A mathematical demand function for new Toyotas sold per year for a dealer is as
ID: 1167915 • Letter: A
Question
A mathematical demand function for new Toyotas sold per year for a dealer is as follows Qr = 200 - 0.001Pt + 0.005Pm - 10Pg + 0.01I +0.003A
where
Qt = quantity purchased,
Pt = the average price of Toyotas,
Pm = the average price of mazdas,
Pg = the price of gasoline,
I = per capita income, and
A = dollars spent annually on advertising
PM = $20000 PG = $1.00 I = $15000 A = $10000
1. Use the above to calculate the arc price elasticity of demand between PT = $15000 and PT = $10000
2. What will the estimated quantity sold and total revenue be at each of the above prices (fill in table below)?
PT
QT
Revenue
$15000
$10000
3. Marketing wants to lower PT from $15000 to $10000. From a revenue viewpoint, explain why you agree or disagree with Marketing. Look at the elasticity calculated in #1 and the behavior of revenues in #2.
4. Calculate the point price elasticity of demand, given that PT = $12500 and QT = 345. Compare the results with the answer to #1. If the results are the same, why does this make sense? If the results differ, why does this make sense? The formula is:
5. Calculate the point cross-price elasticity of demand between gasoline price (PG) and Toyota demand. Assume the OPEC lowered petroleum production quotas and caused the price of gasoline to triple to PG = $3.00. Calculate a new QT for PG = $3.00 and PT = $15000. Other variables and their values are given at the top, before question #1. Explain what this elasticity value implies for the responsiveness of the demand for Toyotas is relative to the price of gasoline?
PT
QT
Revenue
$15000
$10000
Explanation / Answer
Qr = 200 - 0.001Pt + 0.005Pm - 10Pg + 0.01I +0.003A
PM = $20000 PG = $1.00 I = $15000 A = $10000
Qr = 200 - 0.001Pt + 100 - 10 + 150 + 30
Qr = 470 - 0.001Pt
When Pt = 15,000, Qr = 455
When Pt = 10,000, Qr = 460
(1) Arc elasticity of demand = [change in quantity / average quantity] / (change in price / average price)
Average Pt = 12,500 & Average Qr = 457.5
So, arc elasticity = (5 / 457.5) / (- 5,000 / 12,500)
= - 0.027
(2)
PT
QT
Revenue
$15000
455
6,825,000
$10000
460
4,600,000
(3)
As we see, the price elasticity is very low, suggesting this is an inelastic market where decrease in price will have little effect on demand. As the revenue table shows, revenue will actually fall if this price change is implemented (decrease in price too high & not offset by a low increase in demand).
So, it is advisable not to lower price.
(4)
dQr / dPt = - 0.001
Point elasticity of demand = (dQr / dPt) x (Pt / Qr)
= (- 0.001) x (12,500 / 345)
= - 0.0036
(5)
Cross price elasticity = (dQr / dPg) x (Pg / Qr)
dQr / dPg = - 10
Pg = 1 (given) for which, Qr cannot be calculated unless Pt value is given. The question does not mention whether to use any previously provided Pt values.
Incomplete information.
PT
QT
Revenue
$15000
455
6,825,000
$10000
460
4,600,000
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