Part 4: Firm Heterogeneity Section 14

Variable Markups

We now study another dimension of firm heterogeneity: variable markups. Firms of different sizes can differ in the markups they charge. This is not the case in the monopolistic competition framework above because of two reasons. First, the demand faced by each producer had constant elasticity, and so the markup was constant. Recall that optimal markups are \(\mu=\frac{1}{\left(1-\frac{1}{\varepsilon}\right)}\). Second, there were no strategic interactions between firms. The number of firms (or products) was assumed to be large enough to make no firm have an effect on the aggregate. If firms are large enough to internalize their effect on others larger firms gain market power and can thus charge larger markups.

The objective in this section is to go over the basics of these two approaches. Each has extensive applications in the economics of misallocation, monetary policy, and international trade.

Variable Elasticity of Demand and Variable Markups

We start extending the framework of Dixit and Stiglitz (1977) to allow for a demand system with variable elasticity of demand. In this we follow Kimball (1995).

Note

Other formulations are reviewed by Costas Arkolakis and Monica Morlacco, see their notes on variable elasticity of demand.

In order to have variable elasticity of demand we need a new formulation of the aggregator. There is an aggregate (or composite) good \(Y\) that is produced by a competitive producer. Production combines a continuum of differentiated goods \(\left\lbrace y_{i}\right\rbrace\). The aggregation technology is implicitly defined by

\[ 1=\int\Upsilon\left(\frac{y_{i}}{Y}\right)di. \]

The properties of the demand for the differentiated goods \(y_{i}\) depend on the function \(\Upsilon\). The function \(\Upsilon\) is either strictly increasing and strictly concave (case where the goods are gross substitutes) or strictly decreasing and strictly convex (case where the goods are gross complements). Note that the function is homogeneous of degree 1 by construction.

This function includes the constant-elasticity-of-substitution (CES) aggregator used above when we let \(\Upsilon\left(x\right)=x^{\frac{\varepsilon-1}{\varepsilon}}\). Other functional forms allow for variable-elasticity of demand. To see this more clearly we need to solve the problem of the aggregator.

The problem of the aggregator is to minimize its cost subject to a minimum level of output \(\overline{Y}\):

\[ \min_{\left\lbrace y_{i}\right\rbrace }\,\int p_{i}y_{i}di\qquad\text{s.t. }1=\int\Upsilon\left(\frac{y_{i}}{\overline{Y}}\right)di \]

The first order conditions are (for each variety)

\[ p_{i}=\Upsilon'\left(\frac{y_{i}}{Y}\right)\frac{\lambda}{Y}, \]

where \(\lambda\) is the Lagrange multiplier. We need to solve for the multiplier. To do this, multiply both sides of the first order condition by \(y_{i}\) and sum across varieties to obtain

\[ \lambda=\frac{\int p_{i}y_{i}di}{\int\Upsilon'\left(\frac{y_{i}}{Y}\right)\frac{y_{i}}{Y}di}. \]

Then, define the price of the aggregate good, \(P\), so that it satisfies \(P\cdot Y=\int p_{i}\cdot y_{i}di\). Finally, replace \(\lambda\) into the first order condition and divide both sides by \(P\) to obtain the inverse demand curve for variety \(i\) expressed in terms of relative prices and relative output:

\[ \frac{p_{i}}{P}=\frac{\Upsilon'\left(\frac{y_{i}}{Y}\right)}{\int\Upsilon'\left(\frac{y_{i}}{Y}\right)\frac{y_{i}}{Y}di}. \]

We can also define implicitly the aggregate price. For this it is convenient to denote \(D=\int\Upsilon'\left(\frac{y_{i}}{Y}\right)\frac{y_{i}}{Y}di\) and then we define a set of two equations that jointly determine \(D\) and \(P\) using the definition of the price index and the aggregator:

\[ P=\int p_{i}\cdot\left(\Upsilon'\right)^{-1}\left(D\frac{p_{i}}{P}\right)di\qquad\text{and}\qquad1=\int\Upsilon\left(\left(\Upsilon'\right)^{-1}\left(D\frac{p_{i}}{P}\right)\right)di, \]

where \(\left(\Upsilon'\right)^{-1}\left(x\right)\) is the inverse function of the first derivative of \(\Upsilon\).

While this looks complicated the key is in realizing that the behavior of the monopolists producing each variety depends only on the elasticity of demand \(\varepsilon_{i}=-\left(\frac{\partial\log p_{i}}{\partial\log y_{i}}\right)^{-1}\). As before, the key is that the actions of a single producer cannot affect the value of aggregate output or prices. The elasticity is

\[\begin{align*} \varepsilon_{i} & =-\left(\frac{y_{i}}{p_{i}}\frac{\partial p_{i}}{\partial y_{i}}\right)^{-1}\\ & =-\left(\frac{y_{i}}{p_{i}}\frac{\partial\Upsilon'\left(\frac{y_{i}}{Y}\right)\frac{P}{D}}{\partial y_{i}}\right)^{-1}\\ & =-\left(\frac{y_{i}}{p_{i}}\Upsilon"\left(\frac{y_{i}}{Y}\right)\frac{P}{D}\frac{1}{Y}\right)^{-1}\\ & =-\left(\frac{y_{i}}{Y}\right)^{-1}\frac{\Upsilon'\left(\frac{y_{i}}{Y}\right)}{\Upsilon"\left(\frac{y_{i}}{Y}\right)} \end{align*}\]

So, knowing the properties of the first two derivatives of \(\Upsilon\) is enough to know the behavior of firms.

When goods are substitutes we have \(\Upsilon'\left(x\right)>0\) and \(\Upsilon"\left(x\right)<0\). When they are complements we have \(\Upsilon'\left(x\right)<0\) and \(\Upsilon"\left(x\right)>0\). So we know that the elasticity is positive (recall we fixed the sign, the demand curve is downward sloping). The key question is whether it is increasing or decreasing in the variety's (relative) output.

In practice, we work with aggregators that imply a decreasing elasticity of demand for larger firms, capturing the fact that larger firms have more market power and charge a higher markup. For example, the aggregator proposed by Klenow and Willis (2016) implies that

\[ \varepsilon_{i}=\varepsilon\cdot\left(\frac{y_{i}}{Y}\right)^{-\frac{\theta}{\varepsilon}} \]

with \(\varepsilon,\theta>0\) parameters, so that as the relative output increases the elasticity decreases.

In equilibrium, more productive firms have lower marginal costs and hence can charge lower prices. This leads them to operate in more inelastic portion of their demand curves and charge higher markups. So, higher markups coexist with lower prices.

Oligopolistic Competition and Variable Markups

Finally, we consider the case of oligopolistic competition, where finitely many firms compete à la Cournot or à la Bertrand in a market. In this setup, it is the strategic interactions between firms that give rise to market power for the more productive firms who end up being larger in equilibrium. These firms charge higher markups. This setup is developed in Atkeson and Burstein (2008) and used in a variety of papers in international trade and misallocation.

Consider a market with \(N\) producers of a differentiated goods competing à la Cournot (so monopolists choose quantities taking quantities of other monopolists as given). Firms are heterogeneous in their marginal cost, \(c_{i}\). The demand for the goods comes from a competitive final good producer that aggregates individual goods using a CES technology

\[ Y=\left(\sum_{i=1}^{N}y_{i}^{\frac{\varepsilon-1}{\varepsilon}}\right)^{\frac{\varepsilon}{\varepsilon-1}}. \]

The final good producer is a price taker and the demand for final goods has constant elasticity of demand \(\eta\), so that \(Y=P^{-\eta}\). We assume that \(\eta<\varepsilon\). This means that the varieties are more substitutable between them than the total output is for other (potential) goods that can be in the economy. This is the case when we have a model with many markets, being aggregated with an (outer) CES aggregator with elasticity \(\eta\).

Just as before, the cost minimization problem of the final good producer implies a demand for variety \(i\) of

\[ \frac{p_{i}}{P}=\left(\frac{y_{i}}{Y}\right)^{-\frac{1}{\varepsilon}}, \]

where

\[ PY=\sum_{i=1}^{N}p_{i}y_{i}\qquad\text{and}\qquad P=\left(\sum_{i=1}^{N}p_{i}^{1-\varepsilon}\right)^{\frac{1}{1-\varepsilon}}. \]

What changes is the behavior of the monopolists. The key is that the elasticity of demand they face depends on how their output affects aggregates. We now have

\[\begin{align*} \varepsilon_{i}^{-1} & =-\frac{\partial\log p_{i}}{\partial\log y_{i}}=-y_{i}\frac{\partial\log\left(\frac{y_{i}}{Y}\right)^{\frac{-1}{\varepsilon}}P}{\partial y_{i}}=-y_{i}\frac{\partial\log\left(\frac{y_{i}}{Y}\right)^{-\frac{1}{\varepsilon}}Y^{-\frac{1}{\eta}}}{\partial y_{i}}=\frac{\partial\log y_{i}^{\frac{-1}{\varepsilon}}Y^{\frac{1}{\varepsilon}-\frac{1}{\eta}}}{\partial y_{i}}, \end{align*}\]

so that we can break the elasticity into two terms, the effect of a change in the variety's elasticity and the effect on aggregate production,

\[ \varepsilon_{i}^{-1}=\underbrace{\frac{1}{\varepsilon}}_{\text{Elast. of Subs.}}+\left(\frac{1}{\eta}-\frac{1}{\varepsilon}\right)\underbrace{\frac{y_{i}}{Y}\frac{\partial Y}{\partial y_{i}}}_{\text{Elast. of Agg. Output}}. \]

The elasticity of the aggregate output is with respect to variety \(i\) is

\[ \frac{y_{i}}{Y}\frac{\partial Y}{\partial y_{i}}=\frac{y_{i}}{Y}\frac{\partial\left(\sum_{i=1}^{N}y_{i}^{\frac{\varepsilon-1}{\varepsilon}}\right)^{\frac{\varepsilon}{\varepsilon-1}}}{\partial y_{i}}=\frac{y_{i}}{Y}\left(\sum_{i=1}^{N}y_{i}^{\frac{\varepsilon-1}{\varepsilon}}\right)^{\frac{\varepsilon}{\varepsilon-1}-1}y_{i}^{\frac{\varepsilon-1}{\varepsilon}-1}=\frac{y_{i}^{\frac{\varepsilon-1}{\varepsilon}}}{\sum_{i=1}^{N}y_{i}^{\frac{\varepsilon-1}{\varepsilon}}}=\left(\frac{y_{i}}{Y}\right)^{1-\frac{1}{\varepsilon}} \]

This gives us the result. But if we observe this expression in detail we can see that we can also express it as

\[ \frac{y_{i}}{Y}\frac{\partial Y}{\partial y_{i}}=\left(\frac{y_{i}}{Y}\right)^{-\frac{1}{\varepsilon}}\frac{y_{i}}{Y}=\frac{p_{i}y_{i}}{PY}=s_{i} \]

which is the sales-share of producer \(i\). This is a key result. The elasticity of demand is increasing in the firm's market share. The sign follows from the assumption that \(\eta<\varepsilon\).

Replacing back we get that the demand elasticity of variety \(i\) is a weighted average between the elasticity of its own variety and the demand elasticity of the market:

\[ \frac{1}{\varepsilon_{i}}=\frac{1}{\varepsilon}\left(1-s_{i}\right)+\frac{1}{\eta}s_{i}. \]

If the firm is small \(\left(s_{i}\to0\right)\), it behaves like in the monopolistic competition setup and only cares about the elasticity of its own variety, taking aggregates as given. As the firm grows large \(\left(s_{i}\to1\right)\), it behaves like a true monopolist, caring about the elasticity of demand for the market output \(\left(Y\right)\) and not for the competition with other varieties.

The markup is then

\[ \mu_{i}=\frac{1}{1-\frac{1}{\varepsilon_{i}}}=\frac{1}{\left(1-\frac{1}{\varepsilon}\right)\left(1-s_{i}\right)+\left(1-\frac{1}{\eta}\right)s_{i}}. \]

It is again the case that larger firms have larger markups, reflecting the decrease in elasticity as market shares increase.

Aggregating markups

The average markup in a market, \(\bar{\mu}\), is defined as the ratio between the market's price \(P\) and the market's marginal cost \(\overline{c}\),

\[ \overline{\mu}=\frac{P}{\overline{c}}. \]

This is equivalent to defining it as the ratio of the market's revenue \(PY\) and the market's total cost, which, under constant returns-to-scale, is \(\sum_{i}c_{i}y_{i}\).

Because the market aggregator has constant-returns-to-scale, the market's marginal cost is equal to the output-weighted average of the individual marginal costs, that we label \(c_{i}\equiv C_{i}^{\prime}\left(y_{i}\right)\),

\[ \overline{c}=\sum_{i=1}^{N}c_{i}\frac{y_{i}}{Y}. \]

Then, the market's markup is obtained as a sales-weighted harmonic mean of individual markups

\[ \overline{\mu}=\frac{P}{\overline{c}}=\left[\sum_{i=1}^{N}c_{i}\frac{y_{i}}{PY}\right]^{-1}=\left[\sum_{i=1}^{N}\frac{1}{\mu_{i}}s_{i}\right]^{-1}=\left[\sum_{i=1}^{N_{m}}\left(1-\frac{1}{\varepsilon_{i}}\right)s_{i}\right]^{-1}=\frac{1}{1-\frac{1}{\overline{\varepsilon}}}, \]

where \(\overline{\varepsilon}\) is the (weighted-harmonic) average elasticity in the market,

\[ \overline{\varepsilon}\equiv\left[\sum_{i=1}^{N}\frac{1}{\varepsilon_{i}}s_{i}\right]^{-1}. \]

In this way the market's average markup has the same expression as the individual markup with the corresponding market elasticity.

In the case of Cournot competition we can go further using the expression derived above for the elasticity of each firm:

\[ \frac{1}{\overline{\varepsilon}}=\sum_{i=1}^{N}\frac{1}{\varepsilon_{i}}s_{i}=\sum_{i=1}^{N}\left(\frac{1}{\varepsilon}\left(1-s_{i}\right)+\frac{1}{\eta}s_{i}\right)s_{i}=\frac{1}{\varepsilon}\left(\sum_{i=1}^{N}s_{i}-\sum_{i=1}^{N}s_{i}^{2}\right)+\frac{1}{\eta}\sum_{i=1}^{N}s_{i}^{2} \]

where we take advantage of the fact that \(\sum s_{i}=1\) and that the sum of square shares is the definition of the Herfindahl-Hirschman index of concentration (HHI). This index gives the probability that two random dollars spent in the market are spent in the same firm. The result is a direct link between concentration in the market and average markups

\[ \frac{1}{\overline{\varepsilon}}=\frac{1}{\varepsilon}\left(1-\text{HHI}\right)+\frac{1}{\eta}\text{HHI}\qquad\text{and}\qquad\frac{1}{\overline{\mu}}=\underbrace{\frac{\eta-1}{\eta}}_{\text{Monopoly Markup}}+\underbrace{\left(\frac{1}{\eta}-\frac{1}{\varepsilon}\right)\left(1-\text{HHI}\right)}_{\text{Concentration Markup}}. \]