Given two single-variable regression equations, one for Demand and one for Supply, both with error terms and a constant. How does one solve for equilibrium quantity, price? Obviously you can set supply and demand equal to each other and rearrange but where does the variance of the error terms come in? Is there a property of simultaneous causality that causes these variance terms to be a part of the final equilibrium equations?
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$$(1) P_i = \beta_0 +\beta_1Q_i + \beta_2X_i +u_i$$ $$(2) Q_i = \delta_0 +\delta_1P_i + \delta_2W_i +v_i$$
Plugging (2) into (1) and simplifying yields,
$$P_i = \beta_0 +\beta_1(\delta_0 +\delta_1P_i + \delta_2W_i +v_i) + \beta_2X_i +u_i$$
$$P_i = \beta_0 +\beta_1\delta_0 +\beta_1\delta_1P_i + \beta_1\delta_2W_i + \beta_2X_i +\beta_1v_i+u_i$$
$$P_i -\beta_1\delta_1P_i= \beta_0 +\beta_1\delta_0 + \beta_1\delta_2W_i + \beta_2X_i +\beta_1v_i+u_i$$ $$P_i(1 -\beta_1\delta_1)= \beta_0 +\beta_1\delta_0 + \beta_1\delta_2W_i + \beta_2X_i +\beta_1v_i+u_i$$ $$P_i= \frac{\beta_0 +\beta_1\delta_0 + \beta_1\delta_2W_i + \beta_2X_i +\beta_1v_i+u_i}{(1 -\beta_1\delta_1)}$$
Define new notation $\gamma$ to be superconstants, that is, $\gamma_0=\frac{\beta_0 +\beta_1\delta_0 }{(1 -\beta_1\delta_1)}$, $\gamma_1 =\frac{ \beta_1\delta_2}{(1 -\beta_1\delta_1)}$, $\gamma_2 =\frac{ \beta_2}{(1 -\beta_1\delta_1)}$. Also define $\varepsilon_i=\frac{\beta_1v_i+u_i}{(1 -\beta_1\delta_1)}$. We have,
$$P_i =\gamma_0 +\gamma_1 W_i +\gamma_2 X_i +\varepsilon_i$$
So the error term in this equation is $\varepsilon_i=\frac{\beta_1v_i+u_i}{(1 -\beta_1\delta_1)}$, this is a combination of the two error terms in the structural equations. I'm not really sure what you're asking, so feel free to follow up and I can edit this.
