1. Two companies (A and B) are duopolists that produce identical products. Demand for the products is given by the following demand function: P = 10,000 - QA-QB where QA and QB are the quantities sold by the respective firms and P is the selling price. Total cost functions for the two companies are: TCA = 500,000 + 200QA + .5QA² TC8 = 200,000 + 400QB +QB² Assume that the two firms act independently as in the Cournot model (that is, each firm assumes that the other firm's output will not change). Determine the long-run equilibrium output and selling price for each firm.
1. Two companies (A and B) are duopolists that produce identical products. Demand for the products is given by the following demand function: P = 10,000 - QA-QB where QA and QB are the quantities sold by the respective firms and P is the selling price. Total cost functions for the two companies are: TCA = 500,000 + 200QA + .5QA² TC8 = 200,000 + 400QB +QB² Assume that the two firms act independently as in the Cournot model (that is, each firm assumes that the other firm's output will not change). Determine the long-run equilibrium output and selling price for each firm.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:1. Two companies (A and B) are duopolists that produce identical
products. Demand for the products is given by the following demand
function:
P = 10,000 - QA-QB
where QA and QB are the quantities sold by the respective firms and
P is the selling price.
Total cost functions for the two companies are:
TCA = 500,000 + 200QA + .5QA²
TC8 = 200,000 + 400QB +QB²
Assume that the two firms act independently as in the Cournot
model (that is, each firm assumes that the other firm's output will not
change). Determine the long-run equilibrium output and selling
price for each firm.
PQA - TCA
%3!
= (10,000 - QA - QB)QA – (500,000 + 200QA +
= - 500,000 + 9800Q3 - QAQ3 – 1.5Qa?
9800 – Qa – 3Qa
%3D
= PQB - TC3
= (10,000 - QA - Q3)QB – (200,000 + 400QB +
= -200,000 + 9600Q3 – 2Qg² – QAQ3
= 9600 – 4Q8 - QA
De
Setting dna/aQA = 0 and atg/aQg = 0 and solving the simultaneous equations yields:
2QA + Q3 = 9800
Qa + 2Q3 = 9600
QA* = 2,691 units
Qa* = 1,727 units
and P* = $5,582/unit
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