Economic Optimization
Chapter 2
Economic Optimization Process
•
Optimal Decisions
•
•
Maximizing the Value of the Firm
•
•
•
Best decision produces the result most
consistent with managerial objectives.
Produce what customers want.
Meet customer needs efficiently.
Greed vs. Self-interest
•
•
Self-indulgence leads to failure.
Customer focus leads to mutual benefit.
Value of the Firm
𝑛
𝑉𝑎𝑙𝑢𝑒 =
𝑡=1
𝑃𝑟𝑜𝑓𝑖𝑡𝑡
(1 + 𝑖)𝑡
Suppose we plan to bid on an asset that is expect to return profits noted
below over the next four years. If we expect to earn at least 10% return
on investment, what is the maximum we can pay for this property?
t
1
2
3
4
Year
2015
2016
2017
2018
Profit $10,000 $11,000 $12,000
$15,000
1.1
1.21
1.331
1.4641
The
answer
+ $9,016 + $10,245 = $37,443
NPV = $9,091 +
$9,091
Revenue Relations
Total Revenue = Price  Quantity.
• Marginal Revenue – a change in total revenue
•
associated with a one-unit change in output.
• Revenue Maximization – Quantity with highest
revenue, MR = 0.
•
Do Firms Really Optimize?
•
•
Inefficiency and waste lead to failure.
Optimization techniques are widely employed by
successful firms.
Revenue Relations
30
25
𝑃𝑟𝑖𝑐𝑒 = 𝑓 𝑄 = $24 − $1.5𝑄
20
15
10
5
0
0
2
4
6
8
10
Revenue Relations
𝑃 = $24 − $1.5𝑄
𝑇𝑅 = 𝑃𝑄 = $24 − $1.5𝑄 𝑄
𝑇𝑅 = $24𝑄 − $1.5𝑄2
𝑀𝑅 =
𝜕𝑇𝑅
𝜕𝑄
= $24 − $3𝑄
Maximize Revenue when MR=0
𝑀𝑅 =
𝜕𝑇𝑅
= $24 − $3𝑄 = 0
𝜕𝑄
$24 − $3𝑄 = 0
𝑄=8
𝑀𝑎𝑥(𝑇𝑅) = $24 8 − $1.5(8)2 = $96
Revenue Relations
120
100
80
𝑇𝑅 = $24𝑄 − $1.5𝑄 2
60
40
𝑃 = $24 − $1.5𝑄
20
𝑀𝑅 = $24 − $3𝑄
0
0
-20
2
4
6
8
10
Cost Relations
•
Total Cost = Fixed Cost + Variable Cost.
• Marginal and Average Cost
•
•
•
Marginal cost is the change in total cost associated
with a one-unit change in output.
Average Cost = Total Cost / Quantity
Average Cost Minimization
•
•
Average cost is minimized when MC = AC.
Reflects efficient production of a given output level.
Cost Relations
𝑇𝐶 = 𝐹𝐶 + 𝑉𝐶
𝐹𝐶 = $8
𝑉𝐶 = $4𝑄 + $0.5𝑄2
𝑇𝐶 = $8 + $4𝑄 + $0.5𝑄2
𝑇𝐶 $8 + $4𝑄 + $0.5𝑄2 $8
𝐴𝐶 =
=
=
+ $4 + $0.5𝑄
𝑄
𝑄
𝑄
𝑀𝐶 =
𝜕𝑇𝐶
𝜕𝑄
= $4 + $1𝑄
$8
+ $4 + $0.5𝑄 = $4 + $1𝑄
𝑄
Minimize AC when AC = MC
$8
+ $0.5𝑄 = $1𝑄
𝑄
Subtracting by -$0.5Q
𝐴𝐶 = 𝑀𝐶
$8
= $0.5𝑄
𝑄
Multiply of 2Q
$8
+ $4 + $0.5𝑄 = $4 + $1𝑄
𝑄
$16 = 𝑄2
$16 =
𝑄=4
𝑄2
𝑄=
$16
𝑄=4
Cost Relations
120
𝑇𝐶 = $8 + $4𝑄 + $0.5𝑄 2
100
80
60
40
Minimum AC (AC = MC)
20
MC = $4 + $1𝑄
AC = $4 + $0.5𝑄
0
0
2
4
6
8
10
Profit Relations
•
Total and Marginal Profit
•
•
•
Profit Maximization
•
•
Total Profit (π ) = Total Revenue - Total Cost.
Marginal profit is the change in total profit due to a
one-unit change in output, Mπ = MR - MC.
Profit is maximized when Mπ = MR – MC = 0 or MR
= MC, assuming profit declines as Q rises.
Marginal vs. Incremental Profits
•
•
Marginal profit is the gain from producing one more
unit of output (Q).
Incremental profit is gain tied to a managerial
decision, possibly involving multiple units of Q.
Profit Relations
𝑇𝑅 = $24𝑄 − $1.5𝑄2
𝑀𝑅 =
𝜕𝑇𝑅
𝜕𝑄
= $24 − $3𝑄
𝑇𝐶 = $8 + $4𝑄 + $0.5𝑄2
𝑀𝐶 =
𝜕𝑇𝐶
𝜕𝑄
= $4 + $1𝑄
Maximize Profit when MR = MC
𝑀𝑅 = 𝑀𝐶
$24 − $3𝑄 = $4 + $1𝑄
𝑄=5
𝑀𝑎𝑥 𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑇𝑅 − 𝑇𝐶 = $24𝑄 − $1.5𝑄2 − ($8 + $4𝑄 + $0.5𝑄2 )
𝑀𝑎𝑥 𝑃𝑟𝑜𝑓𝑖𝑡 = −$8 + $20𝑄 + $2𝑄2 = −$8 + $20 5 + $2(5)2 = $42
Profit Relations
50
Maximum Profit (MR = MC)
40
𝑃𝑟𝑜𝑓𝑖𝑡 = −$8 + $20𝑄 + $2𝑄 2
30
20
MC = $4 + $1𝑄
10
𝑀𝑅 = $24 − $3𝑄
0
0
-10
1
2
3
4
5
6
7
8
9
10