Investments in productivity and quality under monopolistic competition

Investments in productivity and quality under monopolistic competition Igor A. Bykadorov Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

1

Introduction

The cross-countries differences in productivity and quality are noticeable and allow for various explanations. Recent empirical papers on international trade show that (1) firms operating in bigger markets have lower markups (see e.g. Syverson (2007)); (2) firms tend to be larger in larger markets (see e.g. Campbell and Hopenhayn (2005)); (3) larger economies export higher volumes of each good, export a wider set of goods, and export higher-quality goods (see e.g. Hummels and Klenow (2005)); (4) within an industry there can be considerable firm’s heterogeneity: firms differ in efficiency, in exporting or not (associated with high/low efficiency), in wages (see review in Reddings (2011)); (5) investment decisions are positively correlated with export status of a firm (see e.g. Aw et al. (2008)). To enrich theoretical explanations of these and other empirical regularities we follow Dixit-StiglitzKrugman’s monopolistic competition model Krugman (1979) and its version with heterogeneous firms, suggested by Melitz (2003). Most of previous explanations are based on specific functional form of the utility function, namely constant elasticity of substitution (CES) function. This functional form was popular for a long time for its technical convenience, but its departure from reality was broadly criticized in the literature (see e.g. Ottaviano et al. (2002), Behrens and Murata (2007)). In particular (as one can see from Table 1), CES modelling makes prices and outputs independent from the market size, that contradict the regularities enumerated and thereby prevent explaining them. One of possibilities is to trying other specific utility functions: quadratic like Ottaviano et al. (2002) or exponential like Behrens and Murata (2007), but we find it more interesting to construct a general theory for unspecified functions, and get robust predictions linking the patterns of market outcomes with certain properties of utility and cost functions. To approach regularities (1)-(5) mentioned, this project models endogenous choice of technology in the diversified sector with free entry the spirit of Vives (2008) and some other papers on “endogenous innovations” discussed in our literature review. Mainly, these papers consider partial equilibrium in free-entry Cournot and Bertrand settings. Instead, we develop the general-utility monopolistic competition models from Zhelobodko et al. (2010) and Zhelobodko et al. (2011) towards endogenous choice of technology. Related general equilibrium of monopolistic competition appears more suitable for applications of such model of the market to international/interregional trade and growth, because here income effects do matter. We start with a basic monopolistic competition model of a closed economy with one diversified sector (subsequently extended to multi-sector economy). Primary question is the impact of the market size on equilibrium prices, outputs, number of firms in the industry, and most importantly – on investment in productivity. Under some version of preferences, these endogenous investments can be interpreted as investment in quality. Thereby we simultaneously study impact of the market size on productivity and quality. The findings amount to complete comparative statics of all equilibrium variables, classified in Table 1 into 3 main patterns and some sub-patterns. Our initial intuitive hypothesis was that when a firm sells to a bigger population (say, comparing Russia and China), it has more incentives to invest in decreasing its variable cost or in increasing its quality to exploit economies of scale. However, actual results show that 1

such positive correlation holds true only when utility from the varieties studied displays increasing “relative love for variety” (RLV) which is the Arrow-Pratt measure of concavity, known as relative risk aversion. In CES case RLV remains constant, so investments do not react to the market situation, and under the decreasing RLV we predict counter-intuitive negative correlation. It can be explained as follows: when utility displays increasing concavity, related demand curve appears so convex that the demand decrease (more competition in larger market) makes the monopolist to increase the price in response, decreasing the output. Output being tightly positively correlated with cost-reducing expenditure, this explains the paradox. Though we are not developing the trade model explicitly, it is easy to conclude from these results that when two or more countries integrate there markets, the economies of scale should work as under increase in the market size and each firm’s investment in productivity/quality increase in those sectors where preferences display increasing RLV (which is an empirical question) but not in others. Respective welfare gains can be as much important as the increase in variety of goods stressed by (Krugman (1979)). Similar but more delicate effect is the selection of best producers by larger markets a’la Melitz (2003), enforced by more active R&D by the strongest firms when firms are heterogenous. To find how such effects depend on the nature of preferences and costs, our project develops the model with firms heterogenous in their ability to invest in productivity/quality. Here the preliminary findings are as follows. Prices and outputs follow the same three patterns as in homogenous sector: bigger market makes prices decreasing under increasing RLV, increasing in the opposite case and does not affect prices/outputs under CES. Since outputs are tightly connected with innovations, this fact tells the direction of changes in investment in all 3 cases. New issue, specific only for heterogenous economy, is the question of composition of firms surviving in the market. Again, under CES nothing changes in this respect with the market size, under incresing RLV the worst firms exit and average productivity increases, under decreasing RLV the opposite holds. Similar patterns are found in Zhelobodko et al. (2011) for an economy without the choice of technology, now we show them robust in our richer context. Moreover, under our choice of the technology the gap between bad and good firms widens under growing market, because stronger firms has bigger output and more incentives for innovations. Additional questions addressed relate to the influence of the technology properties on the market outcomes. Table 1 shows that curvature of the cost reduction function determines the sign of changes in the equilibrium markup and some other details. For policy-making our topic may be interesting not only because of new understanding of gains from trade, technological changes in response to trade liberalization and best firm selection. For modernization and active industrial policy practiced in some countries including Russia, it can be interesting, what equilibrium outcome in the sectors of different kind follow from any stimulating measures like tax reductions for R&D. If touching this topic, we should compare social optimum and equilibria in the spirit of Dixit and Stiglitz Dixit and Stiglitz (1977) and find whether they become closer under governmental stimulation. Summarizing, the goal of the project is the theoretical study of investments in productivity and quality under assumptions relevant for trade and development issues: monopolistic competition in general equilibrium with variable elasticity of substitution. The sequel presents (1) The literature review; (2) The homogenous model; (3) The comparative statics in homogenous case; (4) The heterogenous model.

2

Literature review

There are several related papers devoted to international trade that study how liberalization of trade (decrease in trade costs) and countries size influence the investment decisions, the size of the firms, prices, and the number of firms in the industry. Here rather close to ours is the paper Tanaka (1995) that considers the Salop’s model of circular city and influence of trade liberalization on quality. The main finding is that trade liberalization decreases the product quality. Another paper Fan (2005) considers competition in the market of intermediate goods and studies influence of the country size on the quality of goods under 2

specific functional forms of preferences and production functions. Several papers are devoted to influence of trade on technology adoption in the case of heterogeneous firms. In Bustos (2011) the classical Melitz model is supplemented by endogenous choice of technology. The set of available technologies is discrete: low marginal costs with high fixed costs or the opposite pattern. The finding is that trade liberalization increases the share of firms using high-investment technology. In Yeaple (2005) the set of technologies is also discrete. Every producer chooses not only technology, but also the quality of labour used (there are two types of the labour, skilled and unskilled). Then exporting firms are larger, employ more advanced technologies, pay higher wages, are more productive, and a reduction in trade frictions can induce firms to switch technologies, leading to an expansion of trade volumes. The main departures of this project from the papers described are: (1) our utility function is of general, unspecified form; (2) our set of technologies is continuous; (3) we study both homogenous and heterogenous industries. To mention analogues in modelling technique, our departure from Zhelobodko et al. (2011) studying the influence of the market size, and from Dhingra and Morrow (2011) studying social optimality, is only that in both papers the technology is fixed. Finally, the closest analogue of our project in questions and answers is Vives (2008). Unlike our general equilibrium and monopolistic competition, he considers partial equilibrium in free-entry oligopoly setting with endogenous technology and general (unspecified) preferences expressed through the demand functions. The main finding, obtained under some complicated “regularity condition” in terms of demand functions and number of varieties, is that investments in productivity increase with the market size. Strategic interactions (present in oligopoly formally but inessential for our question) make all expressions cumbersome, that prevents formulating such condition for increasing investment in clear and interpretable form or find necessary and sufficient conditions (like we do). Instead, our more tractable model allows for: (1) complete comparative statics; (2) comparison with social optimum; (3) studying heterogenous industry, i.e., much richer analysis. To conclude, we answer the main objection to our previous EERC application: why to abandon CESfunction is important for our goal? We have already explained it to a large extend, but let us support our view by two typical critics of CES from two different grounds: (1) Realism: “Although the CES utility model has been the workhorse of the vast majority of contributions based on monopolistic competition, it is fair to say that this model suffers from several major drawbacks. First, individual preferences lack flexibility since the elasticity of substitution is constant and the same across varieties. Second, the market outcome is not directly affected by the entry of new firms. In particular, markups and prices are independent of the number of competitors. This runs against empirical evidence, which shows that firms operating in bigger markets have lower markups” (Syverson (2007)). (2) Robustness: “As a theorist, I’m not used to relying on particular functional forms for results. These are usually called ‘examples’, not ‘theorems’. If one talks with people working in general equilibrium theory (I do this as an antidote to geographers), they think that this type of model is pretty silly. How can we draw general conclusions about urban policy from these models if the conclusions change when the utility functions or functional form of transport cost change?” (Berliand (2006), page 108).

3

Basic model: homogenous sector with endogenous technology

We model now the closed economy with endogenous investments in technology, like in Vives (2008) but in general equilibrium and monopolistic competition. Departuring from the standard Dixit-Stiglitz model Dixit and Stiglitz (1977) we generalize it in two respects: enable investments in productivity and allow for general VES utility function. We consider one production factor interpreted as labour and a sector using this labour (we show later on that the same model can also describe a sector within a multi-sector economy). There are two types of agents: big number L of identical consumers/workers and an endogenous interval [0, N ] of identical firms producing varieties of some “differentiated good.” Our goal is to show how different preferences for variety determine different outcomes of competition.

3

3.1

Consumer

Each consumer maximizes her utility under the budget constraint through choosing an infinite-dimensional consumption vector (integrable function) X : [0, N ] → R+ . All consumers behave symmetrically, so consumer’s index is redundant. As in Krugman (1979), Vives (1999) and Zhelobodko et al. (2011) , the preferences are described by unspecified additive-separable utility function maximized under the budget constraint: ˆ X

N

u(xi )di,

max 0

ˆ

ˆ

pi xi di ≤ w +

s.t.

N

πi di

N

0

0

L

= 1.

(1)

Here N is the endogenous mass of firms or the scope (the interval) of varieties. Scalar xi is consumption of i − th variety by any consumer and X = (xi )i≤N . Bernulli utility function u(·) satisfies u(0) = 0, u0 (xi ) > 0, u00 (xi ) < 0, ru0 (xi ) ≡ −

xi u000 (xi ) < 2, u00 (xi )

i.e., it is everywhere increasing, strictly concave and its concavity is restricted to guarantee global strict concavity of producer’s profit, which ensures equilibria symmetry and uniqueness (see Zhelobodko et al. (2010)). For our results we need not additional restrictions like homothety or CES. Instead, as in Krugman (1979), Vives (1999) and Zhelobodko et al. (2011), our classification of markets uses the Arrow-Pratt measure of concavity defined for any function g as rg (z) = −

zg 00 (z) . g 0 (z)

Such characteristic of utility is known as “relative risk aversion” in the risk theory and is named “relative love for variety” (RLV) in monopolistic competition (Zhelobodko et al. (2011)). Actually, the above consumer’s program is formally equivalent to the portfolio program. So, wellknown fact apply: concave u results in balanced mixture of varieties, which becomes symmetric (xi = xj ) under identical prices of varieties. Under symmetry, ru (z) = 1/σ(z) is the inverse to the elasticity of substitution \sigma among varieties and to the elasticity of demand for each variety (standardly). This explains its important role for market outcomes. In particular, for CES utility u = xρ , RLV is constant: ru (z) ≡ 1 − ρ and therefore equilibrium prices and outputs remain indifferent to the market size, that shadows the links which we would like to uncover. In the budget constraint, pi is the price and E is the consumer’s fixed income or labor supply equal to expenditure for all varieties. Profits are neglected, becoming zero at the equilibrium. With Lagrange multiplier λ, the first order condition (FOC) or the inverse demand for i − th variety is u0 (xi ) . (2) λ Obviously, p decreases in consumption xi and higher marginal utility of income λ also leads to a decrease in demand. p(xi , λ) =

3.2

Producer

On the supply side, we standardly assume one-to-one correspondence: each variety is produced by one firm that produces a single variety. However, unlike classical setting, each producer chooses the technology level. Namely, if she spends f units of labour as fixed costs, then total costs of producing y units of output is c (f ) y + f units of the labour. It is natural to assume decreasing and convex costs: c0 (f ) < 0, c00 (f ) > 0,

4

lim c(f ) > c0 > 0.

f →∞

Thereby more expensive factory would have smaller marginal costs, investment in productivity shows decreasing returns to scale and cost cannot fade to zero, being higher than some positive c0 .1 Using the inverse demand function p(xs , λ) from (2), the profit maximization of s − th producer can be formulated as2  0  u (xs ) πs (xs , fs , λ) = (p(xs , λ) − c(fs )) Lxs − fs = − c(fs ) Lxs − fs → max . xs ≥0,f ≥0 λ Under our assumption about continuum of producers, it is standard to prove that each producer s has a negligible effect on the whole market, i.e. the Lagrange multiplier λ can be treated as (positive) constant by each s. This Lagrange multiplier is the natural aggregate statistic of the market: the bigger is marginal utility of income λ, the smaller is the demand and therefore smaller the profit of producers. Thereby this λ can be interpreted as the degree of competition among the producers of differentiated goods, like a price index in standard Dixit-Stiglitz model (see Zhelobodko et al. (2011)). Profit maximization w.r.t. supply x and investment f yields FOC: u00 (xs )xs + u0 (xs ) − c(fs ) = 0, λ

(3)

c0 (fs )Lxs + 1 = 0.

(4)

u000 (xs )xs + 2u00 (xs ) < 0 ⇔ ru0 (xs ) < 2,

(5)

These equations are valid under SOC:

−


2 (u000 (xs )xs + 2u00 (xs )) c00 (fs )xs − c0 (fs ) > 0. λ

(6)

Since for each producer these conditions are the same, further we consider only symmetric equilibria and denote xs = x, fs = f ∀s.

3.3

Equilibrium

Entry. Standardly, we assume that firms freely enter the market until their profit remains positive, that implies zero-profit condition u0 (x) f − c(f ) = . λ Lx Labour balance. Under symmetric equilibrium (fi = f, xi = x) labour market clearing means ˆ

(7)

N

(c(fi )xi L + fi ) di = N (c(f )xL + f ) = L.

(8)

0

Summarizing, we define symmetric equilibrium as a bundle (x∗ , p∗ , λ∗ , f ∗ , N ∗ ) satisfying: 1) rationality in consumption (2); 2) rationality in production (3)-(4) and (5)-(6); 3) free entry condition (7) and labour balance (8). It is straightforward to exclude λ and rewrite the equilibrium equations in terms of the Arrow-Pratt measure of concavity rg (x): 1

This assumption is needed for nonnegative cost function, Second Order Condition (SOC) and existence of maximum in profit maximization. 2 Standardly, maximization of monopolistic profit w.r.t. price or quantity gives same results.

5

Proposition 1. Equilibrium consumption/investment couple (x∗ , f ∗ ) in one-sector model with endogenous technology is the solution to the system ru (x) x f = , 1 − ru (x) Lc(f )

(9)

(1 − rln c (f ) + rc (f )) (1 − ru (x)) = 1,

(10)

under the conditions (2 − ru0 (x)) rc (f ) > 1.

ru (x) < 1, Subsequently equilibrium mass of firms

N∗

is determined by equation N=

3.4

(11)

L . c(f )xL + f

(12)

Another interpretation: quality

Consider now similar setting but for producer choosing investment in quality instead of investment in productivity. We shall see that these two settings are equivalent, generate ´ N the same equilibrium equations. Let the utility of each consumer take the form (cf. Tirole (1988)) 0 u(qi xi )di, where qi is the quality level of i − th variety and xi is related consumption volume, so that zi = qi xi is the satisfaction volume. Then, from FOC similar to previous one, the inverse demand function for i − th variety with quality qi is p(xi , qi , λ) =

qi u0 (qi xi ) . λ

Under output yi = Lxi the cost function of i − th producer is c˜(qi )yi + f˜(qi ), where, as before, c˜(qi ), f˜(qi ) are marginal and fixed costs respectively. It is natural to assume that both derivatives with respect to qi are positive, c˜0 (qi ) > 0, f˜0 (qi ) > 0 because to increase the quality of the production the producer should spend more labour per unit and also use more expensive technology. Thus, the profit maximization of i − th producer amounts to  0  qi u (qi xi ) − c˜(qi ) Lxi − f˜(qi ) → max . xi ≥0,qi ≥0 λ Let us introduce auxiliary variables fi = f˜(qi ). Due to monotonicity of function f˜, there exists one-toone correspondence between quality of production qi and the value of fixed costs fi : qi = qi (fi ) = f˜−1 (fi ). Besides, we define c˜(qi (fi )) , zi = qi (fi )xi . qi (fi ) Then the problem of i − th producer can be rewritten in the following equivalent form:  0  u (zi ) − c (fi ) Lzi − fi → max . zi ≥0,fi ≥0 λ c (fi ) =

Obviously, this program of choosing quality appears mathematically equivalent to the program with endogenous productivity introduced previously. Only quantity consumed is measured now in satisfaction level z. Therefore, our further study of comparative statics relates both to endogenous productivity or endogenous quality; these are two similar manifestations of endogenous technology. 6

4

Preliminary comparative statics of homogenous market

Market size. The primary goal is to study the impact on productivity/quality induced by the increasing market size L. The latter can be interpreted as markets’ integration or population growth, or comparison between cities different in size. We study changes in all equilibrium variables: prices p, firm size Lx and mass of firms N , investment of each firm f , and total investments in the economy (N f ). Our equilibrium equations determine (x, f, N ) as an implicit function of L. After total differentiation and long rearrangements we obtain the elasticities of main variables (in terms of concavity of basic functions) and find the elasticities signs. These allow to classify different market outcomes according to increasing/decreasing concavity measure ru (x) (Increasing Elasticity of the Inverse Demand – IEID or Decreasing Elasticity of the Inverse Demand – DEID) and concavity of ln c. Proposition 2. Elasticities of the equilibrium variables w.r.t. market size L in one-sector economy are Ex = Ef =

ru0 x , ((2 − ru0 ) rc − 1) ru

(1 − rln c ) (1 − ru ) , (2 − ru0 ) rc − 1 EN f =

Ep = −

ELx =

rc ru0 x ((2 − ru0 ) rc − 1) ru

(1 − rln c )2 (1 − ru )2 1 + + ru , ((2 − ru0 ) rc − 1) rc rc

rc ru0 x , (2 − ru0 ) rc − 1

E p−c = p

EN = ru −

(1 − rln c ) (1 − ru )2 (2 − ru0 ) rc − 1

(1 − rln c ) ru0 x , (2 − ru0 ) rc − 1

and their signs satisfy classification Table 1:

Ex ELx Ef EN f EN Ep E p−c

DEID ru0 < 0 rln c ≤ 1 rln c > 1 6∃ − 6∃ − 6∃ − 6∃ + 6∃ + 6∃ + 6∃ +

CES ru0 = 0 rln c 6= 1 − 0 0 + + 0 0

rln c > 1 − + + + + − −

IEID ru0 > 0 rln c = 1 0 + + + + − 0

rln c < 1 + + + + ? − +

p

In Table 1, Ex is the elasticity of individual consumption of each variety, Ef - is the elasticity of investment per firm, EN - the elasticity of mass of firms, ELx - the elasticity of total output of one variety, EN f - the elasticity of total investment, Ep - the elasticity of price, E p−c - the elasticity of mark-up (we p

drop the proofs). Commenting, we would say that generally these results remindconclusions in Vives (2008) obtained for oligopolistic model. However, our table shows in more details the influence of market size on all variables; Vives mainly focused on the influence of market size on investment. As we can see from the table, there can be five different types of the equilibria. For increasing/decreasing investments, utility characteristic ru0 is the criterion. Namely, standard CES case (isoelastic demand) is the borderline between markets with decreasing (DEID) or increasing (IEID) elasticity of inverse demand. Main finding is that DEID class show decreasing investments whereas under IEID individual investments increase in response to growing market and investment is always positively correlated with the size of the firm. The latter fact has clear interpretation: bigger output planned motivate higher cost-reducing investment. But more intriguing is the fact that bigger output is not guaranteed for larger market. Decrease of both Lx and investment f happens under DEID because the mass of firms grows too fastly, excessive 7

competition makes the output shrinking, outweighing the market-size motive to invest in marginal productivity/quality. Nevertheless, total economy investment N f always grows because growing mass of firms dominates even when f decreases. As to prices in this model, they respond to market size in the same way as under exogenous technology (f, c) Zhelobodko et al. (2010); there prices also increased under DEID but decreased under IEID preferences. This discrepancy got a clear explanation. In all cases increasing market has positive effect on profits of existing firms, that invites new firms into the industry, so N increases. Then growing competition pushes marginal utility of income λ up, so consumption x of each variety decreases. Paradoxically, under IEID too convex inverse demand function (shifting down with growth of λ) pushes the price up in response. Such regularity generally remains valid under our exogenous technology, though in essence it somehow combines comparative statics w.r.t. market size L with comparative statics in costs c and f . Table 1 shows that market size effects on price generally prevail, though consumption and mass of firms may behave in a new fashion in the exotic case (ru0 > 0, rln c < 1). Interestingly, the nature of the cost function turns out to be the criterion only for increasing/decreasing individual consumption of each variety and related variable called markup M = p−c p . Under sufficiently big elasticity of cost to investment expressed in condition rln c (f ) > 1 individual consumption decreases, otherwise it does not. The first column of the table was proved to be empty; equilibria here inexist (Table 1 also do not mention the case ru0 = 0, rln c = 1 where equilibria are indeterminate). Existence of equilibria in the last two columns is an open question, whereas numerical examples for the middle columns are already constructed and confirm valid equilibria with all effects described by Table 1. Extensions. In the future research we plan to expand these findings for homogenous closed economy in several respects. First, we supplement them by more examples and economic interpretations, study the comparative statics w.r.t. income w and parameterized cost-reducing technology c(.). The hypothesis is that income has no influence on prices in firms’ outputs, but influences number of varieties positively. Supposingly, more productive technology should bring higher investment. But more delicate question is social optimality: should the government stimulate higher investment or not? Second, we can normalize income to 1, introduce a numararie good x0 , and study two-sector model (encompassing partial equilibrium setting as a special case): ˆ N ˆ N max U ( u(xi )di, x0 ) s.t. pi xi di + x0 ≤ 1. X,x0

0

0

Here our expenditure w for all varieties becomes a variable dependent upon the upper-tier utility function U and price situation. However, our findings concerning prices, outputs and investments should remain valid in such two-sector (or multi-sector) model, being independent from the expenditure. Only the mass of firm can display somewhat new patterns depending upon how expenditure w behaves. When U shows strong complimentarity between the sectors, the budget share should remain weakly affected by the market ˆ size, and our findings about mass of firms may remain approximately true. In contrast, when N

u(xi )di)+x0 , with some concave function V as in quasi-linear setting by Vives (2008), the shift

U =V( 0

in budget share towards differentiated sector can be very substantial, that may explain the main source of the discrepancy in effects between our model and Vives. Understanding this discrepancy is a task. Third, for the version of the model with endogenous quality we plan to supplement the classification xL table with the behavior of more variables: q(f ) - the size of the firm adjusted for quality, pq(f )- the price of each variety adjusted for quality. To simplify analysis, we can assume, for example, linear dependence between quality and fixed costs in the form f˜(q) = aq .

5

Heterogeneous firms

We plan to expand our analysis onto the case of firms with heterogeneous cost a’la MelitzMelitz (2003), but with variable elasticity of substitution and endogenous investment.

8

As before, the main question is the influence of the market size and other parameters on firms’ investments in productivity/quality, behavior of prices and outputs, but now not only average variables matter, but their distribution among firms also. Is it true that more efficient firms make more investments to decrease their costs? Is it true that more efficient firms produce better goods? How these activities (investments and the quality of output goods) change when the market size increases as the result of market integration? We have mentioned some papers were these questions have been considered for CES-function, that does not allow for natural price effects. Our novelty is the absence of the specific functional forms. As in homogeneous version of the model, we expect here various patterns of market reactions subject to preferences and cost functions. The model includes one sector. On the production side, there are N firms that can enter the market. Each of them produces one variety. A firm’s individual marginal cost function depends on firm’s identity i and investments f . We assume that firms are ordered by increasing of marginal costs: c(f, i) > c(f, j) ⇐⇒ i > j. Firms appear in the market randomly, and the relative frequency is determined by distribution function Γ(i) on [0, ∞) with probability density γ(i) . Prior to entering market, each firm does not know its actual total costs c(f, i)y + f + fe , where y is output, fe is expenditure to study expected profit (it is fixed and known constant; for example, the cost to order business plan). The expected profit of firm i estimates by the profits of the firms already entered market. Thus the condition for firm i to enter market is that the expected profit (or, which is the same, average profit of the firms already entered market) must be non less than the cost to order business plan. Note that all firms are unsure, to enter market or not, are in the identical situation since each of them does not know her marginal costs c(f, ·) . Though the varieties are different from consumer’s point of view, the firms with equal costs will charge equal prices. Thus the firms with the equal costs type i , i.e. equal marginal costs c(f, i) , can be identified. So the problem of representative consumer is ´ ˆi ´ ˆi ˆ s.t. U = 0 u(xi )N dΓ(i) → maxxi 0 pi xi N dΓ(i) = w ≡ 1,where [0, i] is the set of different (available on market) varieties, while xi is individual consumption of the variety of type i ∈ [0, ˆi] . In the short run, we assume that the cutoff number ˆi is exogenous and such that firms having a marginal ˆ earn zero profits. All firms, operating in the market, thus have a type smaller than cost equal to c(f, i) ˆ or equal to i, while firms having a type exceeding ˆi do not produce. Hence, the mass of operating firms is given by N Γ(ˆi) ≤ N . In the long run, ˆi is endogenously determined by free entry.

5.1

The equilibrium

The inverse demand function now becomes pi (xi ) = u0 (xi )/λ.

(13)

The operating profit of a type i-firm is given by  0  u (xi ) − c(fi , i) Lxi − fi . π e(i, xi , fi , λ, L) = λ The operating producer maximizes this operating profit w.r.t. xi and fi . Thus in optimum point we obtain the dependence xi and fi as functions of λ and L. By substituting these xi (λ, L) and fi (λ, L) in π e(i, xi , fi , λ, L), we obtain for every producer of type i the profit π(i, λ, L) as function of λ and L. Using this function, it is easy to obtain the condition for “boundary” producer, i.e. such producer of type ˆi, whose operating profit equals to zero: π(ˆi, λ, L) = 0. 9

This notation ˆi allows us to formulate the described above the condition for firm to enter market, “the expected profit (or, which is the same, average profit of the firms already entered market) must be non less than the cost to order business plan”, this way: ˆ

ˆi

π(i, λ, L)dΓ(i) = fe . 0

Thus we have the system of two equations with two variables, ˆi and λ. Remark that, by implicit function theorem, the behaviour of profit π(i, λ, L) is as follows: it decreases w.r.t. i (let us recall that, due to the ordering mentioned above, the increasing of i leads to the increasing of marginal costs c(·, i)), decreases w.r.t.λ, increases w.r.t. L. Therefore, considering second equation, it is easy to see that λ increases when L increases, i.e. ∂λ > 0. ∂L This property allows us to make the research on comparative statics of equilibrium values w.r.t. market size L.

Project timetable January-May: Additional study of literature on the project topic, finalizing the results on one-sector closed model, then starting study the heterogeneous case, writing interim report. July-Novembre: finalizing study the heterogeneous case, writing a paper and the final report.

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