Economic Optimization
Chapter 2
Chapter 2
OVERVIEW
 Economic
Optimization Process
 Revenue Relations
 Cost Relations
 Profit Relations
 Incremental Concept in Economic Analysis
Chapter 2
KEY CONCEPTS
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optimal decision
spreadsheet
Equation
dependent variable
independent variable
marginal revenue
revenue maximization
cost functions
short-run cost functions
long-run cost functions
short run
long run
total costs
fixed costs
variable costs
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marginal cost
average cost
average cost minimization
total profit
marginal profit
profit maximization rule
breakeven points
incremental change
incremental profit
breakeven point
average cost minimization
multivariate optimization
constrained optimization
Lagrangian technique
Lagrangian multiplier, λ
Economic Optimization Process
 Optimal
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Decisions
Best decision produces the result most
consistent with managerial objectives.
 Maximizing
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Produce what customers want.
Meet customer needs efficiently.
 Greed
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the Value of the Firm
vs. Self-interest
Self-indulgence leads to failure.
Customer focus leads to mutual benefit.
© 2009, 2006 South-Western, a
part of Cengage Learning
Revenue Relations
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Price and Total Revenue
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Marginal Revenue
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Change in total revenue associated with a one-unit
change in output.
Revenue Maximization
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Total Revenue = Price  Quantity.
Quantity with highest revenue, MR = 0.
Do Firms Really Optimize?
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Inefficiency and waste lead to failure.
Optimization techniques are widely employed by
successful firms.
Where do demand equations come
from?
 Companies
compile data over time on
specific variables. (Time series analysis)
 Companies
compile data across different
areas at a point in time. (Cross sectional
analysis)
Demand Equation
A firm estimates the demand for its product to be related to the price it charges
(P), the amount of advertising it does (A), and the temperature (T). Gathering
data over quarters from the past ten years, it runs a regression over the 40
observations and the computer spits out the following regression equation:
D = 193.0 - 49.5 P + 292.5 A + 17.8 T
(12.1) (105.0)
(9.9)
R2 = 0.74
Standard error of the estimate = 16
Standard error of the coefficients are in the ( ).
Interpreting the equation
• What percentage of variation of demand did the model explain?
- 74%
• What other independent variables could have been included that probably
would have increased the explanatory power of the model?
- the prices of related goods, per capita income
• Based on the regression results, if the price of the good is increased by one
unit, what will be the impact on demand?
- demand would be expected to fall by 45.9 unites
• What if advertising is increased by one unit?
- demand would be expected to increase by 292.5 units
Demand Equation Observations
Based on the sign of the temperature coefficient,
would this product more likely be hot chocolate or Pepsi?
- the positive sign indicates that as the temperature increases so will the
demand for this good. Hence, it’s more likely to be a product such as Pepsi.
In which coefficients would you have at least 95% confidence that you have a
good estimate of the marginal relationship between the independent variable
and the dependent variable?
- each one for which the estimated coefficient is at least twice as great as the
standard error of the coefficient (P and A; not T)
Predicting Demand given some
economic starting points
What is the point estimate of demand if P= 75, A=10, and T=70?
- plug in these values of the independent variables in our regression equation
and out falls D = 921.50.
One can predict with a 95% confidence of being correct that the demand of this
good next period will lie in the range of 889.50 units to 953.50 units. The 95%
confidence interval would be 921.50 + 2(16). This range is 921.50 + 32 =
953.50 on the upper end and 921.50 – 32 = 889.50 on the lower end.
What does the coefficient of P
really mean?
- that if the price of the good is increased by one
unit while every other independent variable is held
constant, the demand for this good will fall by 45.9 units.
- you should recognize that the coefficient in a regression equation is nothing
more than the partial derivative of the dependent variable with respect to a
particular independent variable. For example:
The Supply side of the equation
 Cost
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Marginal Cost
Average Cost
Fixed Cost
Variable Cost
Cost Relations
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Total Cost
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Marginal and Average Cost
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Total Cost = Fixed Cost + Variable 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
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Average cost is minimized when MC = AC.
Reflects efficient production of a given output level.
Goal is to minimize costs
• A business can find the quantity where
costs are minimized by going to the point
where MC = AC The typical cost curve.
© 2009, 2006 South-Western, a
part of Cengage Learning
Maximizing Profit
 Total
Revenue = P * Q
 TR= $24Q – $1.5Q2
 TC
= $8 + 4Q + .5Q2
© 2009, 2006 South-Western, a
part of Cengage Learning
L. Pantuosco Course Notes
(бD)/(б/P) = -45.9 (бD)/(б/A) = 292.5
(бD)/(б/T) = 17.8
Suppose the independent variables have the values given in part (g).
What impact would there by on demand if the price of the goods were
increased by 1%? - to answer this we need to use the concept
of point price-elasticity of demand:
eP = (бD/бP)/(P/D) = (-45.9)
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- this means that a 1% increase in price will
\result in a 3.736% decrease in demand
- compare this to part (i) where we saw that a
one unit (not 1%) increase in price would lead
to a decrease in demand of 45.9 units (not 45.9%).
© 2009, 2006 South-Western, a
part of Cengage Learning
(75/921.50) = -3.736
Profit Relations
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Total and Marginal Profit
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Profit Maximization
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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 v. Incremental Profits
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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.
© 2009, 2006 South-Western, a
part of Cengage Learning
L. Pantuosco Course Notes
What is the difference between a “point estimate of demand” and the “point
elasticity of demand”?
- compare the parts (g) and (j)
- the “point estimate of demand” shows what demand would be at a particular
point along the demand function (when the independent variables are all given
particular values).
- the “point elasticity of demand” shows the percentage change in demand from
the point estimate of demand when there is one percent change in price.
© 2009, 2006 South-Western, a
part of Cengage Learning
L. Pantuosco Course Notes
LEAST-COST COMBINATION OF INPUTS – AN EXAMPLE
A firm has the production function: Q =40L .80 K .20. The prices of
labor and capital are $1,000 and $2,000 respectively. What
combination of L and K would produce an output level of 5800 units
at the lowest total cost?
There are two approaches to this problem: (a) the optimal input ratio approach
and (b) the Lagrangian Multiplier Approach.
© 2009, 2006 South-Western, a
part of Cengage Learning