Econometrics using aggregate market data

The problem with estimating structural models

Structural supply and demand functions

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?

$$cov(P, \epsilon_1)= E[(P-E[P])(\epsilon_1-E[\epsilon_1])]$$ $$cov(P, \epsilon_1)= E[(P-E[P])(\epsilon_1-E[\epsilon_1])]$$

Estimating reduced-form models

Structural supply and demand functions

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.

Reduced form equations

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.

More on reduced form

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?

Using 2-stage OLS

Using IVs

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 $$P$$.

Power of IVs

need variation. if factor v important, little price movelemtn. may be hard to esimate that fact.

Locality of elasticity

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.

Examples of IVs

Product characteristics

We can use the characteristics of products, and other products.

Price increases in other geographies

Assume only correlated with marginal cost in other geography.

Using marginal cost data

Monopoly

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

The Lerner index is:

$$\dfrac{p-MC}{p}$$

Cournot model

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:

$$\dfrac{p-MC}{p}=-\dfrac{\mu}{\epsilon }$$

Measuring market concentration

Herfindahl–Hirschman Index (HHI)

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:

$$H*=\dfrac{H-\dfrac{1}{n}}{1-\dfrac{1}{n}}$$

Concentration ratio

Proportion of output from given firms.

For example $$CR_5$$ is the proportion of output from the $$5$$ largest producers.

Lerner index

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

$$L=\dfrac{P-MC}{P}$$

From $$0$$ to $$1$$.