Comparative Advantage Without Tears: a Cobb-Douglas Version of HOS

Comparative Advantage Without Tears: a Cobb-Douglas Version of HOS

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)

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 =

(14) FB =

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.

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.

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) or

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

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)

Given that 0<,<1, the absolute value of the denominator is smaller than 1, so the elasticity is larger than 1 in absolute value. This is the magnification effect expressed in the Stolper-Samuelson Theorem. Contrary to the Rybczynski theorem, this is a general property of the HOS model. It should be recalled, however, that is only applies within the range of diversification.

2.4. Solving the model: The closed economy case

In order to derive goods and factor prices, we need to introduce the demand side of the model. Still keeping things as simple as possible, we use a Cobb-Douglas utility function:

(21) U = Aa B1-a

where A and B are consumption levels for A and B goods, with a and (1-a) as the consumption shares. Hence we must have

(22)

In a closed economy, we must have consumption = production; A=FA and B=FB. Now substituting into (22) from (13-14), and replacing pA/pB with r/w using (19), we obtain the equation:

(23)

which gives the solution

(24)

Hence the factor price ratio is (inversely) proportional to the K/L ratio; an increase in K leads to a relative reduction in r. Observe that since ,  and a are all positive and smaller than 1, the fraction to the right is positive.

The equation can also be re-written as

(24a)

which shows that w/r is a weighted average of the factor price ratios defining the range of diversification. Hence the single economy is diversified.

The numerator and denominator of the fraction to the right of (24) are weighted averages of the factor cost shares of the two sectors, with the consumption shares as weights. Since these expressions will appear several times in the following calculations, we shall use the notation  = a+(1-a),  = a(1-)+(1-a)(1-), hence we may write

(24b)

For later use, it is useful to observe that  +  = 1 (this is frequently used for simplifying expressions).

If any of the two sectors become more K-intensive, it will lead to a relative increase in r (a reduction in w/r). We have (the reader may easily check these results):

(25) and

If sector A is K-intensive, an increase in its consumption share will lead to a relative increase in r. We have

(26) if >and if <

Corresponding to these factor price effects, we have corresponding changes in goods prices. Using (19) and the solution (24), we have

(27)

with  and  defined as above. It is then easy to verify that an increase in K leads to a relative decline in the price for the K-intensive good. It is also unambiguously the case that an increase in the consumption share for a sector will lead to a relative increase in its price. The price impact of changing technology; i.e. changing factor intensity in the sectors, is more complex: As seen from (27), changes in  or  will affect all the three components of the expression (remember that ZA and ZB are functions of these parameters). In order to limit the length of this note, we drop a closer examination of this issue.

2.5. Factor use, production and the lost Rybczynski effect

Using (7), the factor market clearing equation (10a) may be rephrased as

(28)

Using (10b) as well (LA+LB=L) we can solve for LA and LB, and by using (7) we also find KA and KB. Now substituting for r/w (or w/r) using the solution (24), we obtain the solutions for factor use at equilibrium:

(29a)

(29b)

(29c)

(29d)

The comparative statics are straightforward (remember to spell out  and  in the calculations for a, α and β), e.g. for KA:

(30)

- An increase in K will increase the use of K in both sectors, without affecting the use of L. Observe that the ratio KA/KB is unaffected by K, so capital use will be scaled up proportionately in both sectors.

- The more K-intensive a sector is, and the less K-intensive is the other sector, the more K will it use. If the two sectors “compete for the same factor”, they will as a result use less of it.

- If the demand for goods from one sector increases, the quantity of both factors used in that sector increases, but shrinks in the other sector.

Observe also that factor use is unambiguously positive as long as all parameters are positive. As noted above, there will always be diversification in the closed economy.

By substituting the factor price ratio (24) in the expressions (13)-(14) for production, or by inserting the solutions (29) into the production functions (1)-(2), we can derive equilibrium levels of production. The resulting expressions may be written in a simple form:

(31) FA = K L1- A(a,,)

(32) FB = K L1- B(a,,)

where A and B are functions that may, for given values of a,  and , be considered as (positive) constants.[3] Hence the production levels in the two sectors are simple functions of the factor stocks in the economy. This stylised outcome is a property of the Cobb-Douglas technology, and will not apply generally.

As noted above, an increase in e.g. the K stock will increase the use of K in both sectors, while their use of L will be unchanged. Hence production must increase in both sectors, so the Rybczynski effect with reduced production in one of the sectors does not apply. The elasticities for production with respect to factor endowment changes can be read from (31-32) directly: If K increases by 1%, production in sector A increases by %, and production in sector B by %. Hence production in the K-intensive sector will increase faster, but there is no “magnification effect” in the sense that production in this sector will increase relatively more than the supply of K. This illustrates that the “Rybczynski Theorem” is a special case. This case is at odds with the general logic of the HOS theory – which is that prices adjust endogenously and depend on technology, demand and factor stocks.

2.6. Welfare in the closed economy

It is intuitively evident that if a country has a factor composition which is very different compared to the cost shares for the sectors, it will be worse off compared to a situation where there is a “match”. For example, if  and  are both high (K-intensive), but the country’s K/L ratio is very low, K will be scarce. In this situation, r will be very high, but the physical use of K will be low in both sectors.

In order to show this formally, we insert the solutions for FA and FB (31-32) into the utility function (21) and obtain:

(33) U = Ka+(1-a) La(1-)+(1-a)(1-) (A(a,,))a (B(a,,))(1-a)

or equivalently

(33a)U = K Lτ

where  and  are as before, and we use τ = (A(a,,))a (B(a,,))(1-a) in order to simplify notation.

Given that  as well as  are positive, any increase in K or L will add to the economy’s welfare. Given that U applies to the whole country, and K as well as L also reflect country size, this is not surprising. It is more interesting to study how the composition of factor endowments affect welfare. We therefore examine welfare with the assumption that K + L = R, where R is a constant. Hence we keep the size of the economy constant and study how factor composition affects welfare. The utility function is then:

(33b)U = K (R-K) 

Now we find:

(34)

The sign of this derivative is ambiguous, depending on the sign of the expression in brackets. This is zero when the K share K/R is equal to  (=a+(1-a)). With some more calculation, we find that

(35)

Since 0<<1, this expression is negative.

This implies that welfare reaches a maximum when K/R=γ=a+(1-a), or equivalently (since L/R must then be 1-γ=θ) K/L=γ/θ. Hence the better the match between factor endowments and the cost shares in production, the more efficient will the economy be.

Diagram 4 shows how production volumes in the two sectors, and welfare, changes with K/L, given that K+L=R is constant. The illustration is based on a “symmetrical” case with =0.7, =0.3 and a=0.5, i.e. good A is the K-intensive:

With a low K/L-ratio, less will be produced of good A, and with a high K/L ratio, less will be produced of good B. With a high K/L ratio, good A will be relatively cheaper. For both goods, production is highest when the K/L ratio is equal to cost shares in production (i.e.0.7/0.3=2.33 for sector A, 0.3/0.7=0.43 for sector B). For the economy as a whole, welfare is highest when K/L is between these two values. Given that consumption shares are equal in our example and +=1, welfare is highest when the K/L ratio is equal to 1 (i.e. ln(K/L)=0).

The property that production is maximised for each sector when K/L is equal to the ratio between cost shares, is easily confirmed analytically: By using K+L=R and substituting for L in (31-32), and calculating the first and second order derivatives with respect to K, we find that FA is maximised when K/R=, or equivalently K/L=/(1-), and FB reaches its maximum when K/R=, or equivalently K/L=/(1-). As shown above, welfare maximum is obtained K/R is a weighted average of  and , with the consumption shares as weights.

Diagram 4 illustrates the importance of a “match” between factor endowments and technological requirements. This result also raises some interesting questions with respect to international trade. Suppose, for example, that a country with a K/L ratio that is optimal for welfare integrates with a country that has an extreme K/L ratio. We would then expect that the latter has more to gain from international trade. An issue is even whether the “optimal” country could even lose from trade. As we shall see, this is not the case.

2.7. Summing up: The HOS closed economy