ECON 4415 International Trade

ECON 4415 International trade

Material for the seminar on 10 May 2004 (Arne Melchior)

Questions:

1.Discuss whether regional trade integration may undermine the WTO.

Hint: Read Chapter 10 in Hoekman and Kostecki, and you may find other supplementary material.

2a. (Optional, not mandatory.) Depart from the “home market effect” model as described in lecture note 5 or Chapter 10, Section 4 in the Helpman-Krugman book, or in the seminar note on the model. Now develop a similar model but for three countries of identical size (Li=L is the same for all the three). Trading costs for X goods are t12 between countries 1 and 2, and t between country 3 and the other two countries, 1< t12t. If t12=t, the situation is symmetrical with no regional integration between countries 1 and 2, and all the three countries must have equal shares of X production. Discuss the changes in production for the three countries if t12 is reduced, i.e. that regional trade integration occurs between countries 1 and 2. Hint: See lecture note 5, or seminar note on the home market effect.

2b. (Optional, not mandatory.) Discuss the welfare implications in problem 2a above. Hint: See lecture notes 5.

Illustration on question 2.

The analysis can be undertaken in three steps:

Define the model, as in the lecture notes, and derive the pricing condition and firm size x. As there, assume that all countries are diversified (i.e. they all produce B as well as X), and then you have pB as well as the wage in all countries equal to 1. Then the total income of each country is simply L.

Then set up the three equations for how individual firms in each country allocate sales across markets. These are:

x11 + x12 + x13 = x

x21 + x22 + x23 = x

x31 + x32 + x33 = x

where x, firm scale, is given by your former solutions. Now you may use the demand equations

xji = Yi p-tji1-Pxi-1

to express all x’s in the first column as functions of x11, all in the second column as functions of x22, and all in the third column as functions of x33. You simply divide demand for x21 by demand for x11 and so on, using the appropriate levels of trading costs (=1 for home market sales). After doing this, you have three unknowns. But you also can use that countries 1 and 2 must be identical (same size, same trading costs), so x11=x22. Then the whole thing reduces to two equations with two unknowns, you may solve for x11 and x33 (also substituting the solution for firm size x), and then derive all the other x’s.

Finally, you set up the equations for market clearing for X goods in each country. The consumption share for X is  and the income in all countries is L, so you have

n1px11 + n2 p x21 + n3 p x31 = L

n1px12 + n2 p x22 + n3 p x32 = L

n1px13 + n2 p x23 + n3 p x33 = L

Here p is known from the pricing condition, the x’s you have derived above, and you know that n1=n2 since countries 1 and 2 are identical (country 3 has the same size, but faces different trading costs). Using this, the first two equations become identical, and you may solve for n1 and n3. Having obtained the solutions for the number of firms, you may analyse how these expressions respond to changes in t12. If t12=t, the three countries are obviously in the same situation. All the way you have to assume that t12 as well as t are larger than 1, but you may have t12<t.

As an option (not required), as question 2, you may also check how regional integration affects welfare for the three countries. Then you have to set up the price indexes, which must take the form

Px1 =

Since n1=n2, the expression for country 2 will be similar. For country 3 we must have:

Px3 =

Using the solutions for the number of firms, you may then analyse the bracketed expressions, as indicated earlier. The larger they are, the higher is welfare. Since p is the same everywhere, you may put it outside the brackets and forget it.

Solution to the trade bloc exercise (Question 2)

Model set-up:

-Three countries, labelled 1, 2 and 3.

-One factor of production, labour.

-The three countries are of equal size, Li=L for all i.

-Two sectors:

-Sector B: “Agriculture”, constant returns to scale, one labour unit required to produce one unit of B.

-Sector X: “Manufactures”, produced with increasing returns to scale, differentiated goods, each firm producing one variety of the good.

-No trade costs between countries for good B.

-“Iceberg” trade costs tij>1 for trade between countries i and j.

-CES sub-utility function for X goods, as in lecture 5.

-Cobb-Douglas utility function Ui=XiCBi1- for B (CBi=consumption of B in country i) and X aggregate, giving consumption shares 1- for B and  for X.

-Simplifying assumption: All countries produce X as well as B. With free trade in B, prices for B as well as wages will then be equalised between countries, and we choose units so that wi (the wage) = PBi (the price for B goods) =1 in all countries. Total income in each country is then L, and total consumption of X goods in all the three countries is L. Consumption of B is (1-)L. Since the price for B is 1, this is also the demand function for B goods.

Now following lecture notes 5, we know that the demand function for an individual variety from country i in country j is

x*ij = L p*ij-Pj-1

where Pj is the price index for X goods in country j, and we have used asterisks * to indicate that here, it is the quantity demanded and the “real price” paid by consumers that enter. Given iceberg trade costs tij, some goods “melt away” in shipment and we have x*ij=xij/tij, where xij is the quantity shipped by the firm. Similarly, if firms receive the price pij for the goods, the real price paid by consumers is higher since they receive less than xij, hence we have pij=pij*tij. Using this to express demand in terms of producer-side quantities and prices, we obtain the “corrected” demand functions

xij = L pij- tij1-Pj-1

(which are important to remember).

X producers have fixed costs f and constant marginal costs c in all countries, hence their cost functions are

Ci = f + c j xij

where you may note that because of iceberg trade costs, trade costs do not enter (since they are reflected in the quantities xij). Profits are then

i = j pij xij - f - c j xij

Maximising profits with respect to xij gives the first-order condition

pij + pij/xij xij – c = 0 or pij + pij/xij xij = c

Putting pij outside brackets on the left hand side gives

pij ( 1 + pij/xij xij/pij ) = c

and we see that the second term in brackets is the inverse of the demand elasticity faced by the firm, i.e.

pij ( 1 + 1/eij ) = c

Now we introduce the assumption about competition; there is monopolistic competition so that firms neglect the impact of their actions on the whole market. The true elasticity of demand with the given demand structure is --(1-)sij, where sij is the market share of a firm from country i in market j for X goods (shown in earlier lecture, remember also that >1). With monopolistic competition, we assume that there are many firms so that sij is small, and the firm neglects the second term of the expression. Hence with monopolistic competition, the perceived elasticity eij is equal to -. Using this, the first-order condition above becomes

pij ( 1 - 1/ ) = c

or equivalently

pij = c/(-1)

i.e. the price is a mark-up on marginal costs, determined by the elasticity of substitution. Given that  and c are the same for all pij, we then know that all producers charge the same producer prices, which we denote p=pij for all i,j. Given that we have

p = c/(-1)

in all cases, we may in the following drop subscripts on prices.

With monopolistic competition, we also assume that there is free entry and exit so at equilibrium, firms earn no pure profits. Setting profits equal to zero and using pij=p we obtain

i = j p xij - f - c j xij = 0

which determines the total output quantity of the firm, which is

j xij = f/(p-c)

Given that f, p and c are the same for all firms, the output quantity must be the same for all firms, and we denote it by x. Using the solution for p, we have

x = f (-1)/ c

The above provides the first steps when solving any model of this kind. Next, we turn to the trade bloc model.

Now we assume that between countries 1 and 2, trade costs are t12. Between country 3 and the two former, trade costs are t. We assume that countries 1 and 2 may form a trade bloc by lowering t12, hence we assume t12t. Observe that this is the only thing that can create any difference between the three countries; they are identical in all other respects. Observe also that countries 1 and 2 are identical in all respects; hence we may simplify in the following by assuming that all solutions are similar for firms in countries 1 and 2.

The next step when solving the model is to determine how firms allocate their sales across the three markets. This is given by the three equations:

x11 + x12 + x13 = x

x21 + x22 + x23 = x

x31 + x32 + x33 = x

where x, firm scale, is given by the solution above (but we use x for simplicity in order to save notation). Now we use the demand functions

xij = Yi p-tij1-Pj-1

to express all x’s in the first column as functions of x11, all in the second column as functions of x22, and all in the third column as functions of x33. We simply divide x12 and x32 by x22 and so on, and obtain:

x11 + x22 t121-+x33 t 1-= x

x11t121-+ x22 +x33 t 1-= x

x11t1- + x22 t 1-+x33= x

Given that countries 1 and 2 are identical, we must have x11=x22, and we may use this to simplify further. We then easily find the solutions

x11 = x22 = x

x33 = x

The solutions for other xij’s are then easily found by using the relationships x12=x22t121- and so on, as above. Hence we have determined how much each firm sells in all markets.

By examining the solutions above, we easily find that if t12<t we must have x33>x11=x22. Hence firms in the trade bloc export more, and sell less in their home market. Formally, it is easily established that the derivative of x11 with respect to t12 is positive, while the derivative of x33 with respect to t12 is negative. These results will also be relevant for the welfare analysis, see the last page of the note.

The final step in solving the model is to determine the number of firms in each country. For this purpose, we set up the three market-clearing equations, denoting the number of X firms in each country by n1, n2 and n3. Also this time, we know that n1=n2, and this can be used to simplify the calculations. Sales by all firms in each market must add up to total consumption in each market:

n1 x11 p + n2 x21 p + n3 x31 p = L

n1 x12 p + n2 x22 p + n3 x32 p = L

n1 x13 p + n2 x23 p + n3 x33 p = L

Using that n1=n2 and x32=x31 due to the symmetry between countries 1 and 2, we may simplify into two equations:

n1 (x12 + x22) + n3 x32 = L/p

n1 (x13 + x23) + n3 x33 = L/p

This gives

n1 = L/p

n3 = L/p

The denominator in these expressions, which we denote by Z, can (using x12=x22t121- and so on, as well as x11=x22) be simplified to

Z = x11 x33 ( t121- + 1 – 2 t2-2)

Also simplifying the numerators and completing the (somewhat tedious) calculations, we obtain the solutions

n1 = n2 =

and

n3 =

If t=t12 we easily find that n1=n2=n3. The total number of firms in the world is 2n1+n3. Using the solutions above, we find this to be 3L/(px)= 3L/(f). f is firm size (in value), L is the size of each market for X, and the world market is three times as large.

In order to examine what happens if t12<t, we may take the derivative

n1/t12 =

which is negative since >0. Hence the lower is t12, the larger is the number of firms in countries 1 and 2. Hence we have shown that regional integration will increase the number of X firms in the integrating countries. Given that world demand for X goods is constant, it is then evident that regional integration among countries 1 and 2 must lead to a lower number of X firms in country 3. Hence regional integration leads to “production-shifting”; X production is transferred from country 3 to the integrating countries.

In order to show the welfare impact, note from lecture note 5 that welfare is inversely related to the price indexes Pi. The lower price level for X goods in a country, the higher is welfare. Now, as a trick to simplify the welfare analysis, write down the demand for xii; i.e. a firm’s sales in the domestic market:

xii = L p- Pi-1

Now, if the welfare in country i is inversely related to Pi, it must also be inversely related to xii, since p may here be treated as a constant (common to all firms in all countries). Hence we may derive the welfare impact of integration directly by studying how changes in t12 affect xii. We have already established that regional integration will decrease home market sales for firms in the integrating countries, and increase home market sales for firms in country 3. Hence regional integration will increase welfare in countries 1 and 2, and reduce welfare in the non-participating country 3.

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