<p><span class="math-container">$$(1)   P_i = \beta_0 +\beta_1Q_i + \beta_2X_i +u_i$$</span>
<span class="math-container">$$(2)   Q_i = \delta_0 +\delta_1P_i + \delta_2W_i +v_i$$</span></p>
<p>Plugging (2) into (1) and simplifying yields,</p>
<p><span class="math-container">$$P_i = \beta_0 +\beta_1(\delta_0 +\delta_1P_i + \delta_2W_i +v_i) + \beta_2X_i +u_i$$</span></p>
<p><span class="math-container">$$P_i = \beta_0 +\beta_1\delta_0 +\beta_1\delta_1P_i + \beta_1\delta_2W_i  + \beta_2X_i +\beta_1v_i+u_i$$</span></p>
<p><span class="math-container">$$P_i -\beta_1\delta_1P_i= \beta_0 +\beta_1\delta_0  + \beta_1\delta_2W_i  + \beta_2X_i +\beta_1v_i+u_i$$</span>
<span class="math-container">$$P_i(1 -\beta_1\delta_1)= \beta_0 +\beta_1\delta_0  + \beta_1\delta_2W_i  + \beta_2X_i +\beta_1v_i+u_i$$</span>
<span class="math-container">$$P_i= \frac{\beta_0 +\beta_1\delta_0  + \beta_1\delta_2W_i  + \beta_2X_i +\beta_1v_i+u_i}{(1 -\beta_1\delta_1)}$$</span></p>
<p>Define new notation <span class="math-container">$\gamma$</span> to be superconstants, that is, <span class="math-container">$\gamma_0=\frac{\beta_0 +\beta_1\delta_0 }{(1 -\beta_1\delta_1)}$</span>, <span class="math-container">$\gamma_1 =\frac{ \beta_1\delta_2}{(1 -\beta_1\delta_1)}$</span>, <span class="math-container">$\gamma_2 =\frac{ \beta_2}{(1 -\beta_1\delta_1)}$</span>. Also define <span class="math-container">$\varepsilon_i=\frac{\beta_1v_i+u_i}{(1 -\beta_1\delta_1)}$</span>. We have,</p>
<p><span class="math-container">$$P_i =\gamma_0 +\gamma_1 W_i +\gamma_2 X_i +\varepsilon_i$$</span></p>
<p>So the error term in this equation is <span class="math-container">$\varepsilon_i=\frac{\beta_1v_i+u_i}{(1 -\beta_1\delta_1)}$</span>, 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.</p>

CC BY-SA 4.0
2022-11-08T09:54:58.023Z
2022-11-08T09:54:58.023Z
Michael Gmeiner
<p>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?</p>

<econometrics><supply-and-demand><regression><causality><variance>
Simultaneous Causality and Variance within a supply and demand model
baronwellingtoniii
# Simultaneous Causality and Variance within a supply and demand model

(Asked by: username_0 on 2022-11-08)

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?

**Tags:** econometrics, supply-and-demand, regression, causality, variance

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## 1 answers

### Answer by username_1 on 2022-11-08. Score: 3

$$(1) P\_i = \beta\_0 +\beta\_1Q\_i + \beta\_2X\_i +u\_i$$
$$(2) Q\_i = \delta\_0 +\delta\_1P\_i + \delta\_2W\_i +v\_i$$

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.


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