A producer faces the demand curve (price in dollars

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A producer faces the demand curve

p(q)=90-0.1q

(price in dollars, quantity in units). Its total cost of producing

q

units is

C(q)=20q+2{,}000+kq

, where

k\ge0

represents an environmental fee imposed per unit produced. Assume

0\le q\le900

to keep the price non-negative. (a) Express the profit

P(q,k)

as a function of

q

and the parameter

k

. (b) Derive an explicit formula

q^*(k)

for the production level that maximizes profit in terms of

k

. State the domain of

k

for which this formula is valid. (c) Compute

q^*(0)

and

q^*(15)

, and find the corresponding maximum profit for each value of

k

. (d) Find

\dfrac{dq^*}{dk}

and evaluate this derivative at

k=0

. Interpret its sign and magnitude in the context of how the optimal production level reacts to a small increase in the environmental fee.