PhD Topics in Macroeconomics

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1 PhD Topics in Macroeconomics Lecture 5: heterogeneous firms and trade, part three Chris Edmond 2nd Semester 204

2 This lecture Chaney (2008) on intensive and extensive margins of trade - Open economy model, many asymmetric countries 2- Intensive vs. extensive margins of trade 3- Implications for gravity equations 2

3 Chaney model: key features Many asymmetric countries, asymmetric trade costs No free entry, number potential firms proportional to country size Pareto distribution for firm productivity Numeraire good, costlessly traded Relative to Melitz (2003), extra structure allows model to be solved in closed form, despite much richer patterns of asymmetry 3

4 Chaney model: overview Countries i =,...,N of sizes L i Labor is only factor of production, inelastic supply Sectors s =0,,...,S. Sector s =0is competitive numeraire. Sectors s =,...,S are monopolistically competitive Sector-specific variable trade costs ij,s for each bilateral pair. Sector-specific fixed trade costs f ij,s 0 for each bilateral pair Trade costs need not be symmetric Numeraire is costless to trade, ij,0 =and f ij,0 =0 4

5 Preferences Cobb-Douglas across sectors log U = µ 0 log C 0 + SX µ s log C s, s= SX µ s = s=0 CES within sectors s =,...,S C s = Z s c s (!) s s d! s s, s > Budget constraint C 0 + SX Z s= s p s (!)c s (!) d! apple Y 5

6 Demand Sectoral demand C s = µ s Y P s Demand for variety! in sector s =,...,S c s (!) = ps (!) P s scs 6

7 Price index Sectoral price index implicitly defined by Z P s C s = p s (!)c s (!) d! s so P s = Z s p s (!) s d! s 7

8 Competitive numeraire sector Numeraire good produced by competitive firms Country-specific labor productivity A i in numeraire sector Real wage in units of the numeraire w i = A i (if country i produces the numeraire, i.e., if µ 0 is large enough) 8

9 Trade barriers and technology Variable ij,s and fixed f ij,s trade costs (in units of labor) For firm in sector s with productivity draw a, labor used to deliver y units of output from country i to country j is l ij,s (y, a) =f ij,s + ij,s a y Firm productivity is Pareto with sector-specific shape parameter s G s (a) :=Prob[a 0 apple a s] = a s and we will need s > s for various moments to be defined 9

10 Pricing Isoelastic demand with elasticity s >, same for all countries Price set by firm in country i for market in j is p ij,s (a) = s s ij,s w i a Hence sectoral price index in destination country j P j,s = X N Z n i,s i= p ij,s (a) s dg s (a) s, n i,s := Z i,s d! with p ij,s (a) =+ for producers that do not sell in j 0

11 Exogenous number of potential producers No free entry into production Measure of producers per sector proportional to country income n i,s = w i L i for all s Since no free entry, will be positive profits in equilibrium i = SX s= NX Z j= ij,s (a) dg s (a) where h ij,s (a) = max 0, p ij,s (a) Ownership structure matters ij,s w i a y ij,s (a) f ij,s w i i

12 Ownership structure Income Y i in country i (in units of the numeraire), consists of labor income w i L i plus profit income Global profit income pooled and paid out as dividends per share Assume that each representative worker has w i shares. Then total income in country i is Y i = w i L i + w i L i where dividends per share are := P N j= jw j L j P N j= w jl j 2

13 Profits Conditional on operating, profits are ij,s (a) = p ij,s (a) ij,s w i a y ij,s (a) f ij,s w i = p ij,s (a) ij,s w i a pij,s (a) s µs Y j,s P j,s P j,s f ij,s w i = B ij,s a s f ij,s w i where B ij,s := s s P j,s ij,s w i s µs Y j,s s From here on, drop s subscript to simplify notation 3

14 Cutoff productivity For cutoff firm ij (a) =0, B ij a = f ij w i Solves for a ij = fij w i B ij = a ijw i P j fij w i Y j () where a is the first of many tedious constants a := µ For each country i only producers with a>a ij export to j. Exporting to j depends on price level in j (and trade costs) 4

15 Solving for P j Price level in j depends on which firms enter that market P j = X N Z n k p kj (a) dg(a) k= a kj so that P j = NX Z n k k= a kj kj w k a dg(a) = NX n k k= kjw k Z a kj a dg(a) 5

16 Need to calculate the integral Aside on the Pareto H(x) := Z x a dg(a) (2) With Pareto this is H(x) = Z x a a ( +) da = ( ( )) a ( ) x =+ ( ) x ( ( )), assuming > 6

17 Solving for P j (cont.) So price index satisfies where P j = NX n k k= kjw k ( ) (a ( ( )) kj ) a kj = a kjw k P j fkj w k Y j Plug in a kj and solve for P j... 7

18 Solution for P j After some horrible algebra, we get P j = P j Y ( j ) (3) where P is another tedious constant P := h µ X N i / w k L k a ( ) k= Index of multilateral resistance to trade flows j := h X N i k ( kj w k ) (f kj w k ) ( / ) k= ameasureofthe tyranny of distance duetofixedandvariable trade costs weighted by shares of world income k := Y k Y = ( + )w k L k ( + ) P N i= w il i = w kl k P N i= w il i 8

19 Cutoff productivity revisited Plugging (3) back into formula () fora ij gives a ij = a P ij w i j f ij w i Yj (4) Can now use this to derive implications for trade flows 9

20 Micro trade flows Exports from i to j by a firm of type a a ij are x ij (a) =p ij (a)y ij (a) = pij (a) P j µy j ij w i = x Y j j a with yet another tedious constant x := µ P 20

21 Micro trade flows ij w i x ij (a) =x Y j j a Elasticity with respect to variable trade log x log ij = same as Krugman (980). A partial elasticity, holds j constant Elasticity with respect to destination country log x log Y j = < Micro-level trade flows are similar to what standard monopolistic competition model would predict 2

22 Macro trade flows Aggregate exports from i to j are X ij := n i Z x ij (a) dg(a) = n i Z a ij ij w i x Y j j a dg(a) ij w i = xn i Y j j Z a ij a dg(a) Evaluate using H(x) from (2) above and expression for a ij from (4) 22

23 Macro trade flows Gives X ij = Xn i ij w i j (fij w i ) ( ) Y j with yet another tedious constant X := x ( ) a P ( ( )) Recall n i = w i L i = Y i /( + ) to turn this into a gravity equation To get a simple expression, recognise that X Y + = µ 23

24 Can then write Gravity equation X ij = µ ij w i j (fij w i ) ( ) Y i Y j Y Elasticity with respect to variable trade log X log ij =, independent of! Elasticity with respect to fixed trade log X log f ij =, decreasing in! Both larger in sectors where is large (dispersion in a is small) Macro-level trade flows completely different to micro-level flows. 24

25 Decomposition Recall X ij = n i Z a ij x ij (a) dg(a) Total differential dx Z ij = n i a ij ij dg(a) d ij x ij (a ij)g 0 (a ij d ij Z + a ij ij dg(a) df ij x ij (a ij)g 0 (a ij df ij Sum of intensive margin and extensive margin effects 25

26 Variable trade cost effects Elasticity with respect to variable trade log X log ij = ( ) {z } standard intensive margin from Krugman + ( ( )) {z } new extensive margin = Higher amplifies intensive margin effect of ij but dampens extensive margin effect of ij Exactly cancel so that net effect is that elasticity with respect to variable trade costs is independent of That said, since >, actual elasticity must be greater than in Krugman model (trade flows more responsive) Different structural interpretation of estimated trade elasticities 26

27 Fixed trade cost effects Elasticity with respect to fixed trade log X ij = log f ij {z } no intensive margin + {z } extensive margin = Higher also dampens extensive margin effect of f ij But now this is the only effect, so elasticity with respect to fixed trade costs is decreasing in 27

28 Intuition Consider sector with very differentiated goods (low ) Intensive margin effect demand insensitive to trade costs intensive margin elasticity small when low this is only effect in Krugman model Extensive margin effect market shares insensitive to trade costs less productive firms still have relatively high market share, despite relatively high price as trade costs ( or f) fall,somerelativelyunproductivefirmsenter low, so entrants relatively large compared to existing exporters extensive margin elasticity large when low! 28

29 Structure of gravity equations Define composite trade friction ij := ( ij w i )(f ij w i ) ( ) The gravity equation can be written X ij = µ " ij P N k= k " kj Y i Y j Y, trade elasticity " = Krugman model likewise has gravity equation X ij = µ " ij P N k= k " kj Y i Y j Y, trade elasticity " = with trade friction ij := ij w i Eaton-Kortum (2002) has similar gravity representation 29

30 Next Heterogeneous firms and international trade, part four Technology and trade frictions in Ricardian models with heterogeneous firms Eaton and Kortum (2002): Technology, geography and trade, Econometrica. 30