Starting with the following model:
Q_d1-Q_s1=0
Q_d1=a_0+a_1 P_1+a_2 P_2
Q_s1=b_0+b_1 P_1+b_2 P_2 (1)
Q_d2-Q_s2=0
Q_d2=α_0+α_1 P_1+α_2 P_2
Q_s2=β_0+β_1 P_1+β_2 P_2
We worked out the step-by-step solution of model 1, we found the following results:
P_1^*=(c_2 γ_0-c_0 γ_2)/(c_1 γ_2-c_2 γ_1 ) ---------- (1.1)
P_2^*=(c_0 γ_1-c_1 γ_0)/(c_1 γ_2-c_2 γ_1 ) ---------- (1.2)
Work out the step-by-step solution to find Q_i^*, where (i=1,2).
- i know to start this problem I would have to plug in P_1 and P_2 into Q_d1 or Q_s1 but I don't know how to solve it from there
To find the equilibrium quantities Q1* and Q2* for the given model, you can substitute the obtained equilibrium prices P1* and P2* back into the respective demand and supply equations.
Let's start with Q1*:
1. Substitute P1* and P2* into Qd1 equation:
Q_d1 = a0 + a1 * P1* + a2 * P2*
= a0 + a1 * [ (c2 * γ0 - c0 * γ2) / (c1 * γ2 - c2 * γ1) ] + a2 * [ (c0 * γ1 - c1 * γ0) / (c1 * γ2 - c2 * γ1) ]
2. Simplify the expression by multiplying through denominators:
Q_d1 = [ a0 * (c1 * γ2 - c2 * γ1) + a1 * (c2 * γ0 - c0 * γ2) + a2 * (c0 * γ1 - c1 * γ0) ] / (c1 * γ2 - c2 * γ1)
This gives you the equilibrium quantity Q1*.
Similarly, you can find Q2*:
1. Substitute P1* and P2* into Qd2 equation:
Q_d2 = α0 + α1 * P1* + α2 * P2*
= α0 + α1 * [ (c2 * γ0 - c0 * γ2) / (c1 * γ2 - c2 * γ1) ] + α2 * [ (c0 * γ1 - c1 * γ0) / (c1 * γ2 - c2 * γ1) ]
2. Simplify the expression by multiplying through denominators:
Q_d2 = [ α0 * (c1 * γ2 - c2 * γ1) + α1 * (c2 * γ0 - c0 * γ2) + α2 * (c0 * γ1 - c1 * γ0) ] / (c1 * γ2 - c2 * γ1)
This gives you the equilibrium quantity Q2*.
Please note that the provided equations and variables in your question are specific to your model and may require specific values for the coefficients (a0, a1, a2, α0, α1, α2, b0, b1, b2, β0, β1, β2) and constant terms (γ0, γ1, γ2, c0, c1, c2) in order to obtain numeric values for the equilibrium quantities.