Supply:
\(Q_s=\alpha_1 + \beta_1P+\gamma_1I + \epsilon_1\)
Demand:
\(Q_d=\alpha_2 + \beta_2P+\gamma_2I + \epsilon_2\)
Can’t estimate because the equations are simulataneous.
To estimate, \(cov(P, \epsilon_1)\) needs to be \(0\), but what is it?
\(cov(P, \epsilon_1)= E[(P-E[P])(\epsilon_1-E[\epsilon_1])]\) \(cov(P, \epsilon_1)= E[(P-E[P])(\epsilon_1-E[\epsilon_1])]\)
Supply:
\(Q_s=\alpha_1 + \beta_1P+\gamma_1I + \epsilon_1\)
Demand:
\(Q_d=\alpha_2 + \beta_2P+\gamma_2I + \epsilon_2\)
Can’t estimate because the equations are simulataneous.
Where supply is demand.
\(Q_s=Q_d\)
\(\alpha_1 + \beta_1P+\gamma_1I + \epsilon_1=\alpha_2 + \beta_2P+\gamma_2I + \epsilon_2\)
\((\alpha_1 -\alpha_2) + (\beta_1- \beta_2)P+(\gamma_1-\gamma_2) I + (\epsilon_1-\epsilon_2)=0\)
\((\beta_1- \beta_2)P=-(\alpha_1 -\alpha_2) - (\gamma_1-\gamma_2) I - (\epsilon_1-\epsilon_2)\)
\(P=-\dfrac{\alpha_1 -\alpha_2}{\beta_1- \beta_2} - \dfrac{\gamma_1-\gamma_2}{\beta_1- \beta_2} I - \dfrac{\epsilon_1-\epsilon_2}{\beta_1- \beta_2}\)
We can construct something similar for \(Q\). The results are reduced-form parameters with reduced-form errors.
reduced form for perfect competition, and imperfect
issue is: supply function only defined for perfect competition.
how do you get equilibrium otherwise? what are the other reduced form equations? strucutral?
Let’s say the demand function is:
\(Q_d=\alpha+\beta P + \epsilon\)
How can we estimate this?
OLS will give biased results if \(P\) is correlated with \(\epsilon\).
We can estimate if we have an instrumental variable for \(P\).
need variation. if factor v important, little price movelemtn. may be hard to esimate that fact.
we measure elsaticities as they are. there may be existing competition barriers which cause current substitution. without current monopolies, there may be more unique markets.
eg monopolist has product with sbustitutve, but it is only substitute because the monopolist has kept the price so high.
We can use the characteristics of products, and other products.
Assume only correlated with marginal cost in other geography.