Balakrishnan, Sivaramakrishnan, & Sprinkle – 2e
FOR INSTRUCTOR USE ONLY
5-46

This simplifies to:
Profit = 492.50(
Price
) – 9.85(
Price
)
2
– 4,250 + 85
P
– 3,360.
Or, the relation between Jessica’s profit and price is:
Profit = 577.50(
Price)
– 9.85(Price)
2
– 7,610.
We can solve for the best
P
using calculus or Excel’s solver function. The exact
answer is
P
= $29.31 and expected profit is $854.63
e.
Taxes affect profit in a relatively straightforward fashion. We can take the
profit model developed in part [d] and add the tax variable – doing so yields:
Profit after taxes = Profit before taxes – taxes paid.
Here, because taxes are proportional to income, taxes paid = (tax rate
pre-tax
profit). Adding this column yields:
Price per cap
Demand
Expected
Profit
After-tax
profit
$20
300
$0.00
$0.00
$25
250
$671.25
$503.44
$28
220
$837.60
$628.02
$30
200
$850.00
$637.50
$32
180
$783.60
$587.70
$34
160
$638.40
$478.80
The optimal price continues to be
$30 per cap
.
Notice that the optimal price does not change – this occurs because (in this
example), taxes are a linear function of pre-tax profit. Thus, Jessica still wishes to
maximize pre-tax profit which, in turn, will also maximize after-tax profit. This
aspect of the problem allows instructors to discuss the relation between pre-tax
income and tax rates, including extending the discussion to non-linear relations
between pre-tax and after-tax income.
Taxes will, of course, reduce the amount of Jessica’s profit. In our example,
Jessica will now earn an after-tax monthly profit of:
$850
.75 =
$637.50.
Note (Can skip without loss of continuity):
In the calculus approach,
incorporating taxes yields
Balakrishnan, Sivaramakrishnan, & Sprinkle – 2e
FOR INSTRUCTOR USE ONLY
5-47

Profit after taxes = (577.50(
Price
) – 9.85(
Price
)
2
– 7,610)
(1 – tax rate).
Since t = .25, we have:
Profit after taxes = (577.50(
Price
) – 9.85(
Price
)
2
– 7,610)
(.75).
Profit after taxes = 433.125(
Price
) – 7.3875(
Price
)
2
– 5707.50.
Again, we can solve for the best
P
using calculus or Excel’s solver function. The
exact answer is
Price
= $29.31
(which is exactly what we arrived at earlier).
f.
While Jessica’s venture has intuitive appeal and our original calculations
seemed to indicate that the venture might be a “go” (after all, selling 22 hats a
day does not seem like a lot) the numbers simply do not add up. Given the
totality of the costs involved and Jessica’s market research, it appears that (at
best) Jessica will earn a very modest profit and will be unlikely to maintain a
reasonable lifestyle with this business.
We can see, though, how modest amounts add up – for example, if Jessica were to
“manage” 10 kiosks in various malls around the state, she could earn a reasonable
sum of money – this is precisely what franchisers seek to do.
5.66
Rick’s English Hut.
a.
Currently, Rick’s is generating $60,000 in sales. For alcohol and food, this
translates to:
Alcohol Sales: $60,000
.55 = $33,000, or $33,000/$4 = 8,250 “alcohol units.”
Food Sales: $60,000
.45 = $27,000, or $27,000/$5 = 5,400 “food units.”
Monthly profit can then be calculated as:
[8,250
($4 – $2)] + [5,400
($5 – $4)] – $10,950 =
$10,950.