suppose the inverse demand and cost functions are
given by P 340 - .8QC 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
34
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 firms 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.
35
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

36
Initial optimum output of microchips
(a)
Marginal Revenue and Cost (Thousands of Dollars)
MR 170 - 40Q
150
MR MC when Q 3.3 lots
100
50
MC 38
3.3
Quantity (Lots)
37
Overhead increases by 12,000
(a)
Marginal Revenue and Cost (Thousands of Dollars)
MR 170 - 40Q
150
MR MC when Q 3.3 lots
100
50
MC 38
3.3
Quantity (Lots)
Notice that the optimal output does not change,
since MC is unaffected by a change in overhead
cost. Profit, however, decreases by
12,000whatever the level of output
38
Silicon prices rise from 38,000to 46,000 per
lot
(b)
Marginal Revenue and Cost (Thousands of Dollars)
MR 170 - 40Q
150
Note this will shift the MC function up the
vertical axis
100
MC 46
50
MC 38
3.3
3.1
Quantity (Lots)
Setting MR MC we obtain 170 40Q 46 Q
124/40 3.1 lots
39
Increased demand for microchips
(c)
Marginal Revenue and Cost (Thousands of Dollars)
Note the increase in demand is manifest in a
shift to the right of the MR function
150
100
50
MC 38
3.3
3.8
Quantity (Lots)
Setting MR MC, we obtain 190 40Q 38 Q
152/40 3.8 lots