Download 14Q-8Q=90 6Q=90 Q=15 A=(2*90)2 = $900 P=$175 MC= 8Q +10

Problem Set #1 Key
Sonoma State University
Economics 494- Seminar in Quantitative Marketing II
Dr. Cuellar
Profit Maximizing Behavior of the Firm
Suppose a firm operating in a differentiated product market has the following demand and cost
functions:
A1/2
Q=
TC=4Q2 +10Q +A
1. Calculate the profit maximizing level of output, price and advertising.
Solve demand for P and multiply by Q to get total revenue:
Π = 100Q – 3Q2 + 4A1/2Q – [4Q2 +10Q +A]
The two necessary first order conditions for optimization are:
1.
90 –14Q + 4A1/2 =0.
2.
2QA-1/2 – 1=0 Solve for A1/2 and substitute into (1).
A1/2 = 2Q
Solving (1) for Q =
+
A1/2
Q=
Q-
14Q-8Q=90
6Q=90
Q=15
A=(2*90)2 = $900
P=$175
MC= 8Q +10 = $130
2. Calculate the price elasticity of demand at the profit maximizing solution.
We can calculate the price elasticity of demand from the demand equation:
Q=
= 3.889
3. Calculate the advertising elasticity of demand at the profit maximizing solution.
The advertising elasticity of demand can be derived from the demand equation.

= 1.33
4. Verify that the Lerner mark-up index is equal to the negative of the inverse of the price elasticity
of demand (
).
= 1/.257 = 3.889
5. Verify that the advertising expenditures-to-sales revenue ratio is equal to the negative of the
ratio of the advertising elasticity-to-price elasticity (
).
Estimating Elasticities
Suppose that a firm believes that its demand is given by the equation: Q=kPηIβaγAδ, where:
P is the price per unit and η is the price elasticity of demand
I is income and β is the income elasticity of demand
a is current advertising and γ is the short run advertising elasticity of demand
A is average advertising in the preceding ten years and (γ+δ) is the long run advertising elasticity
of demand
1. The data for problem set #1 contains data for all relevant variables except price. However, you
can still estimate all relevant elasticities by estimating the associated total revenue function.
Estimate the total revenue function and solve for the price, income, the short-run and long-run
advertising elasticities of demand. Are your estimates statistically significant? Show all your
work.
Solving Q=kPηIβaγAδ for price, multiplying by Q and taking logs on both sides produces:
lnTR=
(
)
The estimating equation:
lnTR=β0 + β1lnQ+ β2lnI+ β3lna+ β4lnA+ui
lnqty/lnprice
lninc
lnadv
lnaveadv
Constant
Question #1
lnrev
-0.339
[0.16]
1.147
[0.00]**
0.185
[0.24]
0.119
[0.05]*
-8.201
[0.00]**
50
0.67
Question #3
lnqty
-0.309
[0.00]**
0.786
[0.00]**
0.363
[0.00]**
0.025
[0.39]
-6.065
[0.00]**
50
0.95
Observations
Adjusted R2
p-value in brackets
* significant at 5% level; ** significant at 1% level
Solving the total revenue equation for the respective elasticities:
η -0.746826
β
0.856609
γ
0.138163
δ
γ+δ
0.088872
0.227035
.2
.19
.18
.17
.16
.15
.14
.13
.12
.11
.1
.09
.08
.07
.06
.05
.04
.03
.02
.01
0
Advertising/Revenue
2. Is the firm advertising optimally? Explain fully and show graphically.
The optimal ratio of advertising expenditures to revenue.
1960
1970
1980
1990
YEAR
2000
2010
It appears that the firm is under advertising.
3. Construct the variable price where price=revenue/quantity and re-estimate the elasticity
parameters directly. Are they similar to the estimates in (1)? Are they statistically significant?
Show all your work.
4. Is the firm advertising optimally? Explain fully and show graphically.
The optimal ratio of advertising expenditures to revenue.
1.2
1.1
1
.9
.8
.7
.6
.5
Advertising/Revenue
.4
.3
.2
.1
0
1960
1970
1980
It appears that the firm is under advertising.
1990
YEAR
2000
2010