Question

The demand curve for Blivets (shown below) has the formula

 P =h(q) = 12-4

Where q is measured in thousand of Blivets and p is measured in dollars. We

Get Total revenue(measured in thousands of dollars) from this curve by the recipe

TR(q)=q*h(q). The total cost (measured in thousands of dollars ) of manufacturing Blivets

Is given by the formula TC(q)= q+1. Due to the limited supply of raw material, we

Cannot manufacture more than 3500 Blivets ( q=3.5)

 

A) write the formula for TR(q) and TR'(q) ( simplify your answer as much as possible)

b) find the value of q in the interval between q=0 and q=3.5 thousand blivets at which profit is greatest

c) find the formula for Total profit p(q) of selling q thousand blivets

d) b) find the value of q in the interval between q=0 and q=3.5 thousand blivets at which TR(q) reaches its largest value

Answer 1

Given;

$p=12-4q$

$TC=q+1$

(a)

$TR=p\times q$

$=(12-4q)q$

$=12q-4q^2$

$MR=TR'$

$=\frac{d}{dq}(TR)$

$=\frac{d}{dq}(12q-4q^2)$

$=12-8q$

(b)

$profit (P)=TR-TC$

$=12q-4q^2-(q+1)$

$=-4q^2+11q-1$

Now,

$\frac{dP}{dq}=0$ [first order condition for maximization of P)

$\frac{dP}{dq}(-4q^2+11q-1)=0$

$-8q+11=0$

$8q=11$

$q=\frac{11}{8}=1.375$

Thus, at the level of q=1.375(in thousand bilvets) maximum profit is realized

(c)

formula for total profit is

$P=TR-TC$

$=p\times q-TC$

$=(12-4q)q-(q+1)$

$=12q-4q^2-q-1$

$=11q-4q^2-1$

$=-4q^2+11q-1$

(d)

$TR=price\times quantity$

$=p\times(12-4q)$

$=12q-4q^2$

$\frac{dTR}{dq}=12-8q=0$

$q=\frac{12}{8}$

=1.5

Thus, at the level of q=1.5(in thousand bilvets) the TR reaches its maximum value.