Keywords: Private information, longterm financial contracts, exporter dynamics, international trade, financial intermediation
Abstract:
There is growing empirical support for the conjecture that access to credit is an important determinant of firms' export decisions. We study a multicountry general equilibrium economy in which entrepreneurs and lenders engage in longterm credit relationships. Financial constraints arise in consequence of financials contracts that are optimal under private information. Consistent with empirical regularities, as firm age and size increase, the model implies decreasing mean and variance of firm growth and increasing firm survival. Exporters are larger, their survival in international markets increases with the time spent exporting, and the sales of older exporters are larger and more stable.
Empirical studies of firms and industries across countries reveal that most exports are produced by a small number of very large firms. In the United States, for example, less than 5 percent of all firms exported some of their production in 2010. More than 97 percent of these exporters were small and mediumsized firms (500 employees or less), which only accounted for about 34 percent of total exports^{4} In contrast to large exporters, there is substantial yeartoyear transition in and out of export markets for smaller firms. Furthermore, new exporters are typically small relative to the average exporter and frequently stop exporting after one year. Continuing exporters are more likely to expand rapidly in export markets (Bernard & Jensen, 2004).
Does access to credit affect a firm's decision to export? How does the nature of the credit relationship between banks and firms shape the growth of new exporters? Small and mediumsized firms tend to be more reliant on external financing to operate, which is mostly debt as equity is typically owned by proprietors. There is substantial empirical evidence that the financial conditions faced by small and young firms play an important role in shaping their growth, which is widely interpreted as indirect evidence of frictional financial markets.^{5} Hysteresis in export markets suggests the presence of a fixed cost of entry (e.g.,(Das et al., 2007), (Paravisini et al., 2011)), thereby suggesting that participating in international trade requires a greater access to working capital financing. This in turn could imply that the export decisions of small and mediumsized firm to export is sensitive to the availability of credit.(Minetti & Zhu, 2011) finds evidence supporting this hypothesis by showing that the probability of exporting and the intensity of export is significantly lower for credit rationed firms in Italy.^{6}
Economic theory posits that small and young firms are generally more opaque to external scrutiny. This opaqueness creates an informational asymmetry between lenders and entrepreneurs leading to adverse selection and moral hazard problems. As a result, competitive banks may ration the supply of credit to young and small firms instead of increasing the price of credit to clear the market (e.g., (Stiglitz & Weiss, 1981)). Banks may also find it profitable to reduce the private information through repeated interaction and monitoring of firms. For example, banks may offer selfenforcing contracts that facilitate insurance, such as continued extension of credit during bad times (Allen & Gale, 1999). Such contracts could also induce young entrepreneurs to establish their trustworthiness and the viability of their business by servicing their loans over time (e.g., (Diamond, 1984,1991), and (Rajan, 1992)).^{7} In line with the above argument, (Clementi & Hopenhayn, 2006) show that longterm financial contracts that are optimal given such informational asymmetry can help account for some of the empirical regularities firm dynamics  firm entry and exit, and the mean and variance of firm growth.
Economic theory is relatively uninformative regarding how private information and longterm credit relationships may affect firms' export decisions and shape their growth in international markets. We propose to fill this gap by studying a general equilibrium multicountry model economy in which entrepreneurs and lenders enter into multiperiod credit relationships subject to an informational asymmetry. In the model, atomistic entrepreneurs are born with a blueprint to start a longlived firm that produces an intermediate input. Starting and operating a firm requires paying an initial fixed cost, which is sunk, and working resources each period. Once started, a firm may ship some of its production overseas by paying additional fixed and variable trade costs. We assume that new entrepreneurs do not have any wealth with which to fund their firm, and must seek financing from competitive financial intermediaries. Financial frictions arise because financial intermediaries cannot directly observe the revenue generated by the firms they are financing, and must instead rely on the reports from creditor entrepreneurs.
With limited liability, entrepreneurs have an incentive to underreport the revenues generated by their firm to avoid repaying their debt. Financial intermediaries mitigate this moral hazard by offering new entrepreneurs a longterm financial contract designed to induce truthful reporting. The contractual arrangement in our model is closely related to that in (Clementi & Hopenhayn, 2006), and a financing constraint emerges as an outcome of the optimal contract. New firms operate below their efficient level, and the financing constraint is relaxed as the entrepreneur's claim to future cashflows increases. Wellperforming firms  those which service their debt for a sufficiently long time  gain access to sufficient credit to pay the export costs. New exporters are less financially constrained than domestic firms, but their growth continues to depend on their performance each period until they become fully unconstrained.
We show that there exists a stationary equilibrium in which firms differ by their capital structure. Even if the aggregate quantities are constant in the steady state, the life cycle of individual firms is not. Each period, new firms are funded, and incumbent firms grow, shrink, begin or stop exporting or exit permanently. Consistent with empirical regularities on firms dynamics (e.g, (Cooley & Quadrini, 2001)), we show that the mean and variance of firm growth and the likelihood of exit are negatively related to firm age and size. We also show that smaller and younger firms pay fewer dividends, borrow and invest more, and that the investment of small firms is more sensitive to cash flows, even after controlling for their future profitability. Furthermore, in line with empirical studies on firm export dynamics (e.g., (Eaton et al., 2007), and (Ruhl & Willis, 2008)), we show that new exporters account only for a small share of total exports, and that a large fraction of new exporters does not continue to export in the following year. Continuing exporters, on the other hand, are less likely to exit export markets as the number of years exporting increases, have larger and more stable sales, and generally reach their efficient size in a few years.
This paper contributes to the theoretical literature exploring the dynamics of firm export decisions. Research in this direction has modeled firm export dynamics as the outcome of learning as in (Eaton et al., 2012), investment in risky R&D as in (Atkeson & Burstein, 2010), and stochastic shocks to productivity as in (Ruhl & Willis, 2008), (Arkolakis, 2011) and (Alessandria & Choi, 2011). Our paper is closest in focus to (Kohn et al., 2011) in which firms face stochastic shocks to productivity and a borrowing constraint arises because of oneperiod contracts that are optimal under limited enforcement. A key difference with (Kohn et al., 2011) is that the stochastic process underlying firm dynamics in our model is an outcome of the long term financial contract, and not persistent productivity shocks. That is, an entrepreneur's financing constraint binds less as its claim to future cash flows increases, and not because she becomes more productive. Furthermore, we conduct our analysis in general equilibrium. In the model, financial intermediaries actively engage in maturity and risk transformation in a competitive financial market using workers' and entrepreneurs' shortterm deposits to fund a portfolio of longterm risky projects. The general equilibrium provides a link between the dynamics of firm balance sheets, aggregate conditions, and the balance sheets of the lenders, thereby relating financial intermediation to international trade.
Our paper is also related to research on the impact of financing constraints on the welfare gains of international trade. Financing constraints in these models arise because of exogenous collateral requirement (e.g., (Chaney, 2005), (Manova, 2008)), oneperiod contracts that are optimal under private information (Feenstra et al., 2011), or longterm contracts that are optimal under limited enforcement (Wang, 2010), (Brooks & Dovis, 2011). Last, our paper is related to studies of the efficient allocations in dynamic economies with private information such as (Atkeson & Lucas, 1992), (Smith & Wang, 2006) and (Verani, 2011).
The rest of the paper is organized as follows: Section 2 presents the model and Section 3 describes the financial arrangement between investors and entrepreneurs and derives the properties of the optimal contract. Section 4 defines the general equilibrium, and section 5.1 analyzes the model numerically. Concluding comments are contained in Section 6; proofs of propositions and derivations are relegated to the Appendix.
Workers are born without wealth, survive into the next period with exogenous probability , and are instantly replaced by new ones when deceased. Workers discount the future at rate and are endowed with one unit of time each period that they allocate between labor and leisure. Labor is paid at wage , and workers use their income to either buy the numéraire consumption good , or to purchase contingent claims at price that pay units of consumption in the next period if the agent is alive, and zero otherwise. Agents do not value bequests, and will thus place all their savings in these claims. Workers assess their consumptionleisure decision according to
(1) 
,  (2) 
New entrepreneurs are born with a blueprint to produce an intermediate good . Entrepreneurs, like workers, are born without wealth, survive into the next period with probability , and are instantly replaced upon death. Entrepreneurs are riskneutral, and discount the future at the rate . We assume entrepreneurs do not make laborleisure decisions, and instead devote a fixed fraction of their time to supervise their firm. Entrepreneurs assess their consumption decision according to
Financial intermediaries are riskneutral and discounts the future at the same rate as entrepreneurs. They raise shortterm deposits from workers via the annuity market, and can offer longterm financing to the entrepreneurs. The assumptions on worker characteristics imply stationary demographics of workers so that annuities can be offered without risk. Deposits from workers and entreprenuers in period are used to fund entrepreneurs' risky projects in period . Repayments from entrepreneurs to the intermediary are used to repay the deposits with interest. Perfect competition in the financial market implies that we can focus on a representative financial intermediary.
The final good is assembled by a large number of firms using domestically produced and imported intermediate goods and a constant elasticity of substitution aggregator. Intermediate goods are imperfect substitutes, and final good producers maximize their profit
Producing an intermediate good requires an initial investment that is sunk, and per period working resources to hire labor and to be used as capital. The th firm produces the th good according to a neoclassical production function , where is capital input and is labor input. We assume that the capital used in production is fully depreciated at the end of the period. An entrepreneur wishing to export must pay a fixed export cost before production begins, and chooses the quantity and of goods to sell domestically and abroad, respectively.^{9} It follows that the allocation of period working resources must satisfy:
,  (5) 
(6) 
The th firm is a monopolist for its differentiated product, but takes the inverse demand function for its product  price as a function of quantity  as given. Project returns of all firms are subject to a sequence of independent and identically distributed idiosyncratic revenue shocks , where High Low. Firm status is indexed by , where and indicate that a firm sells to the domestic market only, or to both the domestic and export market, respectively. The maximum revenue a firm can generate with resources is:^{10}
(7)  
s.t.  
Revenues are only observed by entrepreneurs, so lenders can only learn about the firm's performance and the realizations of the revenue shocks through entrepreneurs' reports, . We denote the history of reports up to period by . A contract is a set of decision rules . Conditional on surviving, a firm is either liquidated, , in which case the entrepreneur receives and the financial intermediary receives , where is the salvage value, or it remains in operation. If a firm is kept in operation, the contract specifies whether or not the firm exports, , and the size of the loan, . After production takes place and revenues are realized, an entrepreneur makes a repayment to the financial intermediary conditional on his expost report . Figure (1) summarizes the timing of events within one period.
A reporting strategy for an entrepreneur is a sequence of reports , where is the true history of realizations of revenue shocks. After every history , the pair implies an expected discounted cash flow and for the entrepreneur and the financial intermediary, respectively. A feasible and incentive compatible contract is optimal if it maximizes for every possible . Following (Clementi & Hopenhayn, 2006), we refer to and as equity and debt, respectively, so that the joint surplus is the value of the firm.
Using the method of (Abreu et al., 1990), the optimal contract can be written recursively by using as a state variable and by defining and as promised continuation values. It follows that equity must satisfy the following accounting identity:
(11)  
s.t.  (8), (9), and (10)  
There exists a region within which a greater value of the joint surplus can be reached by allowing for a lottery on the export decision:^{13}
where is the probability of becoming an exporter and and are the respective continuation values (within the same period) if the firm sells purely domestically or also exports.^{14} Furthermore, takes into account the option of liquidating the firm
(13)  
s.t.  
Panel (a) of Figure 2 plots the optimal value of the firm, , and the value to the intermediary, , as a function of equity, . A firm faces a binding borrowing constraint whenever its equity is below , where is the unconstrained level of resources. That is, is the level of resources that solves the static profit maximization of the firm such that argmax. New firms start at , so that expected profits of the intermediary cover the cost of the initial investment . Smaller firms take on more debt than larger firms, and firms with equity less than cannot borrow enough to pay the trade costs.
Firms' access to credit and growth are determined by the evolution of their capital structure. Using constraints (8), (9) and (10), and solving for next period's equity conditional on the revenue report yields the following law of motion for equity:
if  (14)  
if 
if  (15)  
if 
Figure 3 plots the decision rules for loans, repayments, and dividends as a function of equity. Due to riskneutrality, joint surplus is maximized when equity grows fastest, so dividends to the entrepreneur are optimally zero until the firm can no longer grow faster by postponing dividends, which is when . This implies that it is optimal for the financial intermediary to set the entrepreneur's repayments to for whenever as it allows for the fastest accumulation of equity toward the unconstrained level. Furthermore, the optimization problem takes place on the convex set , which implies whenever . From constraints (8) and (9):
(16) 
(17) 
Panel (c) of Figure 3 plots the investment rate conditional on receiving a high and low revenue shock as a function of equity. Investment by constrained firms is always positive after receiving a high shock, and always negative after recieving a low shock for constrained firms. The investment of small firm, and therefore cashflow, is also more sensitive to revenue shocks than that of larger firms. Furhermore, there is a large increase in investment once the firm becomes an exporter, with subsequent very high possible disinvestment should the firm receive a low shock and exit export markets.
Perfect competition in the financial sector implies that annuities are priced at the workers' survival rate .^{16} The assumptions on worker characteristics ensure that there exists a stationary demographic with constant aggregate deposits and labor supply. Let and be the deposits and hours worked of a period old worker. Setting the mass of workers to one, it follows that aggregate deposits by workers each period are given by
Perfect competition in the financial sector also implies that financial intermediaries break even on new contracts with entrepreneurs, or that in equilibrium. As discussed in the previous section, firm equity evolves according to the conditional continuation values specified in the contract, and , and the sequence of revenue shocks. Let be the state space for firm equity so that , be the Borel algebra generated by , and the measure defined over . Proposition 4.1 follows:
Proposition 4.1 There exists a unique stationary distribution of firms that is ergodic.
The intermediary uses the capital it has accumulated through entreprenuers' repayment, , and workers' deposits, , to finance the initial setup cost and the wage and capital expenditures of all firms before production takes place. It follows that the capital market clears when
Furthermore, the intermediary's budget must be balanced each period. That is, the intermediary's receipts from entrepreneurs plus the scrap value from liquidating poorly performing firms and the return on their own equity must be large enough to finance the cost of borrowing funds on the capital market. A stationary distribution of firms implies that in equilibrium, and it follows that:
Labor market clearing requires that the labor supply from workers is equal to the demand for labor by firms, so that:
(25) 
(26)  
(27)  
.  (28) 
Definition 4.2 (Worldwide Stationary Equilibrium) A stationary equilibrium consists of decision rules for labor supply , consumption , and deposits for workers in each country; a contract policy in each country, consisting of: promised values and , period resource advancements , liquidation lottery , export lottery , and repayments ; an initial contract state in each country; wages and interest rates ; prices and for intermediate goods; and an exchange rate , such that
Proposition 4.3 There exists a worldwide stationary equilibrium.
The contract needs to be solved numerically.^{22} Once the value of the firm and the decision rule for loan size are known, the remaining decision rules can be expressed in closed form as functions of . Given the initial firm size and the law of motion for , we can simulate the lifecycle of a large number of firms to estimate the stationary distribution of firms.^{23}
Let the instantaneous utility function for the workers be .^{24} We simplify the analysis by considering the case of symmetric countries and a constant returns to scale CobbDouglas production technology for intermediate goods: , with . The final good is produced according to a constant elasticity of substituion (CES) production function with constant returns to scale.
A period in the model is 1 year. We begin by fixing five parameters: The worker death rate is chosen so that the average life of workers is 50 years. The iceberg cost of exporting is set to 40 percent, which is in line with previous studies such as (Anderson & van Wincoop, 2004). The probability of a high revenue shock is 0.5, which produces investment volatility roughly in line with studies of firm such as (Cooper & Haltiwanger, 2006); and the salvage value is set to 80 percent of the setup cost . We set the elasticity of substitution between intermediates to , which is consistent with (Broda & Weinstein, 2006).^{25}

Given the above, the six remaining parameters are jointly chosen to match the following six moments:^{26} a labor income share of 60 percent, an average working time of 35 percent, an interest rate of 4 percent, an exit and entry rate of 6.3 percent (in line with (Lee & Mukoyama, 2008)), a share of exporters of 27 percent in line with (Bernard et al., 2007), and we require that new firms start at a size that is 15 percent of the unconstrained firm size.^{27} Table (1) summarizes the calibration. After solving the model, we simulate the life of firms from which we compute the statistics reported in Table (2) and the figures discussed in the next subsection.

Table 2 shows that, in the aggregate, the consumptiontooutput and investmenttooutput ratios are roughly in line with data, which principally follows from targeting the labor income share and labor hours. The exporttooutput ratio is 8.7 percent, which is in line with the US over the last four decades.^{28}
Our results on firm dynamics are consistent with the empirical regularities reported in (Cooley & Quadrini, 2001) and similar to those in (Clementi & Hopenhayn, 2006). Panel (a) of Figure 4 shows that the hazard rate of exit first increases for new firms and then decreases with firm age. On average, 1.2 percent of all firms are liquidated every period, which accounts for about 20 percent of all exiting firms. Panels (b) and (c) of Figure 4 plots the mean and standard deviation for firms of a given age, respectively, and show that younger firms experience a faster albeit more volatile growth than older ones.^{29}
Comparing exporters to domestic firms, Table 2 shows that the average exporter is four times larger (in terms of labor and capital) than the average domestic firm. On one hand, the contract requires that the entrepreneur have sufficient stake into the firm (by accumulating equity) to obtain a loan that is sufficiently large to pay the export costs and generate the additional revenue. On the other hand, the financial position of a firm improves and its access to credit increases after it begins exporting (Figure 3). Thus, while less financially constrained firms are able to export, the financial health of exporters is substantially higher than that of domestic firms because of their activities. This observation highlights the great difficulty of disentangling these two effects in the data.
Let us begin our discussion of exporter dynamics with an example. Panel (a) of Figure 5 plots the lifecycle of three firms taken from our sample of simulated firms. It takes 13 years for Firm 1 to accumulate enough equity to beginning exporting, but it exits export markets after its first year. Firm 1 gains access again to export markets at age 15, from which time it continues to grow until it reaches its efficient size at age 21 and finally exits at age 25. Firm 2 reaches the export lottery region after 12 years but initially fails to secure funding to export. Firm 2 successfully become an exporter 2 years later and continue to export until it exits at age 17. Firm 3 is the least successful of our three firms, and was never able to grow nearly large enough to export, and was liquidated by the bank at age 23.^{30}
The longterm financial contract plays an important role in shaping both the extensive as well as the intensive margin of trade. New exporters face a high probability of exit from export markets during their first year. Panel (a) of Figure 6 plots the hazard rate of exit from export markets. A third of new exporters exit after their first year. Continuing exporters become less likely to exit as their export spells increase, until they only face the exogenous exit rate of 5 percent. To see this, note that new exporters start out with equity that is close to the export lottery region, so that a low shock leads to exit from export markets if the firm loses the lottery. But since firms grow on average, older exporters, who have more equity, are further away from the export lottery region; and unconstrained exporters only cease to export when the entrepreneur dies.
Young exporters grow faster than established ones, but their growth is more volatile. Panels (b) and (c) of of Figure 6 plot the mean and standard deviation of investment of exporter conditional on the length of their export spell. The average growth rate of a two year old exporter is 1.2 percent, and is close to 0 after ten years. The standard deviation of investment, however, is about five times higher for a two year old exporter than a ten year old one.
Few firms start exporting every year, and continuing exporters expand rapidly. Panel (a) of Figure 7 shows that new exporters start with about two thirds of the resources used by an unconstrained firm, and operate close to their unconstrained size (95 percent) after exporting for five years (on average).
New exporters are small, and most exports are produced by very large firms. Only 3.3 percent of domestic firms start exporting every period, and this cohort accounts for about 9 percent of all exporters (panel (b) of Figure 7). Exporters that have been exporting for up to five years account only for approximately 25 percent of all exports (panel (c) of Figure 7). Therefore, the model predicts that the bulk of all exports is produced by established firms that have been exporting for five years or more.
There is widespread empirical evidence that financial fictions play an important role in shaping the growth of small and young firms. There is also growing empirical evidence that the export decisions of firms are sensitive to the availability of credit. This paper investigated how private information and longterm credit relationships may affect firms' export decisions and shape their growth in international markets. We proposed and analyzed a general equilibrium multicountry model economy in which entrepreneurs and lenders enter into multiperiod credit relationships subject to an informational asymmetry.
We showed that the model is consistent with empirical regularities on firms dynamics from the industrial organization literature, and with the models proposed to account for them. Furthermore, in line with recent empirical studies on firm export dynamics, our model predicts that new exporters account only for a small share of total exports, and that a large fraction of new exporters does not continue to export in the following year. Continuing exporters, are less likely to exit export markets as the export spell increases, experience faster and more volatile growth, and generally reach their efficient size in a few years.
The firstorder condition for the maximization of equation (4) with respect to variety yields
(29) 
(30) 
(31) 
(32)  
(33) 
Let us redefine the inverse demand functions as and similarly , where
, and  (36)  
.  (37) 
(38) 
.  (39) 
, and  (40)  
.  (41) 
, and  (42)  
.  (43) 
.  (44) 
,  (45) 
.  (46) 
The workers problem can be written recursively as
,  (48) 
(49)  
(50)  
(51)  
(52)  
To show this condition holds, we start from the zero profit condition for final goods producers and invoke the market clearing condition for intermediate goods (Equation 23):
(53) 
.  (54) 
.  (55) 
.  (56) 
.  (57) 
.  (58) 
.  (59) 
Proposition E.1 There exists a point such that for all , and for all .
Proof It is optimal to reach the unconstrained value in the shortest time possible because the joint surplus is maximized there and both the entrepreneur and the financial intermediary are riskneutral and share the same discount factor. Hence, repayments should be set equal to revenues as long as . This follows the argument set forth in (Clementi & Hopenhayn, 2006). Let us thus rewrite the value of a firm with a given export status as
When the equity of a domestic firm goes to zero, its value goes to : the first constraint, together with the fact that continuation values have to be nonnegative, forces continuation values to go to zero, as equity approaches zero. It follows that the spread between and goes to zero, so has to go to zero to maintain incentive compatibility. Therefore, the optimal resource advancement of a domestic firm will approach zero and thus its value will go to the discounted scrap value. In the case of an exporting firm, the logic is very similar, except that the resource advancement approaches the fixed cost of exporting , which cannot be seized by the entrepreneur. The value of an exporting firm with equity zero will hence be the discounted scrap value minus the cost of paying the export cost, . As both firm value functions are increasing and concave, and strictly so for , the fact that and implies a unique crossing.
Proposition E.2 The function contains an interval on which it is not concave. This implies together with risk neutrality that it is optimal to use an export lottery.
Proof As shown in the previous proposition, there exists a unique equity value where the two value functions cross. For any given , the slope of the exporting firm's value function is steeper than the slope of the nonexporting firm, i.e. . This follows from the fact that the same is true for the underlying revenue functions. Therefore, by continuity of the value functions, . This implies that the function is not concave on some interval . Since the marginal profit of an extra unit of resources goes to infinity as approaches zero, , and as the slope of the exporting value function is zero at the unconstrained level, .
Proof of Proposition 3.1 As outlined above, is increasing and concave for , so any convex combination of the two functions and is too. Since and for some (otherwise it would not pay to finance firms at all), and by assumption, . We know that , so by concavity of the functions , it follows that . Finally, because , it has to be that .
The first point follows immediately from the fact that the expected value of the liquidation lottery is equal to the equity with which an entrepreneur enters it, thus pinning down the probabilities of liquidation and survival. To prove the second point, we have to show that . From the above, it follows that the interval is nonempty. By definition, , which implies together with concavity that . This means that no company with finds it profitable to export, and all companies with do. A company with equity is offered a lottery with expected value equal to the equity the entrepreneur had before. The probabilities of getting and are determined thereby. An entrepreneur wins the lottery with probability , receiving and thus exporting; with probability , he gets and will hence not export.
Concerning the third point, it is clear that is linear in the two lottery regions. When , the value of the firm does not change anymore, so it will stay constant at . The functions are strictly increasing as long as , since is strictly increasing in that region. Therefore the firm value function is strictly increasing for all .
Proof of Proposition 3.2 Partition the domain of the contract in five parts . From the above when and when .^{32} When , the firm is unconstrained and there is no need to provide any incentives to report the truth (as all revenues will go to the entrepreneur), and hence , so . Whenever or , the firm enters a lottery and will either end up with or , with the expected value of the lottery being exactly equal to the promised value or . The expected next period equity for each of these is and , so that . Similarly, when , the lottery for liquidation yields the expected payoff , and the expected equity for next period is then just .
In the stationary steady state, given interest rate and wage , perfect competition in the financial market implies financial intermediaries earn zeroprofit on a new contract. This implies new entrepreneurs receive an initial equity , for which the lender earns just enough to break even. The initial equity is endogenous in the general equilibrium.
Consider the sequence of equity levels from a single firm indefinitely replaced by a new one when it dies. It is clear is a sequence of random variables, and its evolution depends on the properties of the contracts and on the sequence of shocks  productivity, death, export, and liquidation. In what follows, we show that is a timehomogeneous Markov chain such that
(61) 
Proposition F.1 is a timehomogeneous Markov chain on a general state space and is globally stable.
Equip the state space with a boundedly compact, separable, metrizable topology . Let be the measure space for the shocks. Let be any subset of . It follows for any and
(62) 
(63) 
(64) 
(65) 
(66) 
(67) 
For each , is a nonnegative function on , and for each , is a probability measure on . Therefore, for any initial distribution , the stochastic process defined on is a timehomogeneous Markov chain. Let denote the corresponding Markov operator, and let denote the collection of firms distribution generated by for a given initial distribution.^{33}
Write the stochastic kernel with the density representation so that for all . The Dobrushin coefficient of a stochastic kernel is defined by
(68) 
(69) 
(70) 
Proposition F.2 (Existence of a stationary equilibrium) The unique and ergodic invariant distribution of is continuous in prices.
The result follows if the conditions of (LeVan & Stachurski, 2007) (Proposition 2) are satisfied and the proof is similar to the one in (Verani, 2011).
We investigate the effect of the elasticity of substitution between intermediate goods firms on firm dynamics and the aggregate by considering another world economy with . To help with the comparison, we calibrate the economy with to the same moments as the economy with . Table 3 summarizes the value of the parameters used for each economy.
Worker's discount rate  0.959  0.959 
Elasticity of leisure  2.300  2.304 
Workers' death probability  0.02  0.02 
Capital share  0.137  0.2 
Setup investment  0.452  0.26 
Salvage value  0.8  0.8 
Iceberg cost  0.4  0.4 
Fixed export cost  0.033  0.012 
Probability of high/low shock  0.5  0.5 
Firm exogenous exit rate  0.047  0.05 
Table 4 reports the results for the two economies. Given the calibration, the wage rate, aggregate output, consumption and investment are roughly the same in the two model economies so that, from a macroeconomic point of view, the two world economies are comparable. However, a higher reduces the market power of intermediate goods producers at home and abroad. This translates into lower prices for all goods, with a comparatively greater decrease in the price of internationally traded goods. Furthermore, a decrease in firms' market power leads to a substantial reduction of the export share of aggregate output. A greater fraction of domestic firms begins exporting every period, and the hazard rate of exit for new exporters after one year is also higher. Last, a reduction in market power also increases the size differential between domestic and exporting firms, while the share of unconstrained firms becomes smaller.
Targeted: Interest rate  0.040  0.040 

Targeted: Hours worked  0.350  0.349 
Targeted: Labor income share  0.597  0.602 
Targeted: Entry/exit rate  0.066  0.063 
Targeted: Relative size of entrants  0.158  0.153 
Targeted: Share of exporters  0.273  0.269 
Not targeted: Wage rate  0.493  0.501 
Not targeted: Domestic goods price index  3.272  3.079 
Not targeted: Imported goods price index  6.033  4.925 
Not targeted: Output  0.289  0.291 
Not targeted: Consumption/Output  0.796  0.792 
Not targeted: Investment/Output  0.204  0.208 
Not targeted: Export/Output  0.138  0.087 
Not targeted: Entry rate in export market  0.028  0.033 
Not targeted: Exit rate from export market after 1 year  0.256  0.320 
Not targeted: New firm size relative to incumbents  0.321  0.335 
Not targeted: Domestic firm size relative to exporters  0.269  0.242 
Not targeted: Share of unconstrained firms  0.177  0.148 
The results for exports are driven by how the calibration for the economy with affects the fixed and variable trade costs, and . We keep the variable trade costs constant across the two economies, which implies that the ratio of exports to domestic sales decreases as market power decreases for each exporter.^{34} It follows that the fixed cost of exporting must be lower in an economy with a higher to keep the number exporters constant. This implies that the export lottery region is smaller making it easier for firms to enter and exit export markets.
Furthermore, unconstrained firms with lower market power use more resources and sell higher quantities of goods. Since large firms are always exporters, the relative size of exporters is also higher. A lower market power implies that firm profit is also lower, thereby reducing the speed of firm growth and leading to a smaller fraction of unconstrained firms.^{35} To see this, note that the incentive compatibility constraint binds for constrained firms, which implies that . It follows that smaller revenue implies that the spread between the continuation values is smaller too; and the number of steps needed to reach the unconstrained level is higher.