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A restaurant faces very high demand for its signature mousse desserts in the evening but is less busy during the day
A restaurant faces very high demand for its signature mousse desserts in the evening but is less busy during the day. Its manager estimates that inverse demand functions are pe = 30 - Qe in the evening and pd = 16 - Qd during the day, where e and d denote evening and daytime. The marginal cost of producing its dessert evening, MCe, is $8. The marginal cost of producing its dessert daytime, MCd, is $4. There is no fixed cost of producing dessert. Create a spreadsheet with the column headings Qe, Pe, ?Re, MRe, TCe, MC, ne, Qd, Pd, TRd, MRd, TCd, MCd, and itd. (note: ne is profit evening and nd indicates profit daytime) a. What are the optimal prices for the dessert that the restaurant should charge during the evening hours? b. What is the optimal quantity for the dessert that the restaurant should produce during the evening hours? c. What is the total cost of producing the optimal quantity for the dessert during the evening hours? d. What is the maximum profit for the dessert that the restaurant should produce during the evening hours? e. What are the optimal prices for the dessert that the restaurant should charge during the daytime hours? f. What is the optimal quantity for the dessert that the restaurant should produce during the daytime hours? g. What is the total cost of producing the optimal quantity for the dessert during the daytime hours? h. What is the maximum profit for the dessert that the restaurant should produce during the daytime hours?
Expert Solution
Spreadsheet of the given parameters
| Qe | Pe | TRe | MRe | TCe | MCe | πe | Qd | Pd | TRd | MRd | TCd | MCd | πd |
| $11 | $19 | $209 | $8 | $88 | $8 | $121 | $6 | $10 | $60 | $4 | $24 | $4 | $36 |
Given,
Pe = 30-Qe
Pd = 16-Qd
MCe = $8
MCd = $4
Now, TRe = Price at evening*quantity at evening
= (30-Qe)*Qe
= 30Qe - 
Calculation of MRe using derivative
MRe = 30- 2Qe
Now, calculating average price by equalizing MR=MC
MRe = MCe
30-2Qe = 8
-2Qe = (-22)
Qe = (-22/-2) = 11
To find out Pe by substituting value of Qe
Pe = 30-Qe
= 30-11 = 19
Now come to,
TRe = Pe*Qe
= 19*11 =$209
TCe = MCe*Qe
= 8*11 =$ 88
Now, profit from the restaurant at evening ( πe) = TRe - TCe
= $209 - $88 = $121
Price at day time (Pd) = 16-Qd
As the evening hour calculated above, the day time also required to calculate basing to the above steps.
TRd = (16-Qd)*Qd
= 16Qd - 
MRd = 16 - 2Qd (By derivative)
MCd = $4
By equals MRd = MCd
16 - 2Qd = 4
- 2Qd = -12
Qd = (-12/-2) =6
Now, Pd = 16-Qd
= 16-6 = 10
TRd = Pd*Qd
= 10*6 = $60
TCd = 4*Qd
= 4*6 = $24
πd = TRd - TCd
= $60 - $24
= $36
Answers to the given questions
a. Optimal prices for the dessert during the evening hours
(Pe) = $19
b. Optimal quantity for the dessert during the evening hours
(Qe) = $11
c. Total cost of producing the optimal quantity for the dessert during evening hours
(TRe) = $209 and (TCe) = $88
d. Maximum profit for the dessert during evening hours
(πe) = $121
e. Optimal prices for the dessert during day time hours
(Pd) = $10
f. Optimal quantity for the dessert during day time hours
(Qd) = $6
g. Total cost for optimal quantity for the dessert during day time hours
(TRd) = $60 and (TCd) = $24
h. Maximum profit for the dessert during day time hours
(πd) = $36
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