Question

Three firms sell related products in a market. Their supply functions are:

qˢ₁ =P₁
​qˢ₂ =P₂
​qˢ₃ =P₃
​
The firms face the following demands:
qᵈ₁ =−101−2P₁ +4P₂ +5P₃

qᵈ₂ =−1+4P₁ −5P₂ +3P₃

qᵈ₃ =−45+5P₁​ +5P₂​ −4P₃

Given the above supply and demand functions, what price must each firm charge to equate supply and demand?
P₁ =
P₂ =
P₃ =
​

Answer 1

To determine the equilibrium prices for each firm, set the quantity supplied equal to quantity demanded for each firm, forming a system of equations. Solve this system algebraically to find the prices P1, P2, and P3 at which the market is in equilibrium.

Step-by-step explanation:

To find the price that each firm must charge to equate supply and demand, we first acknowledge that when a market is in equilibrium, the quantity supplied (Qs) is equal to quantity demanded (Qd). Given that the supply and demand functions are set for each of the three firms, we form three equations by equating supply functions to their respective demand functions.

For Firm 1:
Qd1 = Qs1

-101 - 2P1 + 4P2 + 5P3 = P1

For Firm 2:
Qd2 = Qs2

-1 + 4P1 - 5P2 + 3P3 = P2

For Firm 3:
Qd3 = Qs3

-45 + 5P1 + 5P2 - 4P3 = P3

These equations form a system of three linear equations with three unknowns (P1, P2, and P3). To solve the system, we apply algebraic methods such as substitution or elimination to find the values of P1, P2, and P3 that satisfy all three equations simultaneously, thus determining the prices at which supply equals demand for each of the three products.