n easier to understand version o Melitz (2003) Daniel Nguyen, University o Copenhagen International Trade, 2 December, 2008 This handout presents a very simpli ed version o Melitz (2003) that ocuses on a key mechanism: sel-selection into exporting by productivity. et s begin by presenting a closed economy version Closed economy continuum o rms produces varieties o a single good using the numeraire actor labor (wage is normalized to ). The endowment o labor is :. Demand Each in nitessimal rm produces a single, unique variety. Given the set o varieties supplied to the market, the consumer s utility unction is 2 where c denotes the quantity consumed o variety. The market is subject to the expenditure = income constraint: p c = 2 By maximizing the utility subject to the income constraint, we can construct the market demand or any variety as a unction o its price:.2 Supply c = c p R2 p Firms are idiosyncratically productive, meaning at birth they draw a parameter rom a distribution with pd g() over support (0; ) : To produce an output q, a rm requires l ( ) units o labor: l ( ) = q (2) The marginal cost o production is : Firms that draw higher are more productive in the sense that they have a lower marginal cost. Given rm 0 s demand unction () and cost unction (2), the rm will maximize pro ts by setting its price as a markup over marginal cost: p = (3) ( ) and garner revenues r ( ) o r ( ) = = 2 ()
determined by substituting equation (3) into () and recognizing that revenue r ( ) = p c : The variable is a measure o the competitiveness o the market: an larger re ects either more supplied varieties or higher productivities o existing varieties. Either way, rm 0 s revenues are lower as a result o higher. Subtracting its labor costs rom its revenues, a producing rm makes a supply pro t o ( ) = (4).3 Decision to supply Given rm 0 s draw o ; the rm will only choose to supply i ( ) > 0: Since ( ) is strictly increasing with and (0) =, there exists a unique such that ( ) = 0 (5) () > 08 > Equation 5 is the CUTOFF condition. Firms drawing < do not produce because it is not pro table to do so. Firms drawing will supply to the market and make nonnegative pro ts. Since is drawn rom a distribution with pd g() over support (0; ), this CUTOFF determines the set o goods supplied to the market and the market competiveness : = = Mg()d 2 where M is the endogenous mass o potential producers that draw. The cuto condition can now be rearranged as R g()d = M (6) relating the two endogenous variables M and :.4 Equilibrium I drawing costs nothing, then new rms will continuously ood the market. Firms drawing < will just exit, re-enter and draw another until they receive a pro tble draw. M will grow in nitely and unsolvably large. Instead, this economy requires a sunk cost o entry D: Think o D as the development cost or a new variety. With this additional cost, a rm s total pro t, as a unction o its productivity, is ( ( ) = D i (7) D i < To determine equilibrium, we must nd the rm s ex ante expected pro t. new rm pays D and a aces a probability R g()d o drawing a pro table : Otherwise, it exits without producing, paying only the development cost D: potential rm s expected pro t can be expressed as E [] = D g()d ( D) g()d (8) 2 0
In an economy where there is ree entry, rms will enter until E [] = 0 With this condition, expression (8)can be rearranged to R = M (9) g()d D This is the zero-expected-pro t condition. ike equation (5), it relates the two endogenous variables in this economy: the mass o entering rms M and the cuto productivity : The cuto and zeroexpected-pro t conditions can be solved to determine the unique solutions to M and : 2 Open economy. Suppose now that there are two countries, Home and Foreign( denotes a Foreign variable) with identical preerences, wages, and labor endowments. Home demand or a variety is now c = p R2\x p g where x is the set o varieties exported rom Foreign to Home. Similarly, x represents the set o varieties made in Home and sold in Foreign, and represents varieties made and sold in Foreign. Exporting involves additional xed and variable costs. For a Foreign based rm with productivity ; the labor requirement l x or exporting a quantity q x to the Home country is l x ( ) = q x where > is the iceberg transport costs o exporting. It still maximizes pro ts by charging a price equal to a constant markup over marginal cost and achieves export revenues o rx ( ) = (0) = () 2 x Note that now includes both domestic varieties and imported varieties. The Foreign rm makes an exporting supply pro t o 2 x ( ) = 2. Exporting Cuto We can now derive an exporting cuto condition similar to (5): x ( x ) = 0 (2) 3
where x is the cuto productivity or Foreign rms to supply to the Home market. Home rms domestic supply decisions are still governed by equation (5), except with our new de nition o market competiveness given by equation () : With the domestic and exporting cuto conditions, we can determine the set o varieties available to a home consumer: and determine the exact nature o : = x = : > x g = : > g Mg () d x M g () d We can repeat this exercise or a Home rm selling in the Foreign market. Since the two countries share identical labor endowments and wages, the Home market competiveness must equal to the oreign competitivenes : I not, all potential rms would simply locate in the market with a lower : Then, by Equation (2), = ) = ) x = x ) M = M That is, the cuto s and mass o entering rms are the same in both countries. The two countries are symmetrical and have identical aggregate market conditions. rm in either country supplies to its domestic market i > ; and exports to the other market i > x : The relationship between and x is determined by combining equations 5 and 2: x = (3) The exporting cuto is higher than the domestic cuto : only the most productive rms export. 2.2 Expected Pro ts Ex ante, rms developing new varieties now have a chance o drawing a productivity high enough to export. rm s total pro t, based on its draw, is now 8 >< D i x ( ) = D i < x (4) >: D i < new rm s expected pro t can be expressed as E [] = x x 0 ( D) g()d D g()d D g()d 4
Free Entry will reduce E [] to zero: E [] = 0 (5) 2.3 Equilibrium Equilibrium is de ned as the triplet ; x ; Mg such that. determines the domestic cuto via equation 5 2. x determines the exporting cuto via equation 2 3. The mass o incoming rms M reduces a new rm s expected pro ts E [] to zero via equation 5 5