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.

In the monopoly model we have:

\(\pi = pq-c\)

\(p[1+\dfrac{q}{p}\dfrac{\delta p}{\delta q}]=MC\)

The price elasticity of demand is: \(\epsilon = \dfrac{p}{q}\dfrac{\delta q}{\delta p}\)

\(p[1+\dfrac{1}{\epsilon }]=MC\)

\(\dfrac{1}{\epsilon }=\dfrac{MC}{p}-1\)

\(\dfrac{p-MC}{p}=-\dfrac{1}{\epsilon }\)

The Lerner index is:

\(\dfrac{p-MC}{p}\)

With competition, the elasticity of demand refers to the whole market, not just a single producer. Instead we have:

\(\epsilon = \dfrac{p}{Q}\dfrac{\delta Q}{\delta p}\)

\(Q=\sum_j q_j\)

We now get:

\(p[1+\dfrac{q}{Q}\dfrac{\delta Q}{\delta q}\dfrac{Q}{p}\dfrac{\delta p}{\delta Q}]=MC\)

\(p[1+\dfrac{\mu }{\epsilon }]=MC\)

Using the firm’s size elasticity: \(\mu = \dfrac{q}{Q}\dfrac{\delta Q}{\delta q}\)

With monopoly this is:

\(\mu = 1\)

In this model this is:

\(\dfrac{p-MC}{p}=-\dfrac{\mu}{\epsilon }\)

The index is the sum of each firm’s market share squared.

For a monopolist this is \(1\), for a completely competitive market it is \(0\), or \(\dfrac{1}{n}\).

\(H=\sum_{i=1}^ns^2_i\)

To normalise this between \(0\) and \(1\) we can use:

\(H*=\dfrac{H-\dfrac{1}{n}}{1-\dfrac{1}{n}}\)

Proportion of output from given firms.

For example \(CR_5\) is the proportion of output from the \(5\) largest producers.

Marginal profit. If high it suggests existing power prevents it from raising output.

\(L=\dfrac{P-MC}{P}\)

From \(0\) to \(1\).