Capital markets and urban unemployment

Capital markets and urban unemployment

Journal of International Economies 15 (1983) 367-385. North-Holland CAPITAL MARKETS AND URBAN UNEMPLOYMENT M. Ali K H A N * Pakistan Institute of Dev...

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Journal of International Economies 15 (1983) 367-385. North-Holland

CAPITAL MARKETS AND URBAN UNEMPLOYMENT M. Ali K H A N * Pakistan Institute of Development Economics and the Department of Political Economy, The Johns Hopkins University, Baltimore, MD 21218, USA Syed N a w a b H a i d e r N A Q V I Pakistan Institute of Development Economics, P.O. Box 1091, lslamabad, Pakistan Received February 1982, revised version received September 1982 In this paper we investigate conditions under which reductions in distortions in one market improve national welfare despite the presence of distortions in another market. The specific distortions we study are a differential between capital rentals and a rigid urban wage. We also introduce a dia~ammatic technique which may find use in the analysis of other problems.

I. Introduction In their perceptive paper on r u r a l - u r b a n migration, Harris a n d T o d a r o (1970) assumed that no factor of p r o d u c t i o n , other t h a n labor, was intersectorally mobile. If capital is assumed to be the only other factor of p r o d u c t i o n , 1 as in the H e c k s c h e r - O h l i n - S a m u e l s o n world, the H a r r i s T o d a r o equilibrium can be viewed as one in which capital earns differential rates of return. As such, in a d d i t i o n to the twin l a b o r m a r k e t distortions arising out of a sector-specific rigid wage a n d the equality of expected wages, a further distortion pertaining to capital m a r k e t s is i n t r o d u c e d into the model. It is of interest to k n o w what impact such a n a s s u m p t i o n has on social welfare, measured in terms of G N P evaluated at i n t e r n a t i o n a l prices. O u r paper reports o n this. *M. Ali Khan would like to acknowledge the support and hospitality of the United Nations Development Program for making it possible for him to visit the P.I.D.E. during the months of May and June 1980. He would also like to thank Bela Balassa and Tatsuo Hatta for encouragement and several helpful comments on the Generalized Harris-Todaro model. Our final thanks to an anonymous referee on whose suggestions this condensed version of Johns Hopkins Working Paper no. 89 was prepared. In particular, the stability result is prompted by his conjecture. 1The Harris-Todaro (1970) model could alternatively be conceived as a three-factor model in which the immobile factors are rural land and urban capital. Our paper has no relevance for such a set-up. 0022-1996/83/$3.00 © 1983 Elsevier Science Publishers B.V. (North-HoUand)

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M.A. Khan and S.N.H. Naqvi, Capital markets and urban unemployment

Our paper can also be seen as a contribution to the general theory of distortions and welfare. On the one hand, we study a situation in which an LDC government cannot directly interfere in the labor markets, but instead concentrates solely on improving the intersectoral allocation of capital; and, on the other hand, takes the distortion in capital markets as given and can only change the urban wage. In either case, we illustrate in a specific context, a more general investigation of conditions under which distortions in one market exacerbate or moderate, in terms of social welfare, the effects of distortions present in other markets. Using, as our relevant backdrop, the following proposition of Bhagwati (1971),2

Reductions in the "degree' of a distortion will not necessarily be welfare increasing if there is another distortion in the system. We show, for example, that with a more capital-intensive urban sector, a decrease in the differential between rentals is always welfare increasing despite the Harris-Todaro distortions. In his brief overview, Bhagwati (1976) focused on three kinds of factor market distortions: the economy-wide rigid wage fi la Brecher-JohnsonLefeber; wage differentials; and the Harris-Todaro distortion. The model presented below is a synthesis in the sense that it simultaneously combines two of the above features by introducing a differential in capital rentals in the Harris-Todaro setting. Of course, such a differential with perfect labor markets goes back at least to Harberger's (1962) piece. Our generalized treatment highlights, for example, the importance of three kinds of factor intensities and it is hoped that it will also be useful for the analysis of Other problems. A final contribution of our paper is to show the relevance of the geometry of the dual so elegantly presented by Mussa (19791.3 We think that it can be compared with advantage to the geometry of the primal originally applied to the Harris-Todaro model by Corden and Findlay (1975)and Khan (1980a). Our paper proceeds as follows. Section 2 presents the model and develops the basic geometry. Section 3 uses this geometry to present a preliminary analysis of the model. In this section we concentrate, in particular, on those results of the Harris-Todaro model which cease to hold when capital markets are no longer perfect. Section 4 is the core of the paper and studies changes in the rental differential and the rigid urban wage. Our geometry relies crucially on there being a rigid, urban wage. In Section 5 we consider the issue of stability of equilibrium. Section 6 concludes the paper with some additional remarks. "See his proposition 6. 3Also see the reference to the work of Burgess and Woodland in Mussa's paper.

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2. The model

Let a country consist of an urban and a rural sector, indexed by u and r, respectively, and be endowed with positive amounts of labor L and capital K. Let the ith sector produce a commodity in amount X~ in accordance with a well-behaved, constant returns to scale production function4 X~ =F~(Li,KJ, where L~ and Ki represent the allocations of labor and capital to the production of the ith commodity. Given marginal productivity pricing, it is a well-known proposition that the technologies F~(') can be equivalently depicted in terms of the underlying unit cost functions C~(') with pi = Ci(w~, Ri),

(1)

and the slope 5 of the C~(.) function given by ( - K d L i ) . For any given international prices p~, one can portray the iso-price curves in the (w-R) plane as in fig. 1. The shape of these curves follows from the wellbehavedness of the individual technologies; what should be emphasized is that they may intersect more than once 6 - - a possibility we disregard in the sequel. w

u

F ~r G

0

"

/~

A

D

C

T

R

Fig, 1. A Harris--Todaro equilibrium (It, p) with perfect capital markets,

"~We shall assume that F~ is twice continuously diffcrentiable in each of its arguments and is strictly concave, 5It is a simple consequence of Samuelson's Envelope Theorem that dC,/~w~=LJX~ and ?CJOR~=KJX~, Additional properties of the unit cost function are that it is positively homogeneous of degree one in its arguments and concave. On all this, see, for example, the references in Mussa's (1980) paper. 6This is simply a standard factor-intensity reversal.

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M.A. Khan and S.N.H. Naqoi, Capital markets and urban unemployment

In the absence of any factor-market distortions, the producers in either sector face the same set of factor prices and hence an equilibrium is given by the intersection at 8 of the iso-price curves uu and rr. We shall now use these iso-price curves to depict a Harris-Todaro equilibrium with the additional feature of imperfect capital markets, i.e. where 2 = U/L,,

w, = ~--- (1 + A)w,; R,=(1 + t)R=;

(2a)

t>0.

(2b)

(2a) formalizes two separate elements: (i) an exogenously given fixed urban wage 7 and (ii) the Harris-Todaro equilibrating condition given by the equality of expected wages. U is the size of the urban unemployed and hence ,~ can be interpreted as an unemployment rate a with (1/(1 +2) formalizing the probability of getting a job. t is an exogenously given differential between capital returns.

2.1. Perfect capital markets We begin by introducing our geometry for the case where there are no capital-market distortions, i.e. t=O. Consider fig. 1. OG is the exogenously given urban wage which determines that the urban producers operate at p and with OA the rental to capital for the economy as a whole. This leads the rural producers to operate at p with Ap the corresponding rural wage. Since #p represents ( ~ - w , ) , it is clear that I~p/pA is the unemployment rate 2. Finally, GpT is the tangent to the rr curve at p and its slope is (-K,/L,), and similarly, /zD is the tangent to the uu curve at /~ and its slope is --(K=/L.). Given the above basics, we can now depict the value of the G N P at a particular Harris-Todaro equilibrium, i.e. the value V = p , X , + p , X . . Fig. 1 shows the value of V/K to be OC and the value V/L to be OF, the aggregate capital-labor ratio being given by the slope of the dotted line FpC. This can easily be seen. Let X,/ be the marginal product of the jth factor in the production of the ith output. Then, by constant returns to scale L

L

£

K

V = prXr L r -I- puXu L= -I- prXr K r -b paX= K..

(3)

Since we have the equations L , + L , ( 1 +2) = L ;

K,+K,=K,

(4)

7In the context of perfect capital markets, Khan (1979, 1980) relaxes this assumption by the more general w . ~ ( w , , ~ , K . ) . SA more conventional definition of the unemployment rate is given by U/(L~ + U), i.e.

~./(I+~).

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371

marginal productivity pricing and (2a) yield: (5)

V = w,L + RK.

Since K / L = p A / A C = w , / ( O C - R), we obtain: (6)

V/K = w,L/K + R = OC.

Similarly we can obtain VL = OF. The above discussion, in particular (5), also makes clear the fact that the coordinates of p measure the social opportunity costs of capital and labor. All that remains is the depiction of the outputs X~ and the allocation of labor and capital to the production of each good. For this we need only concentrate on the lines GpT and pD and, accordingly, fig. 2 focuses only on them. However, for ease of later reference, p is designated by H, the intersection of HD with the w-axis by J, and the point A by l,C.The principal point to keep in mind here is that the slope of the line p D ( = J H D ) is given by (-K,,/(I+2)L,,). This easily follows from the fact that pA/AD=I~A/AD (1 + 2). Now let K J K be denoted by oft, the aggregate labor-capital ratio by I, and l~ the labor-capital ratio in the production of good i. From (4), it is then easy to see that /-l.(1 +2) ~g"=l,-l,(l+2)

and

l,--I .tC,=I_I,(I+2).

(7)

w

••x

Ku/(l+x)Lu

G

K/L

H I ~ 0

V

D

~ K r ILr C Fig. Z

T

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M.A. Khan and S.N.H. Naqvi. Capitalmarketsand urban unemployment

But CV/y=I, DV/y=I,(I+2) and TV/y=l,, and thus the reader can check for himself that

3g', = CD/TO

and

.xe. = CT/TD.

(8)

Since XdK=.av~f(li), the above magnitudes can also be used as measures of the output per capital stock in each sector. A perfectly symmetrical discussion shows that

L,/L=JF/JG

and

(I+2)LdL=FG/JG.

(9)

Let the line passing through p and parallel to DY intersect the R-axis at L (see figs. 6 and 7 below). It is then easy to see that the size of the urban unemployment U is measured by DL in the sense that U=(Ku/~)DL. In summary, the above discussion of how physical magnitudes (outputs and factor allocations) are determined only serves to reinforce the point made by Mussa (1979) that the unit cost function diagram is more suited to illustrating value rather than physical magnitudes. It is also worth mentioning in passing, that the cone DpT (DHT in fig. 2) represents in the context of the Harris-Todaro model, what Chipman calls the 'cone of diversification'. Any aggregate capital-labor ratio whose slope is greater than that of JHD and less than that of GHT, represents a parameterization that is consistent with an unspecialized equilibrium. Further, the Harris-Todaro equilibrium depicted in fig. 1 is one for which the urban sector is more capital-intensive in employment-adjusted terms 9 than the rural sector, i.e.

D - K . L , - K,L.(1 + 2)= K,K.(i~-I.(1 +2))>0.

(10)

A final point to be noted is that the tangent at any point on an iso-priee locus can be used to give the ratio of factor shares at that point. Thus, in fig. 1, TA/AO is O,t./O,r, where 0,L is the share of labor in the value of total rural output. This follows from the simple observation that OA is R, and A T is W~L,/K,. Analogously, the intersection of GT with the y-axis would give

0,d0,L. We thus have in figs. 1 and 2 a diagrammatic technique that exhibits all the endogenous variables associated with a Harris-Todaro equilibrium in a setting where there is no differential between capital rentals. In the next subsection we show how the geometry can be modified to handle this complication. 9These intensitieswere first introducedin Khan (1980); also see Neary (1981).

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373

2.2. Imperfect capital markets

Fig. 3 depicts a Harris-Todaro equilibrium when there is a positive differential t between capital rentals. As before, AI~ represents the exogenously given urban wage with OA the corresponding rental to urban capital. OA is, of course, no longer the rental to rural capital. The angle QOA measures the rental differential t and given R~, R, is measured by ON. ON transferred to the R-axis determines that the rural producers operate at p with B p = A Y as the rural wage. Since (~(,-wr) is now represented by #Y, it is clear that I~Y/YA is the unemployment rate 2. This has also as an immediate corollary that the slope of H Y D is given by ( - K J(1 +2)L~). A comparison of fig. 3 with fig. 1 bears emphasizing. In the absence of a rental differential, R ~ = R r and B in fig. 3 becomes identical with A. This causes Y, H and p to become identical. The 'cone of diversification' is now given by DHT, or equivalently by GHJ. Fig. 2 applies without any change; W.

U

0

Fig. 3. A Harris-Todaro equilibrium(ll,p) with imperfectcapital markets.

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M.A. Khan and S.N.H. Naqvi, Capitalmarketsand urban unen,ployment

OC and OF measure V/K and V/L, respectively, and it is now the coordinates of H that measure the social opportunity costs of labor and capital. However, the last two assertions need somewhat more justification than the corresponding one for perfect capital markets. Let the coordinates of the point H be (R*, w*). Then the equation of the line Hp is given by

(11)

(w* -- w~)/(R* -- R~) = ( -- K,/L,) =L,w* + KrR* = p,X,,

and that of the line HYD by (12)

(1 + 2)L~w* + K~R* = p~X~.

Addition of (11) and (12) yields that V=w*L+R*K. Since the slope of the dotted line passing through H measures the capital-labor ratio, the second assertion follows. The third assertion follows directly from Diamond and Mirrlees (1976) since (R*,w*) represents the wage and rental that equate the shadow profit rate to zero in each sector, as can be seen from eqs. (11) and (12). Following Srinivasan and Bhagwati (1978), a more direct proof can also be given. The social opportunity cost of (say) labor is given by aV OX, aX~ O L = p ~ - ~ + p~ - ~ .

(13)

Given constancy of aggregate capital and labor and international prices fix KdX~, LdXi and 2, we obtain: K, 0Xr K~ dX~ X, O~ ~ X~ OL = 0

and

the fact that

L, 3X~ Lu(1+ 2) t~X~ X~ ~ - q X~ O----L-= 1.

(14)

On substituting the solution of OXJaL and aXr/dL in (13), we obtain a value of aW/aL which is precisely the w* obtained by solving the eqs. (11) and (12) or, in geometric terms, is the w coordinate of the intersection of the line GHp and J H Z A completely analogous argument can be provided for OV/aK. We began this paper with the assertion that a Harris-Todaro equilibrium in a setting with intersectoral immobility of capital can be viewed as one in which capital earns differential rates of return. We conclude this subsection by presenting fig. 4 which substantiates this. It combines the Corden-Findlay geometry with ours and a version of it has been used by Neary (1982). The upper left-hand quadrant is fig. 1 of Corden-Findlay and determines, in particular, ~ and wr. Once we have these, the upper right-hand quadrant determines the rentals R r and Ru. If these rentals pertain to the same factor of production, the lower right-hand quadrant enables us to calculate the

M.A. Khan and S.N.H. Naqoi, Capital markets and urban unemployment W

A

375

u

tz

q I r

~A' U

0 l

Q N Fig. 4. A Harris-Todaro equilibrium (p, p) with capital immobility.

implicit differential t. Fig. 4 also draws the relevance of our geometry to a setting where labor is the only mobile factor of production.

3. A preliminary analysis In this section we use the geometry developed in section 2 to derive some recent results for the Harris-Todaro model with perfect capital markets 1° and show that some of these results no longer hold when a rental differential is introduced. In the absence of a rental differential, R= = R , and Y and H shift to p, resulting in a simplification that is a source of several results that, we now find, do not carry over to a more general setting. Consider, to begin with, the proposition that a more capital-intensive rural sector implies non-existence of a Harris-Todaro equilibrium. 11 This can be seen rather simply by interchanging the uu and rr curves. In this case the urban sector is less capital-intensive than the rural sector for all urban wages higher than the one given by the undistorted equilibrium s. Thus, any urban wage on the ue segment of the uu curve leads to a higher rural wage and hence implies a ~°This is the setting studied by Corden and Findlay (1975) in their pioneering paper; also see Stiglitz (1978), Khan (1980) and Neary (1981). IISee Khan (1980a, p. 540). JIE--H

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M.A. Khan and S.N.H. Naqoi, Capital markets and urbm, unemployn,ent

negative rate of unemployment 2, precluding the existence of a HarrisTodaro equilibrium as defined. 12 With imperfect capital markets, no such existence problems necessarily arise, as can be seen from the equilibrium exhibited in fig. 5. A second consequence of p being identical with Y is that employmentadjusted factor intensities coincide with value factor intensities, i.e. sign D = sign [1, - l~(1 + 2)] = sign A = sign [0,L0,x - 0,,~0,K].

(15)

In the geometry of fig. 1, a more capital-intensive (in employment-adjusted terms) rural sector implies that the line pD will always intersect the R-axis

M

Y

0

B

TC

D

R

N Ic--a-=--~ 0 Fig. 5. A Hards-Todaro equilibrium (I~,P) with the rural sector more capital-intensive than the urban sector in terms of employment-adjusted factor intensities and less capitalqntensive in terms of value factor intensities. t2It is important to bear in mind that once we admit reversal of factor-intensity rankings, it is quite possible for a Harris--Todaro equilibrium to exist in which agriculture is relatively capitalintensive. We are indebted to an anonymous referee for drawing our attention to this possibility. Note that this is also explicitly stated in Khan (1980a, p. 540, paragraph 3).

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M.A. Khat, and S.N.H. Naqvi, Capital markets and urban tmemployment

between T and A leading to TA/OA > D A / O A implying that A > 0. This is the reason why value factor intensities have no role to play in the treatment of the Harris-Todaro model given, for example, in Khan (1980a, b). As figs. 5 and 6 make clear, this may no longer be true when a capital differential is introduced. In algebraic terms, A

=

(L'wrR"K=-L=w"R~K') wrR,,K,K~ 1 = ~ (t, - ,,(1 +),)(1 p,p=X~X . p,p.,.a , a .

+ t)).

(16)

A third consequence of p being identical with Y is that the social opportunity costs of labor and capital are the rural wage and rental, respectively, t3 This is just the observation that H is identical with p and hence the coordinates of H, (R*, w*) are indeed (R, w,). The fact that this is no longer true in the general setting can be seen from fig. 6 in which the social opportunity cost of capital is negative. It is important to note that negative social opportunity costs are not a consequence of a reversal of factor intensities as figs. 5 and 7 make clear. Given the impossibility of negative social opportunity costs in the HarrisTodaro model with perfect capital markets, we have the concurrent result of the impossibility of immiserizing growth of labor and capital. 14 In the context of fig. 1, an increase in the aggregate endowment of labor flattens the

w r

p. p' P Y

0

A B' B D

L'

TL

R

Fig. 6. Negativesocial opportunitycost of capital at the Harris-Todaro equilibrium(It,p). taSee proposition4,1 in Khan (1979). l'See Khan (1982).

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M.A. Khan and S.N.H. Naqoi, Capital markets and urban unemployment

curve pC and, by shifting the point C outwards, increases welfare. Similarly, the fact that an increase in aggregate capital leads to an increase in welfare can be seen by the shifting of the intersection of pC and the w-axis. These results no longer hold when capital markets are imperfect. It is easy to check in fig. 6 that an increase in aggregate capital is immis'erizing. Fig. 7 shows a situation when an increase in labor is immiserizing. A further point to be noted is that fig. 6 exhibits reversal of factor intensities and fig. 7 does not. W

r

0

U

A

Fig. 7. Negative social opportunity cost of labor at the Harris-Todaro equilibrium (p, p) with the urban sector more capital-intensive than the rural sector in terms of both employmentadjusted and value factor intensities.

Another consequence of Y being identical with H is that changes in welfare are accurately measured by changes in 2, i.e. by changes in the ratio I~Y/YA. The reason for this is simply that the diversification cone has its vertex at Y. Thus, in fig. 9, 2 is getting progressively larger as the urban wage is getting progressively higher, leading us to conclude that there is welfare deterioration at each stage. So far we have concentrated on the changes brought about in the HarrisTodaro model with the introduction of a differential between capital rentals. It is also worth pointing out that some basic properties of the model remain unchanged. It is an important property of the Harris-Todaro model that the unemployment rate depends solely on the international prices and the

M.A. Khan and S.N.H. Naqvi, Capital markets and urban unemployment

379

exogenously given urban wage. Is It is independent of the aggregate endowment provided the economy is incompletely specialized. This is just the observation that in fig. 1 the ratio lzp/pA is independent of the slope of the line pC provided this line is rotated within the cone of diversification DpT. As can be seen from fig. 3, this result survives to a setting with imperfect capital markets with the only qualification that 2 is now also made to depend on t. If the urban sector is more capital-intensive in employment-adjusted terms than the rural sector, i.e.

Ir/(l + 2)-l.>O,

(17)

an increase in capital increases urban output and decreases rural output, provided the economy remains unspecialized. An analogous statement holds for increases in labor. This is Rybczynski's theorem as generalized to the Harris-Todaro model and it is equally applicable to the rental-differential case as it is to the situation when capital markets are perfect. In either case, it can be proved using fig. 2 and with the use of eq. (8). For example, as L increases, CF becomes flatter, and hence, d~", and X, increase and o,Y"u and X, fall. As we would expect, value intensities play no role in the validity of Rybczynski's theorem, even in the presence of a rental differential. Finally, we consider changes in the given international prices Pr and p,. For such changes, the relevant iso-price locus just shifts uniformly outward. In the case without a rental differential, it is easy to see from fig. 1 that an increase in the rural price leaves the rental unchanged, increases the rural wage, and decreases the rate of unemployment. As can be seen from figs. 5 and 6, this result carries over to the rental-differential case with the only qualification that both rentals remain unchanged. It has to be emphasized that this result is independent of either kind of factor intensities. It can also be easily checked that an increase in the urban price, on the other hand, increases rentals, depresses the rural wage, and increases the unemployment rate.

4. Reductions of distortions in one market

In this section we get to the main object of this paper, namely to give conditions under which a ceterus paribus reduction in the capital-market distortion improves welfare despite the labor-market distortions.

tSKhan (1980a) terms this the "factor-price and unemployment rate equalization theorem'. Note that this property is true only of the Harris-Todaro model with intersectoral capital mobility; it does not hold in the original Harris-Todaro specification.

380 4.1.

M.A. Khan and S.N.H. Naqvi, Capital markets and urban unemployment A reduction in the capital differential

In terms of the geometry of Sections 2 and 3, we have to find conditions under which a flattening of the line OQ or, equivalently, a shortening of the distance AB leads to the line HC moving outward. Begin with fig. 3. As the differential is reduced and B moves towards A, Y moves upwards towards p' leading to a steepening of DY. Since pG is also becoming steeper, the cone DHp is pulled upwards resulting in the line FHC monotonically moving outward. In the limiting case of a zero differential, p, H and Y are all at p' and welfare is at its maximum. Since the urban sector is more capitalintensive in employment-adjusted terms than the rural sector, we have Proposition 1. If the urban sector is more capital-intensive than the rural sector in employment-adjusted terms, a decrease in the differential between capital rentals always improves welfare. To complete the proof of proposition 1 we have yet to show that an increase in the differential may be immiserizing when the factor intensities are reversed. The reader can check for himself that this is indeed true from fig. 5. However, fig. 5 exhibits the phenomenon of reversal of factor intensities and we have to make sure that this is not relevant to our result. This can be seen from fig. 8 in which there is welfare improvement despite the reversal of factor intensities. Our proposition 1 can be usefully compared with an analogous result for the wage differential setting as described, for example, in Batra (1973). There, a rise in the wage differential leads to an improvement in welfare if and ohly if physical and value factor intensities give identical rankings of the two sectors. 16 Our proposition 1 has nothing definite to say for situations when the rural sector is the more capital-intensive one. However, when the economy is in a position of laissez-faire, i.e. t = 0, we have a much sharper result. Proposition 2. I f capital markets are perfect, a tax on rural capital always leads to a deterioration in welfare. Such a proposition is not surprising in view of the fact that a HarrisTodaro equilibrium with perfect capital markets does not exist when the rural sector is more capital-intensive than the urban sector and as such, we are always under the hypothesis of proposition 1. At any rate, the relevant diagrams for checking the validity of proposition 2 are figs. 5 and 6.

t6See, for example, Batra (1973, section 10.9).

M.A. Khan and S.N.H. Naqvi, Capital markets and urban unemployment

381

~ I~'

r

0

N'I/

~ C BD C' ~]

Fig. 8. An increase in the urban wage leads to welfareimprovement without reversal of factor intensifies.

4.2. An increase in the urban wage

It is a known result 17 that with perfect capital markets, an increase in the urban wage decreases welfare if and only if the urban sector is more capitalintensive than the rural sector in employment-adjusted terms. Such a result is illustrated in fig. 9 which also makes explicit the fact that this result is contingent on the economy remaining unspecialized. As can be checked, this is indeed so for the wage'increase/~ to #' or #' to #" but not for # to #". Unfortunately, this result is no longer valid with a differential between capital rentals. A convincing demonstration of this is given in fig. 7 where the urban sector is indeed more capital-intensive in employment-adjusted terms than the rural sector and, furthermore, there is no reversal of factor intensities. The reason for this becomes clear once we realize that, in the presence of a rental differential, a change in welfare can be decomposed into two parts. The first is a change in the size of urban unemployment valued at t~Th/s is a simple consequenceof theorem 6.1(0 in Khan (1980b).

382

M.A. Khan and S.N.H. Naqvi, Capital markets and urban unemployment W

jii

0

D"

D'

D

R

Fig. 9. An increase in the urban wage leads to a deterioration in welfare.

the rural wage and the second is the change in rural capital stock valued at the absolute differential in rentals Rut. More formally, 18

OV/O~ = - w,L, 02t/Off + R~t OK,/O~.

(18)

Both terms in our decomposition measure the resources being wasted a~ a result of the distortions. With t = 0, we are in the Harris-Todaro world with perfect capital markets and there is no problem. This is also true if the response OKr/a~ is 'well-behaved', i.e. negative. One would expect this to be so on the ground that an increase in the urban wage would prompt urban employers to lay-off labor and substitute it by capital causing the capital stock in the rural sector to fall. However, ever since Corden and Findlay's (1975) pioneering paper, we know that these first-round tendencies may be counteracted upon to give an increased employment of labor with an increased urban wage. This paradox has been intensively investigated recently 19 and there is little point in going over the same ground. We should mention though that it hinges crucially on the elasticities of substitution in the two sectors and one would expect these same factors to give a perverse aK,/O~ response. 8This is obtained by making use of the marginal productivity conditions and the fact that the endowments L and K are independent of ~. 19See eq. (42) in Khan (1980a) and Neary (1981).

M.A. Khan and S.N.H. Naqvi, Capital markets and urban unemployment

383

5. An adjustment process

Neary (1981) and Khan (1980b) show that under a reasonable adjustment process, a Harris-Todaro equilibrium is stable if and only if at that equilibrium the urban sector is more capital-intensive in employmentadjusted terms than the rural sector. Their result is for a world with perfect capital markets and it is of some interest that the capital market distortion does not alter this stability condition. This section reports on this finding which has important implications for proposition 1 above. Denote by P an adjustment process which is defined by the following differential equations:

DKr=(O{(Rr/Ru(l+t))-l},

q~' > 0,

DU=O{w,(I+2)I~,)-I},

~b(0) = 0,

7s'>0, 0(0)=0,

where D is the time derivative operator. These equations are hardly novel and need little comment. We can now present Proposition 3. An equilibrium of the adjustment process P "is locally, asymptotically stable if and only if the urban sector is more capital-intensive in employment-adjusted terms than the rural sector, i.e. if and only if k , - k , (1 +2) >0.

The above proposition can be proved easily using the methods of Neary (1978). Assume without any loss of generality that ¢'(0)=~b'(0)=l. Then, linearization of the differential equations around their equilibrium values yields the following matrix:

o,, (K.L.)

U O,L L, a,(1 +t)

a,(1 + t) LK,

IO,rfK.

L,,'~

U

wr

(

w, UCK

]]

"

(19)

where a i is the elasticity of substitution in the production of good i and 0u is the share of the jth input in the value of the ith output. It is now easy to show that the sign of the determinant is given by the sign of k./(1 + 2 ) - k r . If this is positive, certainly (k.-k,) is positive and the trace of the above matrix is negative, completing the proof. It is worth underscoring why the introduction of a rental-differential makes no difference to the stability properties of a Harris-Todaro equilibrium in the presence of an exogenously given urban rigid wage. Given international prices, a rigid urban wage fixes the urban rental at its

384

M.A. Khan and S.N atl. Naqvi, Capital markets and urban unemployntent

equilibrium value, even in disequilibrium situations. As a consequence, the differential appears only in the first row of the determinant (19) and hence can be factored out. We conclude this section by reminding the reader that our proposition 3 considerably narrows the range of possible welfare "paradoxes'.

6. Concluding remarks Throughout the paper we have limited ourselves to an exogenously given urban wage. A natural question arises as to the consequence of replacing this assumption by a more general specification in which the urban wage is endogenously determined as in K h a n (1980a, 1982). A geometric treatment is obviously inadequate for such a setting and it is an open question as to the form in which the stability result of K h a n (1980b) would continue to hold. Throughout this paper our emphasis has been on the international value of G N P as the relevant indicator of welfare. We could have alternatively used the size of unemployment as our measure of social welfare, In this case, the geometry is no longer illunainating z° but the algebra can be used to derive results that are neither surprising nor dear-cut, It is worth pointing out that our paper has an obvious relevance to situations in which non-distortionary methods for raising revenue are no longer available, zt Thus, issues raised by the financing of an urban wage subsidy through a differential tax on capital or alternatively by the financing of a subsidy to urban capital through a lowering of the urban wage naturally led us to the model studied here. O u r final remark relates to our constant usage of the term 'capital' to refer to K . This is m o r e in deference to the conventional treatments of the twosector model rather than a presumption that we have anything to say on the intertemporal allocation of r e s o u r c e s , zz This is obviously outside our scope here, Z°An exception to this are changes in the differential t as plotted in fig. 6. Since U =(KJff)DL, the geometry is not usual for any comparative static changes involving K. or ~'. ZtFor a general dis~atssion of the consequences stemming from the financing of subsidies, see section Ill in Naqvi (1969). ""But see section 11 in Naqvi {1969).

References Batra. R24. 1973, Studies in the pure theory of international trade (The Macmillan Press. London). Bhagwati, J~ 1971. The generat2zed theory of distortions and welfare, in J.N. Bhagwati et at. eds~ Trade. balance of payments and growth (North-Holland Publishing Company, Amsterdam.) Bhagwati, J~ 1976, Preface to S,P. Magee, International trade and distortions in [:actor markets (Mnxce~Dekker, Inc.,New York).

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