Competition Policy in a Simple General Equilibrium Model

A. Holding n^\j (or n) Fixed

To implement the first stage, we employ a modification to the budget constraint analogous to that used in deriving corollary 1.1, where we held n^j fixed. Even though we are now in a multisector model in which prices and quantities will adjust in all sectors (notably, including sectors

i \neq j

), we wish at this stage to hold n^\j constant. To implement this restriction, we use entry subsidies σⁱ in each sector

i \neq j

, set so as to hold n^\j constant. Hence, in each sector

i \neq j

the entry condition iswhereas in sector j we allow free entry:The resulting budget constraint is¹³

$\begin{matrix} (15) & \sum_{i = 1}^{I} p^{i} x^{i} n^{i} = y - \sum_{i \neq j} σ^{i} n^{i} . \end{matrix}$

The appendix demonstrates the next proposition.

Proposition 2.

In the multisector model, when there is also a subsidy on entry set to keep the number of firms in all sectors

i \neq j

constant, the effect of strengthening competition policy (raising γ^j) in sector j on social welfare is given by

$\begin{matrix} (16) & {\frac{d U}{d γ^{j}} |}_{n^{\ j}} = \sum_{i = 1}^{I} (p^{i} - \frac{d c^{i}}{d x^{i}}) λ n^{i} {\frac{d x^{i}}{d γ^{j}} |}_{n^{\ j}} + (\frac{X_{2}^{j}}{X_{1}^{j}} - \frac{x^{j}}{n^{j}}) λ p^{j} n^{j} {\frac{d n^{j}}{d γ^{j}} |}_{n^{\ j}} . \end{matrix}$

This expression resembles expression (8) in proposition 1 for the one-sector model with an outside good. The latter term, indicating the effect of reduced variety in sector j, is identical except that the change in n^j holds only n^\j fixed, whereas in proposition 1 the corresponding derivative also took the prices (p^\j) and quantities (x^\j) in all other sectors to be unchanged. Consider the case in which tougher competition policy in sector j (raising γ^j), which reduces p^j, causes expenditures to flow out of all other sectors and the resulting fall in demand has the effect of reducing prices in those sectors (p^\j).¹⁴ This general equilibrium adjustment will dampen the outflows from sectors

i \neq j

, which will reduce demand in the targeted sector j and thereby dampen the fall in p^j and, accordingly, the fall in n^j. So in this basic case, the reduction in variety in sector j will be moderated.

Now compare the first terms of expressions (8) and (16). The most obvious difference is that we now have a summation of effects on deadweight loss, covering all sectors and not just the targeted sector j. This core difference will be elaborated momentarily. The other difference is that the change in quantity demanded (now, again, for every sector and not just in sector j) is likewise the general equilibrium price effect because here the prices (p^\j) and quantities (x^\j) in all other sectors adjust. In the case just noted, the price effect in sector j of toughening competition policy (γ^j) in that sector will likewise be moderated. Regarding sector j as a whole, the general equilibrium effects moderate both the reduction in deadweight loss and the reduction in variety, leaving us with a trade-off qualitatively similar to that in the one-sector model.¹⁵

Before further elaboration of the first term in proposition 2, it is helpful to state the following analogue to corollary 1.1.¹⁶

Corollary 2.1.

In the multisector model, when there is also a subsidy on entry set to keep the number of firms in all sectors (including sector j) constant, the effect of strengthening competition policy (raising γ^j) in sector j on social welfare is given by

$\begin{matrix} (17) & {\frac{d U}{d γ^{j}} |}_{n} = \sum_{i = 1}^{I} (p^{i} - \frac{d c^{i}}{d x^{i}}) λ n^{i} {\frac{d x^{i}}{d γ^{j}} |}_{n} . \end{matrix}$

Comparing this expression to proposition 2’s expression (16), we obviously no longer have the effect of changing variety in sector j, and also the derivatives here reflect that all of the nⁱ, including n^j, are held constant. There are still general equilibrium effects due to adjustments in prices and quantities in other sectors embodied in these derivatives.

This summation, like that in the first term of expression (16) in proposition 2, indicates that deadweight loss changes not only in sector j, where x^j rises so deadweight loss falls, but also in all other sectors

i \neq j

. Taking the simple case just above in which xⁱ falls for all

i \neq j

, the implication is that deadweight loss rises in all other sectors as a consequence of forgone purchases at prices that are in excess of marginal cost. Hence, focusing solely on deadweight loss, we have a cross-sector trade-off. Moreover, even if the effect in each sector

i \neq j

is small, the aggregate effects in all such sectors will be of the same order as the effects in the targeted sector j because the total expenditure outflows from sectors

i \neq j

necessarily equal the inflow to sector j.

To explore this trade-off further, we can develop a more intuitive condition for when tougher competition policy in a given sector reduces total deadweight loss in the economy. In doing so, it is helpful to define the Lerner index (introduced in n. 11) as

$\begin{matrix} (18) & ℒ^{i} \equiv \frac{p^{i} - (d c^{i} / d x^{i})}{p^{i}} . \end{matrix}$

Further analysis in the appendix shows the next corollary.

Corollary 2.2.

In the multisector model, when there is also a subsidy on entry set to keep the number of firms in all sectors (including sector j) constant,

$\begin{matrix} (19) & sign ({\frac{d U}{d γ^{j}} |}_{n}) = sign (ℒ^{j} - \sum_{i \neq j} α^{i} ℒ^{i}), \end{matrix}$

where

$\begin{matrix} (20) & α^{i} \equiv - \frac{p^{i} n^{i} {(d x^{i} / d γ^{j}) |}_{n}}{p^{j} n^{j} {(d x^{j} / d γ^{j}) |}_{n}} . \end{matrix}$

The interpretation of these weights is that the αⁱ, for $i \neq j$ , indicate the relative outflows of expenditures from each of the other sectors, which collectively fund the inflow into sector j. (These αⁱ are related to diversion ratios that are sometimes used in competition analysis.) Note that, for substitutes in the relevant sense, $α^{i} > 0$ (xⁱ and x^j move in opposite directions, and there is a negative sign in the definition of αⁱ). Also, $α^{j} = −1$ .

The right-hand side of expression (19) indicates that strengthening competition policy in sector j reduces (raises) total deadweight loss when the Lerner index (which indicates marginal deadweight loss) in that sector is greater (less) than a knockout weighted average of the Lerner indexes in all other sectors. This statement is in accord with intuition in light of the aforementioned description of the weights. As price falls in sector j, deadweight loss is reduced in that (distorted) sector as expenditures flow in. However, those expenditures necessarily flow out of other sectors that, in general, are also distorted. Hence, marginal deadweight loss rises in sectors

i \neq j

. The magnitude of that aggregate is given by the Lerner index in each other sector, weighted by the extent of the outflow from the corresponding sector.

Corollary 2.2 states that, in a world in which there are multiple sectors and general equilibrium effects—but the number of firms in all sectors n is held fixed—the welfare-maximizing prescription is to strengthen (weaken) competition policy in sectors with relatively “high” (“low”) markups. The use of the quotation marks reflects that there is not, in general, a single value of the Lerner index,

L^{*}

, such that it is optimal to raise (lower) γ^j whenever

L^{j}

is greater (less) than

L^{*}

. The knockout weighted-average Lerner index omits a different Lerner index that depends on the sector under consideration, and, moreover, the αⁱ weights, defined in expression (20), depend on which sector is targeted.

It was convenient to develop the analysis of multisector deadweight loss for the version of the competition policy experiment in which we fixed nⁱ for all i. Returning to the experiment that was the subject of proposition 2, in which free entry was allowed (only) in sector j, the overall welfare effect of competition policy also included the effect of induced exit in reducing variety in sector j. The first term of that condition would yield precisely the same expression in terms of the comparison of Lerner indexes, the only adjustment being that, in defining the αⁱ, the derivatives of the xⁱ would reflect the corresponding policy experiment (allowing free entry in sector j).

Reflection on corollary 2.2’s condition—which refers to the case in which the number of firms is held fixed in all sectors—provides another important insight. Specifically, it is suggestive of Lerner’s (1934) important claim (largely neglected in industrial organization economics) that the level of the markups in an economy is irrelevant; more precisely, if all markups are in the same proportion, there is no distortion.¹⁷ To prove this point in the present model (and, as per corollary 2.2, with entry subsidies that hold the number of firms in all sectors constant), it suffices to demonstrate that the weights in the other sectors sum to 1, that is,

\sum_{i \neq j} α^{i} = 1

when

ℒ^{i} = ℒ

for all i. That is done in the appendix, which establishes another corollary.

Corollary 2.3.

In the multisector model, when there is also a subsidy on entry set to keep the number of firms in all sectors (including sector j) constant and, moreover, if the Lerner indexes in all sectors are equal (i.e., there exists $L$ such that $ℒ^{i} = ℒ$ for all i), then strengthening competition policy (raising γ^j) in sector j has no effect on social welfare—that is, ${(d U / d γ^{j}) |}_{n} = 0$ .

B. Free Entry and Exit in All Sectors

Let us now move to the second stage of the analysis: general equilibrium with free entry and exit in all sectors. The analysis of this case uses methods and formulations similar to those already employed, especially for proposition 2, which differed only in holding n^\j fixed using an entry subsidy. As we will see below, however, the results diverge in important ways. As a preview, note that most conventional effects are absent. After all, gains (losses) to consumers are offset by losses (gains) to firms in the first instance, the representative individual’s reallocation of consumption across sectors has no direct effect on utility as a result of the envelope condition, and all firms have zero profits in equilibrium when there is free entry. Therefore, welfare impacts arise primarily as a consequence of pecuniary externalities due to general equilibrium effects as well as firms’ entry and exit decisions.

The basic difference in the setup is that we now omit the subsidy from the entry condition in all sectors (so there is free entry and exit) and, correspondingly, from the budget constraint. We are back to the budget constraint ofand the free-entry condition in every sector ofThe appendix derives the following proposition.

Proposition 3.

In the multisector model with free entry in all sectors, the effect of strengthening competition policy (raising γ^j) in sector j on social welfare is given by

$\begin{matrix} (23) & \frac{d U}{d γ^{j}} = \sum_{i = 1}^{I} [(p^{i} - \frac{d c^{i}}{d x^{i}}) λ n^{i} \frac{d x^{i}}{d γ^{j}} + (\frac{X_{2}^{i}}{X_{1}^{i}} - \frac{x^{i}}{n^{i}}) λ p^{i} n^{i} \frac{d n^{i}}{d γ^{j}}] . \end{matrix}$

This condition bears superficial resemblance to prior results. One difference throughout is that each derivative here takes nothing as fixed and hence is the full derivative in general equilibrium with free entry. As explained in connection with proposition 2, in a basic case (in which goods in all sectors $i \neq j$ are substitutes in the relevant sense with those in sector j), the general equilibrium price effects in other sectors (p^\j falls) dampen effects in sector j. Now we have, in addition, that the reduction in the number of firms in other sectors (n^\j also falls, because of the reduction in demand) mitigates this dampening effect (it mitigates the fall in price and also reduces variety, both of which make expenditures in other sectors less attractive), so the net, overall offset in sector j is attenuated relative to what it was before.

The two types of effects, which these derivatives on the right-hand side of expression (23) weight, are as before. The sum of the first terms across sectors indicates the overall change in deadweight loss, which could be interpreted in terms of how the weighted average of the Lerner indexes,

L^{i}

, in all sectors

i \neq j

compares to

L^{j}

(where now the αⁱ weights in an analogue to expression [20] would instead be defined using the full derivatives, without the nⁱ being fixed). The sum of the second terms across sectors indicates the overall change in utility due to changes in the number of varieties in all sectors. In our simple case, variety falls in every sector.

Alternatively, one can interpret this expression sector by sector, noting that the two terms for each sector i are the same as the two terms in proposition 1’s expression (8) for the one-sector model of sector j (except for the derivatives being for a different policy experiment). Hence, we can think of first determining the net welfare effect in each sector due to effects in that sector on deadweight loss and on variety and then summing these effects across all sectors—or, perhaps, as with corollary 2.2, we could compare the net effect in the targeted sector j with the total effects in all other sectors.

Such interpretations, however, are potentially misleading. One must carefully account for the fact that the pertinent derivatives now embody all of the general equilibrium effects with free entry. Moreover, even though the expressions for sector j and sectors $i \neq j$ are the same, they are not in substance symmetric, particularly regarding the first (deadweight loss) term in the summation. We have already encountered this point with regard to proposition 2 and its corollaries: increasing γ^j, in a typical case that has been the focus of prior discussion, causes the representative individual’s expenditures to flow into sector j but out of sectors $i \neq j$ . When n^\j was held constant, this meant (in our benchmark case) that ${(d x^{i} / d γ^{j}) |}_{n} < 0$ for $i \neq j$ , so that the outflows from sectors $i \neq j$ caused deadweight loss to rise. However, with free entry and exit, it is possible—and in important cases true—that $d x^{i} / d γ^{j} > 0$ for some or all $i \neq j$ , so that outflows from some or all sectors $i \neq j$ cause deadweight loss to fall.

A sharp illustration that more broadly illuminates the interpretation of proposition 3 involves the case of homogeneous goods. Suppose that, for some sector

k \neq j

, goods are homogeneous. As shown after proposition 1 (see expression [9]), this implies that

X_{2}^{k} / X_{1}^{k} = x^{k} / n^{k}

, so the second term for sector k equals zero. It might appear from our earlier analysis that this leaves a positive first term, indicating a rise in deadweight loss due to expenditures flowing out of sector k, which we are assuming is a sector in which price exceeds marginal cost. However, in typical cases—when allowing for free entry and exit in sector k—the outflow from sector k instead causes overall deadweight loss to fall, and to an extent that is reflected by this first term, but with

d x^{k} / d γ^{j} > 0

To see why, suppose further that, for this homogeneous-goods sector,

c^{k} (x^{k}) = F^{k} + φ^{k} x^{k}

: that is, there is a fixed cost and constant marginal cost for each firm in sector k. Taking again the case in which raising γ^j reduces expenditures in sector k, we have a lower price, p^k. In addition to inducing exit—which has no direct effect on welfare because goods in sector k are homogeneous, implying that the second term in expression (23) for sector k equals zero—it must also be true that average cost falls: the free-entry condition,

p^{k} x^{k} - c^{k} (x^{k}) = 0

, implies that

p^{k} = c^{k} (x^{k}) / x^{k}

, and, as mentioned, p^k falls. That, in turn, implies that

d x^{k} / d γ^{j} > 0

Taken together, free entry has, in a sense, reversed the result from proposition 2 with regard to sector k. When n^k was held fixed, the reduction in expenditures in sector k simply reduced sales that had been made at a price in excess of marginal cost, so deadweight loss rose. Now, with free entry and exit, we have exit in sector k, which involves a savings in fixed costs. Because firms (which are excessive in number with homogenous goods) that still operate now produce more output, production is overall more efficient in this sector. Note in particular that the greater the excess of price over marginal cost (the larger is $p^{k} - d c^{k} / d x^{k}$ ), ceteris paribus, the greater was the excessive incentive for entry. Here, the excess of price over marginal cost does not translate into profits because they are fully dissipated by entry. To sum up the welfare effects associated with sector k in the case of homogeneous goods: the movement of expenditures from sector k to sector j leaves the representative individual indifferent, as a consequence of the envelope condition; leaves firms in sector k indifferent because they earn zero profits regardless, as a consequence of free entry; and avoids wasted resources associated with fixed costs in sector k, as a consequence of induced exit. As per the preceding discussion, this latter effect, at the margin, is indicated by the excess of price over marginal cost in sector k because that profit margin is precisely what induces entry and is consumed, in terms of fixed costs, in the process.

One way to drive home the lesson from the homogeneous-goods case is to state this corollary.

Corollary 3.1.

In the multisector model with free entry in all sectors, if the goods in each sector

i \neq j

are homogeneous, the effect of strengthening competition policy (raising γ^j) in sector j on social welfare is given by

$\begin{matrix} (24) & \frac{d U}{d γ^{j}} = \sum_{i = 1}^{I} (p^{i} - \frac{d c^{i}}{d x^{i}}) λ n^{i} \frac{d x^{i}}{d γ^{j}} + (\frac{X_{2}^{j}}{X_{1}^{j}} - \frac{x^{j}}{n^{j}}) λ p^{j} n^{j} \frac{d n^{j}}{d γ^{j}} . \end{matrix}$

Taking the concrete case in which goods in all sectors $i \neq j$ are substitutes for the goods in sector j and, moreover, have a cost function of the form in the preceding example, we have that all of the terms in the summation are positive, contributing to an increase in welfare. The fact that this expression is almost identical to expression (16) in proposition 2 yet has almost opposite implications drives home the point that the pertinent derivatives must be interpreted with care. It can be very misleading to examine a world that ignores effects that arise in general equilibrium with free entry.

To gain further intuition about homogeneous goods in sectors

i \neq j

, one can rewrite the corresponding terms in the summation by substituting for

d x^{i} / d γ^{j}

from a differentiated free-entry condition (22) to yield the following:

$\begin{matrix} (25) & \frac{d U}{d γ^{j}} = (p^{j} - \frac{d c^{j}}{d x^{j}}) λ n^{j} \frac{d x^{j}}{d γ^{j}} + (\frac{X_{2}^{j}}{X_{1}^{j}} - \frac{x^{j}}{n^{j}}) λ p^{j} n^{j} \frac{d n^{j}}{d γ^{j}} + \sum_{i \neq j} (- λ x^{i} n^{i} \frac{d p^{i}}{d γ^{j}}) . \end{matrix}$

The first two terms, for the targeted sector j, correspond to the full welfare effect in the one-sector model with an outside good, except, of course, that here we instead have multisector general equilibrium derivatives.

The third term sums the effects associated with the homogeneous-goods sectors $i \neq j$ . Each $x^{i} n^{i} (d p^{i} / d γ^{j})$ constitutes the fall in expenditures in such a sector specifically as a consequence of the general equilibrium reduction in pⁱ (i.e., not including the more direct substitution effect of expenditures flowing out of this sector on account of the reduction in p^j). The preceding minus sign indicates a welfare gain, which λ converts from dollars to utils. The more direct effect from expenditures flowing out of each sector $i \neq j$ does not appear because it is associated with two equal and offsetting forces: a loss in sales that were at a price in excess of marginal cost (a welfare loss, via a reduction in profits) and the induced exit (saving the same amount of resources and hence an offsetting welfare gain). The remaining effect, measured by the third term, reflects that the general equilibrium reduction in price causes both a net null effect as producer surplus becomes consumer surplus and also additional induced exit that saves further resources (a social gain), which has a magnitude equal to the profit reduction from this price effect. As a consequence, the contrast with proposition 2 is even starker. When n^\j was held fixed, the outflow from sectors $i \neq j$ was an unmitigated welfare loss. With free entry and exit in the homogeneous-goods case, not only is this loss fully offset but also there is a further general equilibrium effect on price that induces further exit, yielding an overall welfare gain rather than a welfare loss.

Having shown that the expenditure outflows from other sectors are beneficial for sectors in which goods are homogeneous, let us now consider the matter more broadly. To stake out another reference point along the continuum of possibilities, consider the case in which, in every sector $i \neq j$ , the value of variety is such that the free-entry equilibrium number of firms just equals the socially optimal number of firms (in the sense discussed above, in connection with corollary 1.2). In this instance, results analogous to those in proposition 2 and corollaries 2.1 and 2.2 are fully restored, even though we now allow the number of firms to be endogenous, that is, we have free entry and exit in all sectors.

Corollary 3.2.

In the multisector model with free entry in all sectors, when competition policy γ^j is such that the resulting number of firms nⁱ in each sector

i \neq j

is socially optimal (in the free-entry equilibrium), the effect of strengthening competition policy (raising γ^j) in sector j on social welfare is given by

$\begin{matrix} (26) & \frac{d U}{d γ^{j}} = \sum_{i = 1}^{I} (p^{i} - \frac{d c^{i}}{d x^{i}}) λ n^{i} {\frac{d x^{i}}{d γ^{j}} |}_{n^{\ j}} + (\frac{X_{2}^{j}}{X_{1}^{j}} - \frac{x^{j}}{n^{j}}) λ p^{j} n^{j} {\frac{d n^{j}}{d γ^{j}} |}_{n^{\ j}}, \end{matrix}$

even though the number of firms in sectors $i \neq j$ , n^\j, is not held fixed.

The demonstration follows the logic of corollaries 1.1 and 1.2, while using the method for proposition 2 that is applicable to the multisector model. First, employ a marginal entry subsidy σⁱ set to keep nⁱ constant for all $i \neq j$ . That yields proposition 2 and its corollaries. Second, remove those marginal subsidies. If the nⁱ in sectors $i \neq j$ were all socially optimal, then the (marginal) removal of the σⁱ, which causes the respective nⁱ to fall, has no effect on welfare. Hence, the characterization for the welfare effect of the policy change is the same.

It is worth elaborating on the intuition for this special case. When we hold the nⁱ constant, we have the simple increase in deadweight loss as expenditures flow out of sectors

i \neq j

. Next, as with the homogeneous-goods case, we have exit, which saves production costs and provides an offset to the reduction in deadweight loss. Finally, however, when variety is valuable, that production cost savings from exit comes at some expense to the utility from variety that the exiting firms had provided. In this special case in which the number of firms in each sector

i \neq j

is optimal in the initial equilibrium, the latter two effects are equal (and opposite), so we are back to just the expression for the increase in deadweight loss (i.e., for sectors whose goods are substitutes with those in the targeted sector j).¹⁸

With the results from the special case of variety being optimal in equilibrium and from the case of homogeneous goods in mind, we can see more generally how effects in each sector

i \neq j

contribute to the overall impact on welfare of strengthening competition policy in sector j.¹⁹ For any sector

i \neq j

that is homogeneous, the outflow of expenditures (continuing to focus on the basic case of substitutes) raises welfare due to the general equilibrium effect on price in that sector. As the value of variety increases from zero, at some point the outflow will have no net welfare consequences: the effects associated with the two terms for the sector in proposition 3’s (expression [23]’s) summation will be of equal magnitude but opposite sign. (Note that if this condition held in every sector

i \neq j

, the conventional approach that confines analysis to the targeted sector j would then be valid, as long as the appropriate general equilibrium derivatives were employed.) From that point, as the value of variety increases still further, the outflows will increasingly reduce welfare. For the particular value of variety such that the initial equilibrium has the socially optimal number of firms, the naïve expression given by ignoring entry and exit is precisely correct (as long as one interprets the

d x^{i} / d γ^{j}

correctly). Past that point, the welfare loss from the outflow of expenditures is even greater than this.

Finally, note that the aggregate effects from sectors

i \neq j

will, in general, depend on which sector j is targeted (which γ^j is being raised). To be sure, if all sectors were fully symmetric with each other, the results would be the same for assessing competition policy in any sector. However, such symmetry does not hold even approximately in actual economies. Furthermore, the first term in proposition 3’s expression (23) is weighted by

d x^{i} / d γ^{j}

and the second term by

d n^{i} / d γ^{j}

, so the weight on each effect in each sector

i \neq j

depends on which sector j is targeted. This point is also clear from expression (20) for the αⁱ (adjusted for the present experiment). Relatedly, the sign of the effect of competition policy on welfare in the case examined in corollary 2.2 makes clear how, in addition to the weights, it also matters which sector j is targeted, because that determines which

L^{j}

is compared to the knockout weighted average of the

L^{i}

, for

i \neq j

(which excludes

L^{j}

from the summation).

In summary, even though the expressions for deadweight loss and variety are similar across all of these cases that differ regarding what (if anything) is held constant, because the pertinent derivatives reflect qualitatively different exercises, the relative magnitudes differ and the signs of the terms (notably, the deadweight-loss terms) can reverse. Just as proposition 2 and its corollaries show how moving from one sector to many sectors changes results dramatically, proposition 3 shows that taking into account free entry and exit in these other sectors produces notably different outcomes even from that more encompassing multisector benchmark.

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A. Holding n\j (or n) Fixed

B. Free Entry and Exit in All Sectors

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A. Holding n^\j (or n) Fixed