Comparative advantage without tears: A Cobb-Douglas ...



ECON 4415 International trade – notes for lectures 2-3

Comparative advantage revisited: A Cobb-Douglas version of the Heckscher-Ohlin-Samuelson (HOS) model

Arne Melchior

January 2004

Abstract

In research on the increased wage gap between skilled and unskilled workers in some rich countries, it is, based on the HOS model, normally assumed that trade between rich and poor countries leads to an increased wage gap in rich countries. In this paper, we show that this only applies unambiguously when factor prices are equalised in the HOS model. If the poor country is fully specialised in production intensive in unskilled labour, the relationship between international trade and factor prices is modified. The paper shows that international trade leads to a welfare gain for both countries also when one country is fully specialised. On the other hand, it is shown that international trade is less important for countries than to have an appropriate mix of production factors. For poor countries, policies aimed at increasing the stock of physical or human capital are therefore more important than free trade, although the latter is beneficial. As a by-product of the analysis, the paper provides a complete analytical treatment of the 2x2x2 HOS model, using simplified functional forms that provide explicit analytical solutions on most issues.

1. Introduction

This note is written in order to provide a complete treatment of the Heckscher-Ohlin-Samuelson (HOS) model of international trade, using functional forms that allow us to derive analytical solutions for production, trade and welfare. For this purpose, the HOS model is formulated with Cobb-Douglas production and utility functions. The choice of these functional forms restricts generality in on sense, but on the other hand we gain by being able to provide a complete analytical treatment of the model. Compared to the so-called “hat calculus” (see, for example, Jones 1965)[1] or the “duality approach” (see, for example, Dixit and Norman 1980)[2], we believe that the Cobb-Douglas version provides a simpler treatment and overview of the model. The simplified approach allows us to focus on some features of the model that have received less attention in the literature. For example, we show that a country’s factor composition is more important for its welfare than international trade. Furthermore, we examine the gains from trade under complete specialisation, and how the impact of international trade is modified when there is complete specialisation.

In section 2, we derive production, trade and welfare in a closed economy, and thereby develop tools that are later, in section 3, used in the analysis of international trade, with or without factor price equalisation.

2. The closed economy case

2.1. Production technology

Consider a single economy with factor endowments K and L (capital, labour, skilled or unskilled labour, etc.). There are two sectors A and B, using K as well as L in production, with KA, KB, LA, LB denoting the factor amounts used in the two sectors. Factors can move freely between sectors. Factor rewards are r (for K) and w (for L), respectively.

Production functions are Cobb-Douglas:

1) FA = KA( LA1-(

2) FB = KB( LB1-(

Total cost in sector A is

(3) CA = rKA + wLA

Maximising production in sector A subject to given costs, we obtain the first-order conditions

(4) (FA/ KA = (A r and (1-()FA/ LA = (A w

where (A is a Lagrange multiplier. Now substituting for KA and LA in (3), we find that

(5) (A = FA/ CA

so the Lagrange multiplier is equal to the inverse of unit costs. Substituting this into (4), we obtain

(6) KA = ( CA/ r and LA = (1-() CA/ w

This also confirms that

(7) [pic]

i.e. the costs shares under the Cobb-Douglas technology are constant. These are given by (6), i.e. ( = rKA/ CA and (1-() = wLA/ CA.

By substituting (6) into (1) and rearranging, we find the cost function

(8) CA = ZA FA r( w1-( where ZA = (-((1-()((-1)

Similarly, we obtain for sector B

(9) CB = ZB FB r( w1-( where ZB = (-((1-()((-1).

2.2. Production and factor use

In the model, production, factor use, goods prices and factor prices are all endogenously determined. In order to understand how the model works, it is nevertheless useful to proceed step by step and examine first how production and factor use depend on prices, if we treat the latter as given. In this paragraph, we shall therefore analyse how production and factor use are related to factor prices – without full solutions for the latter.

We assume that factors are fully utilised:

(10a) KA + KB = K

(10b) LA + LB = L

Substituting into (10) from (6) and (8-9), we obtain

11) ( ZA FA (r/w)(-1 + ( ZB FB(r/w)(-1 = K

12) (1-() ZA FA (r/w)( + (1-() ZB FB(r/w)( = L

Solving for FA and FB, we obtain

13) FA = [pic]

14) FB = [pic]

Consider, for example, that L becomes relatively more expensive so that the factor price ratio w/r increases. In both sectors, there would be an incentive to use less L and more K. But because L must also be fully utilised, this is not possible; one of the sectors has to expand in order to “absorb” the L stock. From (14), we can find that this has to be the L-intensive sector. Similarly, if r increases, production in the K-intensive sector will expand. The higher is w/r, the more will be produced of the L-intensive good, and the lower is w/r, the more will be made of the K-intensive good. This reasoning is for given K and L; in the general equilibrium analysis, w/r is determined by K/L. An increase in K will cause production of K-intensive goods to increase, but the corresponding increase in w/r will modify this effect.

For the later analysis, it is important to observe that production is positive for both sectors only for a specified range of w/r. By setting (13) and (14) equal to zero, we find that

(15) FA=0 if w/r = K/L * (1-()/(

(16) FB=0 if w/r = K/L * (1-()/(

Hence if the factor price ratio falls outside the range defined by these two values, the economy will specialise in one of the two sectors. In the closed economy, we shall see that the equilibrium factor prices are within this range. With international trade, however, this needs not be the case, and countries may become specialised.

Diagram 1 illustrates this “range of diversification” in the model.

[pic]

Diagram 1: Factor prices and factor composition

(Note: Due to software limitations, ( and ( are written as a and b, respectively.) Diagram 1 shows how the K/L ratio in the two sectors responds to the factor price ratio. From (7), we see that the curves KA/LA and KB/LB must be rays from the origin, with slopes (/(1-() and (/(1-(), respectively. In the diagram, we have assumed that sector A is more K-intensive. When w/r=(1-()/(, we know from (14) that the production of A goods is zero. Then all the resources in the economy are used in sector B, so we must have KB/LB=K/L. Similarly, when w/r=(1-()/(, we have FB=0 and KA/LA=K/L.

Diagram 2 shows how production in the two sectors varies with w/r, simulating the model with (=0.7 and (=0.3, and K=L.

[pic]

Here the minimum value of w/r (for the range of diversification) is 0.43, while the upper bound is 2.33. The two curves intersect with the horizontal axis at these values.

Finally, observe from (13-14) that if we treat factor prices as constant, the impact of an increase in the K stock depends on the sign of (, which again depends on whether (>( or not. An increase in the K (L) stock will increase production in the K-intensive (L-intensive) sector, and reduce production in the other sector. This corresponds to the Rybczynski “theorem”, stating that increasing one factor will lead to a more than proportionate increase in production in the sector using that factor intensively, and reduced production in the other sector. This conclusion, however, is not valid in the general equlibrium situation, where goods and factor prices are allowed to vary. In general, an increase in K will lead to a reduction in the r/w ratio, and this will – according to the analysis above – contribute to lower production in the K-intensive sector. Later, we shall examine the outcome in this more general case, and we shall se that there is no “magnification effect”.

2.3. Goods prices and factor prices

In standard textbook treatment of the HOS model, a second “magnification effect” is the Stolper-Samuelson theorem: A change in relative goods prices leads to an even stronger change in relative factor prices. For example, if the price for K-intensive goods increases, the factor price ratio r/w will increase even more.

With perfect competition, unit costs must equal the price, hence we have

17) CA/ FA = ZA r( w1-( = pA

18) CB/ FB = ZB r( w1-( = pB

It should be observed that these relationships only apply when production is positive; hence the following result only applies when the economy is diversified.

Dividing (6) by (7), we obtain

19) [pic] or [pic]

This describes the relationship between relative goods and factor prices. Diagram 3 simulates the relationship, with varying values for ( and (.

[pic]

In the graph, we only show the curves within the range of diversification defined by (15)-(16). If the two sectors are very different, this range is large (cf. the curve for (=0.1, (=0.9, where the permissible segment is even larger than shown). When sectors become more similar, the non-specialisation range becomes more narrow (the two other curves).

If sector A is more K-intensive, w/r is a decreasing function of pA/pB. If A is more L-intensive, the curves are upward sloping. We have (arbitrarily) chosen values so that (+(=1, therefore all the curves pass through the point (1,1). Around this point, the curves are steeper, the more similar the two sectors are in terms of factor intensity. If the sectors are very similar in this sense, small price changes may lead to large changes in the factor price ratio, and consequently also in output for the two sectors. But if the sectors are similar, changes in factor endowment will have little impact on their relative prices, so not too much emphasis should be put on this effect.

The elasticity of the factor price ratio with respect to pA/pB is equal to

20) (9) [pic]

Given that 0 β |α < β |

|K1 |+ |- |

|L1 |+ |- |

|K2 |+ |+ |

|L2 |- |- |

The response to changes in K2 and R2 is standard; an increase in the amount of a factor in the diversified country 2 will lower its relative price. From (A1), we know that the same is the case in the specialised country 1, with respect to w1/r1. The special thing with non-diversified cases is that the international links will be different. We already know that the factor price ratio in country 1 is unaffected by factor endowment changes in country 2. And for country 2, Table A1 shows that factor endowment changes in country 1 has a “perverse” effect: An increase in any of the two factors in country 1 will have the same effect on factor prices in country 2. If A is K-intensive, for example, an increase in K1 as well as L1 will lead to increased production of A in country 1. This will increase country 2’s imports of A, and reduce production of A. Since A is K-intensive, this will lower the relative price of K. If sector A is L-intensive, an increase in K1 or L1 will lower the relative factor price w2/r2. This is different from the case with factor price equalisation, where an increase in endowments of a factor will have a similar impact in both countries.

The result has interesting implications. For example, consider poor developing countries that are specialised in L-intensive goods, and have a relatively large L stock. If these grow and accumulate physical capital (by investing) or human capital (by education and learning), this may at first be to the detriment of unskilled labour in rich countries. At the point when the developing countries become diversified, further growth will reduce the w/r ratio in rich countries. Hence specialisation may lead to a “discontinuity” in the impact of growth in developing countries.

The outcome outlined above will not be sustainable if the countries become too different. At some stage, it is possible that country 2 will specialise in B. From (16) we know that this occurs when w2/r2= (K2/L2)*(1-α)/α. Inserting this into Φ, we obtain the condition K1αL11-α = K2αL21-α. This defines the “borderline” between factor allocations that result in specialisation in good A in country 1 only, and allocations that also create full specialisation in country 2 (in sector B). Corresponding to the other three cases where one country is diversified and the other specialised in one sector, there are three other “borderlines”, as shown in Diagram 7 in the main text.

In the main text, we also examine the case when country 1 is specialised in sector B, while country 2 is diversified. The equilibrium condition corresponding to (A7a) is then

(A7b) ΦB = [pic]

This is used for the simulations in the main text.

Both countries specialised

We continue assuming that country 1 is specialised in good A; hence equations (A1)-(A3) still apply. For country 2, which now specialises in B, with production F2B=K2 β L21-β, we obtain similarly

(A8) [pic]

The consumption level for B in country 2 (derived similar to A3) will be

(A9) B2 = (1-a)F2B

In value, exports of A from country 1 must equal exports of B from country 2. This gives

(A10) (F1A - A1) pA = (F2B - B2) pB

or equivalently, using the results above and the earlier results for A,

(A10a) (1-a) F1A pA = a F2B pB

Observe that an implication of this, since Y1= F1A pA and Y2 = F2B pB, is that the income ratio is equal to

(A11) [pic]

Hence consumers use a certain share of their money on each good, and these money accrue to the respective producing countries. The income share of a country is independent of its size; it depends only on the consumption shares! If a small country is the sole producer of a highly demanded good, it can become very rich due to the relatively high price for that good.

Equation (A10) also defines the price ratio between the goods, which is

(A12) [pic]

With perfect competition, unit costs must equal the price in the respective countries, as in equations (17)-(18). Observe that not only will factor price ratios be different in the two countries, but also levels. Replacing pA and pB in (A12) with unit costs, using (17)-(18), (A1) and (A8), we obtain:

(A13) [pic]

Hence w1 is relatively larger, if demand for good A is high, if production of good A is relatively more L-intensive (low α or high β), or if the L stock in country 1 is relatively small. Observe that the K’s do not enter here, so e.g. a large poor country with a large L stock can have a very low wage, irrespective of its K/L ratio.

In order to examine welfare, we may observe that consumption of the non-produced good in either country is equal to its imports. Using earlier results, we obtain

(A14a) U1** = a F1Aa F2B1-a

(A14b) U2** = (1-a) F1Aa F2B1-a

The ratio between welfare in the two countries is therefore equal to the income ratio (A11), or a/(1-a). Hence how the “pie is shared”, depends only on consumption shares. On the other hand, the “pie may be enlarged” by means of factor endowment increases in either country. With complete specialisation, it is an advantage to be a small country: If e.g. country 2 becomes ten times larger than country 1, country 1 will still have a share of world real income equal to the consumption share of its product. Hence income per factor unit in country 1 must be much higher.

A comparison with welfare in autarky could be possible by using the conditions defining the two “borderlines” for this case with specialisation in both countries, for studying e.g. U1**-U10. It is, however, difficult to obtain an unambiguous conclusion for the whole permissible range.

-----------------------

[1] Jones, R.W. (1965), The Structure of Simple General Equilibrium Models, Journal of Political Economy 73: 557-572. Reprinted in Bhagwati, J. (1981), International Trade: Selected Readings, first edition, MIT Press.

[2] Dixit, A. and V. Norman (1980), Theory of International Trade, Cambridge University Press.

[3] We do not report these expressions. The reader may derive their form; most easily by inserting the solutions for factor use into the production functions.

[4] In Bhagwati, J., A. Panagaryia and T.N. Srinivasan (1998), Lectures on International Trade, Second Edition, MIT Press.

[5] Samuelson, P.A. (1962), The Gains from International Trade Once Again, The Economic Journal 72: 820-829, reprinted in Bhagwati, J. (1981), International Trade. Selected Readings. First edition. MIT Press.

[6] The first order derivative is ³Æ1-³-1 (Æ1-1) (1-³) and the second order derivative is equal to

³ (1-³) (1+³) Æ1-The first order derivative is γφ1-γ-1 (φ1-1) (1-γ) and the second order derivative is equal to

γ (1-γ) (1+γ) φ1-γ-2.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download