2. Setup

2.2. Demand side

Assume first that products are differentiated and the demands for each product are smooth. Demand for product $j$ is $q_{j}=D_{j}(p),$ where $p$ is the vector of $N$ prices. We assume that the demands are downward sloping, $ \frac {\partial D_{j}}{\partial p_{j}}\lt 0,$ and the products gross substitutes, $\frac {\partial D_{j}}{\partial p_{k}}\geq 0$ for $k\neq j$ . If the Jacobian of the system has a dominant negative diagonal, it can be inverted, giving the inverse demands $p_{j}=P_{j}(q),$ with $\frac { \partial P_{j}}{\partial q_{j}}\lt 0$ and $\frac {\partial P_{j}}{\partial q_{k}} \leq 0$ for $k\neq j$ (see Vives, Reference Vives1999) $.$ In addition, to simplify things, we assume that all demands are concave.

We also want also to compare the outcomes with product differentiation and without. To do this with two demands that represent a relatively comparable context, we consider a homogeneous market with a demand that we call, for product $j$ , with ”equivalent price-effect” to the system. We will write $ Q=D(p),$ where $p$ is here the unique price.Footnote ⁵

Definition 1. The demand for homogeneous goods has an equivalent price-effect for good $j$ to the system of differentiated goods if it has a price derivative that equals the effect of the change of $p_{j\text { }}$ on the sum of the differentiated demands $\sum \nolimits _{k}D_{k}(p).$ That is,

\begin{equation*} \frac {\partial Q}{\partial p}=\frac {\partial D_{j}(p)}{\partial p_{j}} +\sum \nolimits _{k\neq j}\frac {\partial D_{k}(p)}{\partial p_{j}}. \end{equation*}

An obvious consequence is that $-\frac {\partial q_{j}}{\partial p_{j}}\equiv -\frac {\partial D_{j}(p)}{\partial p_{j}}\gt -\frac {\partial Q}{\partial p}.$ Another, less obvious, is given by the following.

Lemma 1. The demand $Q=D(p)$ with equivalent price-effect for product $j$ to the system $q_{j}=D_{j}(p)$ implies $-\frac {\partial p}{ \partial Q}\gt -\frac {\partial p_{j}}{\partial q_{j}}.$

Proof. See Appendix.

The consideration of a homogeneous demand with equivalent price-effect gives us two specific alternative market demands for which we can compare the outcomes resulting from the actions of firm $j.$ The price-effect in the differentiated demand for firm $j$ is greater in absolute value than in the homogeneous demand, and the lemma establishes that price is less sensitive to quantity than it is in the homogeneous demand.

2.3. Behavior and outcomes

We are going to consider four situations that we depict in Table 1. We draw on Vives (Reference Vives1999), who treats exhaustively each one of these situations. Firms may compete alternatively in a market with perfectly substitutable products or with differentiated products, characterized by the system of demands that we have assumed above. In both situations, firms can compete either in prices (Bertrand competition) or in quantities (Cournot competition). Moving clockwise in Table 1, we have the equilibria of Bertrand with homogeneous product, Bertrand with differentiated product, Cournot with differentiated product, and Cournot with homogeneous product. When we need clarity, we use the shorthands H, B, C, and Q, respectively.

Table 1. Competition and market power outcomes $^{a}$

^a $\eta _{j}=-\frac {p_{j}}{q_{j}}\frac {\partial D_{j}}{\partial p_{j}}$ , $\varepsilon =-\frac {Q}{p}\frac {\partial P}{\partial Q}$ , and $\varepsilon _{j}=-\frac {q_{j}}{p_{j}}\frac {\partial P_{j}}{\partial q_{j}}$ .

In what follows, we briefly describe behavior and outcomes. Product differentiation softens price competition and quantity competition too. Despite this, a ranking of competitive outcomes among the four situations that includes all cases is not possible. However, the rest of this paper shows that the situations can be clearly ranked according to the impact of productivity advantages on the market shares and hence on the structure of the market. The most radical impact happens in the top-left corner, when a productivity advantage drives the firm to gain the entire market, becoming milder as we move clockwise to the outcome of Cournot.

The toughest situation for firms happens when products are perfectly substitutable and firms compete in prices. Let $\omega =\{\omega _{1},\omega _{2},\ldots, \omega _{N}\}$ be the vector of productivities. The unique Nash equilibrium is the firm with the greatest productivity pricing a little below the marginal cost of the firm with the second greatest productivity (Tirole, Reference Tirole1989; Vives, Reference Vives1999). Suppose that firm $j$ has the greatest productivity and firm $k$ the second. In this case, $p_{j}=c(1-\omega _{k})-\varepsilon, $ $q_{j}=D(c(1-\omega _{k})-\varepsilon )$ , $q_{k}=0$ and $q_{l}=0$ for all $l\neq j,k$ . Firm $j$ becomes the only firm in the market ( $S_{j}=1$ ), with (approximate) market power $\frac {p_{j}-C_{j}^{\prime }}{ p_{j}}\simeq \frac {\omega _{j}-\omega _{k}}{1-\omega _{k}},$ its relative productivity advantage $.$ If no firm has a productivity advantage, the only Nash equilibrium is the competitive outcome of all prices equal to marginal cost.

We locate this radical outcome in the top-left corner of Table 1. The equilibrium is usually known as Bertrand with homogeneous product.

With product differentiation, when firms compete in prices, the first-order conditions for profit maximization give $\frac {p_{j}-C_{j}^{^{\prime }}}{ p_{j}}=\frac {1}{\eta _{j}},$ where $\eta _{j}=-\frac {p_{j}}{q_{j}}\frac { \partial D_{j}}{\partial p_{j}}$ is the absolute value of the elasticity of demand for product $j.$ And when firms compete in quantities we write, following Vives (Reference Vives1999), $\frac {p_{j}-C_{j}^{^{\prime }}}{p_{j}}=\varepsilon _{j},$ where $\varepsilon _{j}=-\frac {q_{j}}{p_{j}}\frac {\partial P_{j}}{ \partial q_{j}}$ is the absolute value of the elasticity of inverse demand. For a given vector of prices $p$ , it happens that $-\frac {\partial P_{j}}{ \partial q_{j}}\geq \frac {1}{-\frac {\partial D_{j}}{\partial p_{j}}}$ and hence $\varepsilon _{j}^{p}\geq \frac {1}{\eta _{j}^{p}}$ for all $j.$ The reason is that the firm is acting at each equilibrium as a monopolist with respect to different residual demands, which imply different elasticities. An implication is that, with the same demand system, the market outcome under quantity competition is less competitive, in the sense that the vector of prices is equal or greater than the prices under price competition (see Vives, Reference Vives1999, p. 156, for the proof). That is, it happens that $p^{C}\geq p^{B},$ associated to $\varepsilon _{j}\geq \frac {1}{\eta _{j}}$ for all $j.$

Although Bertrand with product differentiation softens price competition, the two market power outcomes of the first row of Table 1 cannot be ranked completely without ambiguity. When firm $j$ chooses the optimal price according to the residual direct demand, the resulting margin may be greater or lesser than its productivity relative advantage, that is, $\frac { p_{j}-C_{j}^{^{\prime }}}{p_{j}}=\frac {1}{\eta _{j}}\gtrless \frac {\omega _{j}-\omega _{k}}{1-\omega _{k}},$ even if all the rest of firms are going to increase their margins with respect to their zero equilibrium margins at Bertrand with homogeneous product. However, in the Cournot equilibrium of the intersection of the second row and second column of Table 1, all firms exercise more market power than in the Bertrand equilibrium with product differentiation.

Let us finally see what happens in the second row of the table, considering as demand for the homogeneous goods a demand that has an equivalent price-effect for $j.$ Competitor $j$ will choose according to the first-order conditions $p(Q)+q_{j}\frac {\partial p}{\partial Q}=C_{j}^{\prime }$ if the product is homogeneous and $p_{j}(q)+q_{j}\frac {\partial p_{j}}{\partial q_{j}}=C_{j}^{\prime }$ if the product is differentiated $.$ At the same price in the homogeneous market as in the differentiated market, $p=p_{j}$ , it can be checked that the Cournot competitor would choose quantities such that elasticities $\frac {q_{j}}{Q}\frac {Q}{p}\frac {\partial P}{\partial Q} =S_{j}\varepsilon$ and $\frac {q_{j}}{p_{j}}\frac {\partial p_{j}}{\partial q_{j}}=\varepsilon _{j}$ are identical. This implies $q_{j}^{Q}\lt q_{j}^{C}.$ In this sense, Cournot is less competitive than Cournot with product differentiation.

However, in general, there is no reason for the two scenarios to give the same price. With the demands having the same choke off price, it is clear that the first marginal revenue falls more rapidly with $q_{j}$ and the choice will imply $q_{j}^{Q}\lt q_{j}^{C}.$ But we have not made any assumption about the level of the demands and in particular the choke off price. In addition, the heterogeneity of the price effects of firms—with the implication of different homogeneous equivalent price-effect demands—makes the comparisons less straightforward.

3. Best response functions

Let us analyze the change of the best response functions with the growth of productivity. We will focus on the change in the productivity of one firm, while the productivity of the rest remains the same. We deal in turn with the three cases with smooth profit functions: Bertrand with product differentiation, B; Cournot with product differentiation, C; and Cournot, Q. In B and C, we assume the same demand, in Q a homogeneous demand with equivalent price-effect to the system of B and C.

Let us start with Bertrand with product differentiation. The profit function of the firm $j$ that sets its optimal price as monopolist of the residual demand can be written $\pi _{j}=p_{j}q_{j}(p)-C(q_{j}(p),\omega _{j}),$ and the first-order condition is $q_{j}(p)+p_{j}\frac {\partial q_{j}(p)}{ \partial p_{j}}-C_{j}^{\prime }(\omega _{j})\frac {\partial q_{j}(p)}{ \partial p_{j}}=0.$ Write the first-order condition as the optimal response of $ j$ to the rest of prices $,$ $p_{j}=R_{j}(p_{-j})$

\begin{equation*} q_{j}\big(R_{j}(p_{-j}),p_{-j} \big)+ \big(R_{j}(p_{-j})-C_{j}^{\prime }(\omega _{j}) \big)\frac {\partial q_{j}\big(R_{j}(p_{-j}),p_{-j} \big)}{\partial p_{j}}=0. \end{equation*}

The change in the best response of $j$ due to a change in productivity is

(1) \begin{equation} \frac {\partial R_{j}(p_{-j})}{\partial \omega _{j}}=\frac {\frac {\partial C_{j}^{\prime }}{\partial \omega _{j}}\frac {\partial q_{j}}{\partial p_{j}}}{ 2\frac {\partial q_{j}}{\partial p_{j}}+ \big(p_{j}-C_{j}^{\prime } \big)\frac {\partial ^{2}q_{j}}{\partial p_{j}^{2}}}\lt 0. \end{equation}

With concave demand, the denominator is negative (in fact, the denominator coincides with the second order condition for the problem of setting an optimal price). Both terms in the numerator are negative, and therefore the product is positive. The sign of the derivative is hence negative.

It follows that an increase in productivity displaces downward the best response function of the firm, and the firm has incentives to decrease the price. Suppose two moments of time, $1$ and $2,$ say. All the rest equal, a new equilibrium should emerge with a vector of prices $p_{2}^{B}\leq$ $ p_{1}^{B}.$ All firms decrease their prices or at most keep the same. Figure 1 panel B illustrates the displacement and the new equilibrium for the case of two firms, $j$ and $i.$

Figure 1. Best response changes with an increase in productivity.

Let us now discuss Cournot with product differentiation. The profit function of the firm $j$ that sets its optimal quantity as monopolist of the residual demand can be written $\pi _{j}=p_{j}(q)q_{j}-C(q_{j},\omega _{j}),$ and the first-order condition is $p_{j}(q)+q_{j}\frac {\partial p_{j}(q)}{\partial q_{j}}-C_{j}^{\prime }(\omega _{j})=0.$ Write the first-order condition in terms of the optimal response of $j$ to the rest of quantities $,$ $ q_{j}=R_{j}(q_{-j})$

\begin{equation*} p_{j}\big(R_{j}(q_{-j}),q_{-j} \big)+R_{j}(q_{-j})\frac {\partial p_{j}\big(R_{j}(q_{-j}),q_{-j} \big)}{\partial q_{j}}-C_{j}^{\prime }(\omega _{j})=0. \end{equation*}

The change in the reaction function is

(2) \begin{equation} \frac {\partial R_{j}(q_{-j})}{\partial \omega _{j}}=\frac {\frac {\partial C_{j}^{\prime }}{\partial \omega _{j}}}{2\frac {\partial p_{j}}{\partial q_{j} }+q_{j}\frac {\partial ^{2}p_{j}}{\partial q_{j}^{2}}}\gt 0. \end{equation}

The denominator is negative for the same reason as before. The numerator is also negative, and hence the derivative is positive.

It follows that an increase in productivity displaces upward the best response function of the firm, and the firm has incentives to increase the quantity put in the market. All the rest equal, a new equilibrium should emerge with more quantity for firm $j$ in detriment of the output of the rivals. Given the usual slope of the reaction functions, total output expands. Figure 1 panel C illustrates the displacement and the new equilibrium for the case of two firms, $j$ and $i.$

Let us finally discuss quantity competition with identical products. Suppose that the firm faces a homogeneous demand with equivalent price-effect for firm $j$ . The profit function of Cournot competitor $j$ in a homogeneous product market is $\pi _{j}=p(Q)q_{j}-C(q_{j},\omega _{j}),$ where what has changed is that the unique price in the market is the result of the total quantity $Q=\sum \nolimits _{k}q_{k}$ of the perfectly substitutable goods. Firm $j$ chooses quantity optimally according to the first-order condition $ p(Q)+q_{j}\frac {\partial p(Q)}{\partial q_{j}}-C_{j}^{\prime }(\omega _{j})=0.$ In terms of the best response $q_{j}=R_{j}(q_{-j})$ , we can write

\begin{equation*} p\left(R_{j}(q_{-j})+\sum \nolimits _{k\neq j}q_{k} \right)+R_{j}(q_{-j}) \frac {\partial p \left(R_{j}(q_{-j})+\sum \nolimits _{k\neq j}q_{k} \right)}{\partial q_{j}} -C_{j}^{\prime }(\omega _{j})=0. \end{equation*}

The change in the reaction function now is

(3) \begin{equation} \frac {\partial R_{j}\left(\sum \nolimits _{k\neq j}q_{k} \right)}{\partial \omega _{j}}=\frac {\frac {\partial C_{j}^{\prime }}{\partial \omega _{j}}}{2\frac {\partial p}{\partial q_{j}}+q_{j}\frac {\partial ^{2}p}{\partial q_{j}^{2}}}\gt 0. \end{equation}

It follows that an increase in productivity displaces again upward the best response function of the firm, and the firm has incentives to increase the quantity set in the market. A new equilibrium will imply, as before, more quantity for firm $j$ in detriment of the output of the rivals, and an expanded total output. Figure 1 panel Q illustrates the displacement and the new equilibrium for the case of two firms, $j$ and $i.$

We can summarize what we know as follows. When firm $j$ experiences an increase in its productivity, its best response function moves in the direction of either decreasing the price or expanding output depending on the type of market competition (price or quantity). This movements meet the following.

4. Distribution of market shares

The results of the previous section suggest that, other things equal, market shares should be closely linked to productivity advantages and that this relationship must be sharper as price competition intensifies. In this section, we directly explore the relationship between productivity advantages and market shares. There is one case in which this relationship is straightforward, and that is the case of Cournot competition under homogeneous goods. The reason is that the price is common to all firms, and hence all advantages are directly conveyed by the quantities. We will see that under this type of competition, market shares can be approximated by an expression that is linear in the productivity advantages.

In the other extreme, it lies the radical case of Bertrad competition under homogeneous product. Market shares are so sensitive to price (and hence productivity) differences that a small advantage becomes enough to dominate the entire market. Bertrand and Cournot competition under product differentiation lie in between. Since the product is differentiated and firms price differently, we need to define market shares in terms of sales, $S_{j}=p_{j}q_{j}/\sum \nolimits _{k}p_{k}q_{k}.$ Then productivity advantages both impact the prices and quantities set by firms. There are no easy ways to separate the influence on each variable, and the relationships that link market shares and productivity advantages are going to become highly nonlinear. However, the relationships developed in the previous section suggest that the introduction of differentiated prices should help to translate more sharply the advantages to shares than in Cournot with homogeneous product and more with Bertrand competition than with Cournot competition.

We start describing the shares in Cournot competition with homogeneous product, which we are going to denote by $S^{Q}.$ Then we will characterize the shares with Bertrand with product differentiation and Cournot with product differentiation, respectively. Finally, we establish a second proposition that ranks the sensitivity of a given share to the productivity advantages under the different competition situations. We will use the notations $S^{B\text { }}$ and $S^{C\text { }}$ for the share corresponding to Bertrand and Cournot under product differentiation.

The first-order condition for Cournot competition can be rewritten as

\begin{equation*} p(1-S_{j}\varepsilon )=C_{j}^{\prime }, \end{equation*}

which allows to compute Cournot price by aggregation as

\begin{equation*} p=\frac {\overline {C^{\prime }}}{1-\varepsilon /N}. \end{equation*}

where $\overline {C^{\prime }}$ is the average of marginal costs, $\overline {C^{\prime}}=\frac {1}{N}\sum \nolimits _{k}C_{k}^{\prime }$ . The combination of both expressions gives the relationship between market shares and relative marginal costs

\begin{equation*} S_{j}^{Q}=\frac {1}{\varepsilon }- \left(\frac {1}{\varepsilon }-\frac {1}{N} \right)\frac { C_{j}^{\prime }}{\overline {C^{\prime }}}. \end{equation*}

The derivative can be written as

(4) \begin{equation} \frac {\partial S_{j}^{Q}}{\partial \omega _{j}} = \left(\frac {1}{\varepsilon }-\frac {1}{N} \right)\frac {C_{j}^{\prime }}{N\overline {C^{\prime }}}\frac {N\overline {C^{\prime }}-C_{j}^{\prime }}{N\overline {C^{\prime }}}\gt 0, \end{equation}

where we use the fact that $\frac {1}{C_{j}^{\prime }}\frac {\partial C_{j}^{\prime }}{\partial \omega _{j}}=-1.$

Using the approximation $\frac {C_{j}^{\prime }}{\overline {C^{\prime }}}=1+ \big(\frac {C_{j}^{\prime }}{\overline {C^{\prime }}}-1 \big)\simeq 1-\frac {\omega _{j}-\overline {\omega }}{1-\overline {\omega }}$ , where $\overline {\omega }=\frac {1}{N}\sum \nolimits _{k}\omega _{k},$ we can write

\begin{equation*} S_{j}^{Q}=\frac {1}{N} + \left(\frac {1}{\varepsilon }-\frac {1}{N} \right)\frac {\omega _{j}-\overline {\omega }}{1-\overline {\omega }}. \end{equation*}

Under Cournot competition in a perfect substitutes market, market shares diverge if productivity is different across firms, and diverge in direct relationship to the magnitude of productivity advantage of each firm with respect to mean productivity. The lower the elasticity of inverse demand (or the greater the direct elasticity of demand $\eta )$ Footnote ⁶ $,$ and the larger the number of firms, the greater the impact of productivity.

Notice that, under common input prices, for a given number of firms and elasticity of demand, the distribution of productivity $f(\omega )$ completely determines the distribution of shares. A change in the distribution of productivity will be transmitted to the distribution of shares. For example, if during some period of time productivity advantages become asymmetric, we expect market shares to become equally asymmetric, and the market becomes concentrated.

In the case of Bertrand with product differentiation and Cournot with product differentiation, we have a different price for each product. First-order conditions give

\begin{eqnarray*} p_{j}^{B}\left(1-\frac {1}{\eta _{j}} \right) &=&C_{j}^{\prime }, \\ p_{j}^{C}(1-\varepsilon _{j}) &=&C_{j}^{\prime }, \end{eqnarray*}

respectively. Aggregation produces formulas that depend on all the elasticities and the firm share in costs

\begin{eqnarray*} S_{j}^{B} &=&\frac {1-\frac {1}{\eta_{j}}}{1-\frac {1}{\eta}}\frac {C_{j}}{ \sum \nolimits _{k}C_{k}}, \\ S_{j}^{C} &=&\frac {1-\varepsilon_{j} }{1-\varepsilon }\frac {C_{j}}{ \sum \nolimits _{k}C_{k}} \end{eqnarray*}

where $\frac {1}{\eta }=\sum \nolimits _{k}S_{k}^{B}\frac {1}{\eta _{k}}$ and $ \varepsilon =\sum \nolimits _{k}S_{k}^{C}\varepsilon _{k}.$ If all elasticities are equal, market shares exactly coincide with the shares in costs.

Notice that cost shares can be rewritten in terms of relative marginal costs

\begin{equation*} \frac {C_{j}}{\sum \nolimits _{k}C_{k}}=\frac {C_{j}^{\prime }q_{j}}{ \sum \nolimits _{k}C_{k}^{\prime }q_{k}}=\frac {S_{j}^{Q}C_{j}^{\prime }}{ \sum \nolimits _{k}S_{k}^{Q}C_{k}^{\prime }}, \end{equation*}

where now we use the notation $S_{j}^{Q}$ for the current quantity shares. The weight of marginal costs with the quantity shares introduces an additional complexity. Note that $S_{j}^{Q}$ is, as in the Cournot case, and even with equal price elasticities, a function of the entire vector of advantages. Productivity advantages reduce the cost of each unit of output but also increase the output amount. How output increases depends on the demand relationship and type of competition.

We can approximate the formula for cost shares as $\frac {C_{j}}{ \sum \nolimits _{k}C_{k}}=S_{j}^{Q}(1-\omega _{j})/(1-\overline {\omega }_{Q})$ , where $\overline {\omega }_{Q}=\sum \nolimits _{k}S_{k}^{Q}\omega _{k}.$ Plugging this expression in the formulas for $S_{j}^{B}$ and $S_{j}^{C}$ , we have the nonlinear analogous of the linear expression for Cournot under homogeneous product.

The expressions for market shares are difficult to manage under arbitrary heterogeneity of firms. However, for a given set of market shares, we can get insights by differentiating directly the market share under Cournot and Bertrand competition (recall that equilibria are associated with different residual demands).

Under Bertrand competition, we have

\begin{equation*} S_{j}^{B}=\frac {p_{j}q_{j}(p)}{\sum _{k}p_{k}q_{k}(p)}, \end{equation*}

and we obtain

\begin{equation*} \frac {\partial S_{j}^{B}}{\partial p_{j}}=-\frac {S_{j}}{p_{j}} \left[(1-S_{j})(\eta _{j}-1)+\sum \nolimits _{k\neq j}S_{k}\eta _{kj} \right], \end{equation*}

where $\eta _{kj}$ are the cross-price elasticities $\eta _{kj}=\frac {p_{j}}{ q_{k}}\frac {\partial q_{k}}{\partial p_{j}}.$ The effect of $\omega _{j\text { }}$ on the share can be now computed as

(5) \begin{equation} \frac {\partial S_{j}^{B}}{\partial \omega _{j}}=\frac {\partial S_{j}^{B}}{ \partial p_{j}}\frac {\partial p_{j}}{\partial \omega _{j}} =S_{j} \left[(1-S_{j})(\eta _{j}-1)+\sum \nolimits _{k\neq j}S_{k}\eta _{kj} \right]\gt 0, \end{equation}

where we use that $\frac {\partial p_{j}}{\partial \omega _{j}}=-p_{j}.$

Under Cournot competition, we have

\begin{equation*} S_{j}^{C}=\frac {p_{j}(q)q_{j}}{\sum _{k}p_{k}(q)q_{k}}, \end{equation*}

and we obtain

\begin{equation*} \frac {\partial S_{j}^{C}}{\partial q_{j}}=\frac {S_{j}}{q_{j}} \left[(1-S_{j})(1-\varepsilon _{j})+\sum \nolimits _{k\neq j}S_{k}\varepsilon _{kj} \right], \end{equation*}

where $\varepsilon _{kj}$ are the absolute value of the cross-quantity elasticities $\varepsilon _{kj}=-\frac {q_{j}}{p_{k}}\frac {\partial p_{k}}{ \partial q_{j}}.$ The effect of $\omega _{j\text { }}$ on the share can be calculated as

(6) \begin{equation} \frac {\partial S_{j}^{C}}{\partial \omega _{j}}=\frac {\partial S_{j}^{C}}{ \partial q_{j}}\frac {dq_{j}}{dp_{j}}\frac {\partial p_{j}}{\partial \omega _{j}}=S_{j}\frac {1}{\varepsilon _{j}} \left[(1-S_{j})(1-\varepsilon _{j})+\sum \nolimits _{k\neq j}S_{k}\varepsilon _{kj} \right]\gt 0, \end{equation}

where we use that $\frac {1}{\frac {dp_{j}}{dq_{j}}}\frac {\partial p_{j}}{ \partial \omega _{j}}=\frac {q_{j}}{\varepsilon _{j}}.$

We can now establish

5. Concentration and inflation

The relationship between market power and inflation has often worried economists, who have speculated that firms with market power may help to sustain the increase in prices over time. Although this is an old concern (see, e.g., Weiss, Reference Weiss1971), recent papers have revisited it as an explanation for recent trends in the US economy. According to, for example, Covarrubias et al. (Reference Covarrubias, Gutierrez and Philippon2020) and Brauning et al. (Reference Brauning, Fillat and Joaquim2023), a presumed rise in concentration and the evolution of prices show the link between inflation, concentration, and market power.

However, productivity gains provide a reason why, in times of technological change, we should rather expect the opposite relationship. Sections 3 and 4 have shown that asymmetric productivity gains may simultaneously determine that the prices fall or increase less than cost at the same time that the market becomes concentrated. In this section, we develop with some detail how this happens in a market with Cournot competition. Proposition2 suggests that this outcome can still be more intensive when price competition is sharper, so we expect it to be a more general phenomenon. We then carry out a simple empirical exercise (in the style of Ganapati, Reference Ganapati2021) to illustrate how recent increases in prices have shown negative relationships with increases in concentration by industries.

5.1. Theory

Let us assume that the productivity advantages $\omega$ of the population of a market are, in a given moment $t,$ distributed normally around the mean $ \overline {\omega }_{t}$ , that is, $\omega \sim N(\overline {\omega }_{t},\sigma _{t}^{2}).$ Footnote ⁷ Take marginal costs as $C^{^{\prime }}=c(w_{t})\exp \!({-}\omega ).$ Marginal cost is then a lognormal variable with mean $ E(C^{\prime })=c(w_{t})\exp ({-}(\overline {\omega }_{t}+\frac {1}{2}\sigma _{t}^{2})).$ Suppose that this is a perfect substitutes market where firms behave Cournot. Assume that demand elasticity is constant and the number of firms doesn’t change. It follows that the log of the price is related to mean marginal cost as

\begin{equation*} \ln p_{t}=-\ln \left(1-\frac {\varepsilon }{N} \right)+\ln c(w_{t})- \left(\overline {\omega } _{t}+\frac {1}{2}\sigma _{t}^{2} \right). \end{equation*}

Inflation $\Delta \ln p_{t}$ (the rate of increase over time of the market price) has two determinants. First, the increase in $\Delta \ln c(w_{t}),$ due to the increase in the prices of the inputs, pushes the price up. Second, the increase in $\Delta (\overline {\omega }_{t}$ , $+\frac {1}{2}\sigma _{t}^{2}),$ because productivity gains, pulls it down. Notice that this second force can operate even with no modifications of the average productivity. It is enough that productivity spreads through the gains in some firms, even if it decreases in others (a mean-preserving change in the variance).

An implication of the formula is that, with Cournot’s exercise of market power, an unequal increase in the productivity of the firms will tend to decrease the output price (or to reduce the increase induced by input price inflation).

To see an example of how this happens, think of the ordered distribution of productivity advantages $\omega .$ Suppose that the upper tail of the distribution, from some arbitrary threshold level of productivity on, has the values systematically increased. We are going to call this type of increase a stochastically dominant productivity change (the cumulative distribution function shifts down). We will see a simultaneous increase of $ \overline {\omega }_{t}$ and $\sigma _{t}^{2}.$

This kind of change in productivity will affect simultaneously the concentration of the market. The relative marginal cost of a firm can be written as $\frac {C^{\prime }}{E(C^{\prime })}=\exp \![{-}(\omega -(\overline { \omega }_{t}+\frac {1}{2}\sigma ^{2}))]=\exp \!({-}\omega ^{\ast }).$ We will call $\omega ^{\ast }$ the relative productive advantage of the firm. The market share of a firm is $S^{Q}=\frac {1}{\varepsilon }-(\frac {1}{ \varepsilon }-\frac {1}{N})\exp ({-}\omega ^{\ast }).$ Note that if $\omega ^{\ast }=0$ (the productivity advantage equals mean productivity), the share of the firm is just the average market share. Let us discuss what happens when there is a change in the distribution of $\omega ^{\ast }.$

The share $S^{Q\text { }}$ is a positive monotonic function of $\omega ^{\ast }.$ Call the distribution functions of $\omega ^{\ast }$ and $S^{Q},$ $ G_{1}(\omega ^{\ast })$ and $F_{1}(S^{Q})$ , respectively. Since $S^{Q\text { } }$ is a monotonic transformation of $\omega ^{\ast }$ , the quantiles of $ F_{1}(S^{Q})$ are monotonic transformations of the quantiles of $ G_{1}(\omega ^{\ast }).$ It follows that, if we consider a change of the distribution of $\omega ^{\ast }$ from $G_{1}(\omega ^{\ast })$ to $ G_{2}(\omega ^{\ast })$ , where $G_{2}(\omega ^{\ast })$ stochastically dominates $G_{1}(\omega ^{\ast }),$ we will have also a new distribution $ F_{2}(S^{Q})$ that stochastically dominates $F_{1}(S^{Q}).$ A consequence is that

\begin{equation*} CRk_{2}-CRk_{1}=\int \nolimits _{Q_{S}(\tau )}^{1}S^{Q}dF_{2}\big(S^{Q} \big)-\int \nolimits _{Q_{S}(\tau )}^{1}S^{Q}dF_{1}\big(S^{Q} \big)\geq 0, \end{equation*}

where $CRk$ is the concentration ratio for the first $K$ firms and $ Q_{S}(\tau )$ is the quantile $\tau =100\frac {N-K}{N}\%$ in the distribution of $S^{Q}.$ Footnote ⁸

Summarizing, if the evolution of productivity is such that the relative productive advantages $\omega ^{\ast }$ do not vary, the distribution of market shares will not change. For example, suppose that all productivities increase in $g$ percentage points, that is, $\omega _{jt+1}=\omega _{jt}+g$ . However, if some firms improve productivities more than others, raising a stochastically dominant new distribution of relative productivity advantages, the market is going to become more concentrated in the sense that concentration ratios are going strictly to increase.

This suggests an important role for processes of innovation and diffusion of technology, where productivity gains are not instantaneous and simultaneous. When some firms experience relatively bigger gains in productivity, and the differences stays for some time, average marginal cost is going to decrease or increase less, moderating price increases. At the same time, the market is going to become concentrated, with an increased share of the most productive firms. This should happen if the market is a Cournot competition with homogeneous product, but section 4 suggests that the process of concentration can be even sharper with more price competition.

5.2. Empirics

We want to relate price variation and changes in concentration. The economic censuses compute concentration ratios ( $CR_{4}$ and $CR_{20}$ ) by industries of US manufacturing. The data for 2002, 2007, 2012, and 2017 are publicly available for the North American Industry Classification System (NAICS).Footnote ⁹ We combine these data with the output prices estimated for the manufacturing NBER-CES database.Footnote ¹⁰ As we employ the NBER-CES version for NAICS 1997, the use of the concentration ratios without further work matches a decreasing number of industries. However, our objective here is only a quick look, and we do not think that this impedes to transmit the main message.Footnote ¹¹ We are able to match to the NBER-CES database 468, 466, 298, and 290 industries in the 2002, 2007, 2012, and 2017, respectively.

The average value of the 4-firm concentration ratio is $0.43$ , and for the 20-firm concentration ratio is about $0.71$ . Both means show almost no changes over the years, and this despite the changing number of industries for which these means are computed. Of course, this is compatible with rich movements in both directions of the individual indices across industries. We are able to work with a total of 1039 5-year changes in prices and concentration, after dropping 15 changes in concentration outside the (−0.50, 0.50) interval. We annualize the log changes in prices dividing by 5, and we keep untouched the 5-year differences in concentration. The coefficients of regression can be interpreted as elasticities. If the coefficient shows a value of $X$ , we can read it implying an annual price change of $10X\%$ for a $10\%$ change in concentration.

Table 2 summarizes a few results. Column (1) reports the regression of the changes in prices on the change of joint share of the first 4 firms ( $\Delta CR_{4}$ ) and the change of the share of the next 16 firms ( $\Delta CR_{next16}=\Delta CR_{20}-\Delta CR_{4}$ ). The concentration of the top four firms tends to raise prices, and the share of the next 16 firms tends to decrease them, but the results are not significant.

Table 2. Effects of concentration in price, 2002–2017. Yearly average price change regressed on the variation in concentration ratios $^{a}$

$^{a}$ Computed for subperiods 2002–2007, 2007–2012, and 2012–2017, for 451, 298, and 290 industries, respectively.

However, all the important things in concentration happen in the interplay between these two shares ( $4$ and next $16$ first firms): in $91.5\%$ of the cases, at least one of these shares increases; in $51.9\%$ , it is the joint share of the 16 that does; and in $16.2\%$ , both of them. We select more than half of the cases in which the joint share of the next 16 firms increases. Column (2) shows the significant association to price reductions (or more moderate increases). Column (3) makes clear that what happens in these cases with the top firms is nonsignificant.

The data point out to rich patterns of change (only $16\%$ of the changes seem to be stochastically dominant), but there are unequivocal with respect to the association of many increases in concentration with a better behavior of prices. The topic deserves further investigation.

6. Market power or efficiency?

The growth of productivity is linked to innovation and technological advances, and recent empirical evidence finds an important role for technological progress that directly affects the marginal productivity of labor or LAP (see footnote 2). This type of technological progress explains the widespread falls in the labor shares in costs and revenue observed in the firms of the USA and other advanced economies. In section 2, we have shown that our framework can approximate without modifications the output productivity effect generated by this kind of productivity. In fact, the combination of the idea of unequal gains in LAP generated by technological change and the insights generated by the propositions of sections 3 and 4 give a convincing interpretation of recent facts observed in the US economy and its manufacturing. This interpretation differs radically from the reasonings based on conventional measurements and ideas about market power that have become popular. In this sense, the discussion revives the old questions raised by Peltzman (Reference Peltzman1977) and others about the separate identification of efficiency and market power.

Table 3. Simulation results for a sample of firms with constant markups and labor-augmenting productivity $^{a}$

$^{a}$ Sample of 1000 firms with identical CES production functions with $ \sigma =0.7,$ which produce to serve identical constant elasticity demands of absolute elasticity value $\eta =6.$ Firms experience over time Hicks-neutral and labor-augmenting productivity shocks and set prices with a constant margin over marginal cost.

$^{b}$ Firms start equal, and we report as Period 1 their values after 10 periods of simulation.

$^{c}$ De Loecker and Warzynski (Reference De Loecker and Warzynski2012) markups, computed as labor input elasticity divided by the firm’s share of labor in revenue. Weighted with revenue weights.

$^{d}$ $\frac {\eta }{\eta -1}=\frac {6}{5}=1.2.$

$^{e}$ Cost of labor over revenue. Weighted with revenue weights.

$^{f}$ Three years moving averages.

Two papers claim similar interrelated facts, for US manufacturing and other industries, in a period that approximately spans from1980 to 2016. De Loecker, Eeckhout and Unger (Reference De Loecker, Eeckhout and Unger2020) look directly at the evolution of markups. Autor, Dorn, Katz, Patterson and Van Reenen (Reference Autor, Dorn, Katz, Patterson and Van Reenen2020) look at the fall of the labor shares of revenue. For them, labor shares fall because are a decreasing function of markups. These are the joint stylized facts that emerge from both papers as presented by themselves: first, average markup (sum of markups weighted by sales shares) rises, while aggregate labor share in income falls; second, there is an important reallocation component in both the rise of the average markup and the fall of the aggregate labor share; third, the process is driven by the biggest firms (”superstar” firms in the second paper); and fourth, there is simultaneous concentration of sales.

Let us see how these stylized facts can be alternatively explained by mismeasurements incurred by ignoring the bias in productivity together with the implications suggested by sections 3 and 4 for the dynamics of the distributions resulting from intense unequal technological change.

De Loecker and Warzynski (Reference De Loecker and Warzynski2012) state that firm-level markups are typically computed as $\widehat {\mu }_{jt}=\widehat {\beta }_{X}/S_{Xjt}^{R}$ , where $\widehat { \beta }_{X}$ is the estimate of the elasticity of a variable input $X$ and $ S_{Xjt}^{R}$ the observed share of the input cost on revenue. It has become customary to use labor. Any rigid estimate of the labor elasticity $\widehat { \beta }_{L}$ will already create under LAP a positive growth bias because of the evolution downward of the observed $S_{Ljt}^{R}$ .

In addition, cost minimization implies that, at any moment of time, there is cross-section distribution of the true elasticities $\beta _{Ljt}=\mu _{jt}S_{Ljt}^{R},$ Footnote ¹² whose variation is systematically related to the level of LAP via the share of labor. The estimate produced for the average markup is $\widehat {\mu }_{t}=\sum _{j}S_{jt}\widehat {\mu }_{jt},$ where $S_{jt}$ are firm market shares. It can be written as

\begin{equation*} \widehat {\mu }_{t}=\mu _{t}+\sum _{j}S_{jt} \left(\frac {\widehat {\beta }_{L}-\beta _{Ljt}}{\beta _{Ljt}} \right)\mu _{jt}, \end{equation*}

where $\mu _{t}=\sum _{j}S_{jt}\mu _{jt}$ is the true average markup.

The second term of the right hand side can be seen as a markup weighted covariance between $S_{jt}$ and $(\widehat {\beta }_{L}-\beta _{Ljt}).$ The term $(\widehat {\beta }_{L}-\beta _{Ljt})$ depends directly of LAP because the smaller the true elasticity $\beta _{Ljt}$ , the greater the LAP. Under the effects reviewed in sections 3 and 4, the market shares $S_{jt}$ are going to be related with the level of LAP, and the greater the price competition, the more related. The expected value for the estimate is then the true value plus a (possibly huge) bias, $E(\widehat {\mu }_{t})=\mu _{t}+bias$ .Footnote ¹³ A huge bias only indicates the importance of LAP and its effects on the distribution of shares. Jaumandreu (Reference Jaumandreu2022), using conventional estimates $ \widehat {\mu }_{jt}=0.29/S_{Ljt}^{R},$ shows the important biases generated in the estimation of an unweighted and weighted average markup with a sample of manufacturing Compustat firms (an average markup close to 1 is multiplied by $1.6$ and more than $3,$ respectively).

The fact of LAP suggests another story. An unequal development of productivity gains is likely to leave markups more or less as they were according to the different competitive environments of firms, to push down the shares of labor cost in variable costs and revenue (as result of the evolution of labor under the displacement/compensation that occurs with an elasticity of substitution less than one), to impulse the growth of the market shares of the firms with productivity gains, and concentrate the market.

A simple proof of the likelihood of this alternative explanation is the following. Let us simulate a sample of firms whose productivity grows through a sequence of Hicks-neutral and LAP shocks. Firms produce with a three-input CES production function with elasticity of substitution $\sigma =0.7.$ The productivity factors $\exp (\omega _{H})$ and $\exp (\omega _{L})$ multiply, respectively, the entire input aggregator and the labor input. The productivity terms $\omega _{H}$ and $\omega _{L}$ are inhomogeneous autoregressive Markovian processes of parameter $0.8$ and normal random innovations. We assume them to be independent for simplicity. During the 30 periods that we examine, Hicksian productivity grows at an average of $1.2$ percentage points and the output effect of labor-augmenting productivity at an average of $2.2$ percentage points.

Firms face identical isoelastic demands of elasticity $\eta =6$ and play Bertrand, setting the price for their products by multiplying the short-run marginal cost by a markup $\frac {\eta }{\eta -1}=1.2.$ Footnote ¹⁴ Because productivity shocks have a random component, firms experience heterogeneous marginal cost reductions, and since they pass these reductions on prices, they experience different demanded quantities and growth. Firms in fact have started completely identical 10 periods before we begin to compute their evolution.

Table 3 summarizes a typical result. The average firm size more than doubles, although employment grows much more moderately. The aggregate labor share falls, and there is an important reallocation component because the firms that experience greater LAP both reduce their labor share and grow more. The concentration as measured by the $CR10$ of sales is big and grows.

Despite firms holding exactly the same market power from the beginning to the end, its measurement with De Loecker and Warzynski (Reference De Loecker and Warzynski2012) method, DLW, to compute markups yields a spectacular increase, with an important reallocation component (the difference between the weighted and unweighted means). Markups and their increase are greater in the 90th percentile. The numbers are roughly similar to what the mentioned papers find for the USA in the years 1980–2016.

What the exercise suggests is that research should focus on the process of productivity growth and its effects on employment and share reallocation, emphasizing on the evolution of efficiency more than on the effects of market power.