Q8. Consolidated Sugar Company sells granulated sugar to both retail grocery cha
ID: 2441035 • Letter: Q
Question
Q8. Consolidated Sugar Company sells granulated sugar to both retail grocery chains and commercial users (e.g., bakeries, candy makers, etc.). The demand function for each of these markets is:
Retail grocery chains: P1 = 90 - 4q1
Commercial users: P2 = 50 - 2q2
where P1 and P2 are the prices charged and q1 and q2 are the quantities sold in the respective markets. Consolidated's total cost function (which includes a "normal" return to the owners) for granulated sugar is
TC = 25 + 10(q1 + q2 )
(a) Determine Consolidated's total profit function.
(b) Assuming that Consolidated is effectively able to charge different prices in the two markets, what are the profit-maximizing price and output levels for the product in the two markets? What is Consolidated's total profit under this condition?
(c) Assuming that Consolidated is required to charge the same price in each market, what are the profit-maximizing price and output levels? What is Consolidated's total profit under this condition?
Explanation / Answer
Solution: A
Total Revenue from retail grocery chain (TR1) = P1*q1
P1 = 90-4q1
TR1=P1*q1= (90-4q1)*q1=90q1-4q12
Total Revenue from commercial users (TR2)= P2*q2
P2 = 50-2q2
TR2=P2*q2= (50-2q2)*q2=50q2-2q22
Total revenue from both markets TR = TR1+ TR2
=90q1-4q12 +50q2-2q22
TC= 25+10(q1+q2)
Consolidated Profit () = TR-TC = 90q1-4q12 +50q2-2q22 -25-10(q1+q2)
= 80q1+40q2-4q12-2q22-25
Solution: B
For profit maximization in market 1, differentiate the profit equation with respect to q1 and equate to zero.
d/dq1=80+0-8q1-0-0 =0
Solving we get q1=10
P1=90-4*10=50
For profit maximization in market 2, differentiate the profit equation with respect to q2 and equate to zero.
d/dq2=0+40-0+4q2-0=0
Solving we get q2=10
P2=50-2*10=30
Total Profits = 80*10+40*10-4*10^2-2*10^2-25=575
Solution: C
In this case we are not able to charge different prices in two markets, first add the demand function of two markets.
(P is same in both markets; we use P instead of P1 and P2)
q1= (90-P)/4 = 22.5 – 0.25P
q2= (50-P)/2=25-0.50P
Total demand can be obtained by adding q1 and q2, we get
Q=q1+q2=47.5-0.75P
Or P= (47.5-Q)/0.75=63.33 -1.33Q
TR=P*Q=63.33Q-1.33Q2
TC=25+10(Q1+Q2)=25+10Q
Profits = TR-TC=63.33Q-1.33Q2 –25-10Q = -25 +53.33Q-1.33Q2
For profit maximization differentiate with respect to Q and equate to zero
We get
0+53.33-1.33*2Q=0
Solving we get Q=20
P=63.33-1.33*20=36.66
q1=22.5-0.25*36.66=13.33
q2=25-0.50*36.66 = 6.67
Total Profits = -25+53.33*20-1.33*20*20=508.33
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.