(a) Suppose the firm sets up one stall in the middle of the beach. (i) Derive the maximal distance from the shop x₁ where a sunbather still buys the crepe, and the maximal price p₁ where the stall can serve the whole beach. (ii) Set up stall profits and derive the optimal price the stall will charge. How many sunbathers will the stall serve? (b) Suppose the firm sets up two stalls. (i) Where should the firm optimally place them if she wants to serve the whole beach? Why? (ii) Derive the maximal price p2 where the company can serve the whole beach with two stalls. Calculate the optimal price the stalls will charge. How many customers will the stalls serve? (c) Suppose the firm wants to serve the whole beach with n ≥ 1 stalls. (i) Derive the maximal price pn a firm with n stalls can charge to serve the whole beach. (ii) Explain why the assumption that the firm will supply the whole beach is reasonable given your results in (b). (iii) Given our reasoning so far how many crepes will the firm sell? Write down firm profits opening n stalls and charging Pn. Derive the optimal amount of stalls n* (rounded to an integer) the firm should open. (iv) Given our discussion in the lecture/class: Do you expect a social planner to choose more or less stalls than n*. Why?
(a) Suppose the firm sets up one stall in the middle of the beach. (i) Derive the maximal distance from the shop x₁ where a sunbather still buys the crepe, and the maximal price p₁ where the stall can serve the whole beach. (ii) Set up stall profits and derive the optimal price the stall will charge. How many sunbathers will the stall serve? (b) Suppose the firm sets up two stalls. (i) Where should the firm optimally place them if she wants to serve the whole beach? Why? (ii) Derive the maximal price p2 where the company can serve the whole beach with two stalls. Calculate the optimal price the stalls will charge. How many customers will the stalls serve? (c) Suppose the firm wants to serve the whole beach with n ≥ 1 stalls. (i) Derive the maximal price pn a firm with n stalls can charge to serve the whole beach. (ii) Explain why the assumption that the firm will supply the whole beach is reasonable given your results in (b). (iii) Given our reasoning so far how many crepes will the firm sell? Write down firm profits opening n stalls and charging Pn. Derive the optimal amount of stalls n* (rounded to an integer) the firm should open. (iv) Given our discussion in the lecture/class: Do you expect a social planner to choose more or less stalls than n*. Why?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. Answer all parts (a)-(c) of this question
A crepe producer considers opening n ≥ 1 stalls on a beach of 1 mile length. The firm
estimates that on an average day there are 1000 sunbathers evenly spread along the beach and
that each sunbather will buy one crepe per day, provided that the price plus disutility cost does
not exceed £5. Each sunbather incurs a disutility cost of getting up to buy a crepe and returning
to their beach spot of £5 per mile (i.e., t is £5). Each crepe costs £0.5 to make and the company
incurs a £40 fixed overhead cost per day to operate the stall.
(a)
Suppose the firm sets up one stall in the middle of the beach.
(i) Derive the maximal distance from the shop x₁ where a sunbather still buys the crepe,
and the maximal price p₁ where the stall can serve the whole beach.
(ii) Set up stall profits and derive the optimal price the stall will charge. How many
sunbathers will the stall serve?
(b)
Suppose the firm sets up two stalls.
(i) Where should the firm optimally place them if she wants to serve the whole beach?
Why?
(ii) Derive the maximal price p2 where the company can serve the whole beach with
two stalls. Calculate the optimal price the stalls will charge. How many customers
will the stalls serve?
(c)
Suppose the firm wants to serve the whole beach with n ≥ 1 stalls.
(i) Derive the maximal price pn, a firm with n stalls can charge to serve the whole beach.
(ii) Explain why the assumption that the firm will supply the whole beach is reasonable
given your results in (b).
(iii) Given our reasoning so far how many crepes will the firm sell? Write down firm
profits opening n stalls and charging Pn. Derive the optimal amount of stalls n*
(rounded to an integer) the firm should open.
(iv) Given our discussion in the lecture/class: Do you expect a social planner to choose
more or less stalls than n*. Why?
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