Keynesian unemployment and the shadow economy

JOHN BENNETI- University College of Swansea

Swansea, United Kingdom

Keynesian Unemployment and the Shadow Economy*

A model of temporary equilibrium with rationing is formulated with both a “formal” sector and a “shadow economy” sector. There is Keynesian unemployment in the formal sector, but the shadow economy, in which there is self-employment of households, produces a commodity for which price is market-clearing. Any household may work in both sectors simultaneously. Policy changes directed at the formal sector may have general equilibrium interactions with the shadow economy that feed back on the formal sector. Among the results obtained is a condition under which the formal sector output multiplier is less than unity.

1. Introduction In recent years the role of the “shadow” (or “hidden”) econ-

omy in Western (and other) countries has attracted increasing at- tention. This paper deals with the general equilibrium interactions of the shadow economy with the “formal” economy. Two types of shadow economy activity appear in the analysis. Our main concern is with “black economy” activity that entails the production and sale of commodities and is concealed from the authorities (and is not taxed). Estimates of the value of black economy output as a proportion of first economy output are summarized by Smith (1986). For the U. S. these estimates range from 6 to 27% and for the U. K. from 2 to 15%. In addition, our analysis is relevant to household production activity where there is no trade with other households; that is, it applies to housework and do-it-yourself. Estimates of the value of such activity in the U.S. have been put in the region of 30 to 40% of GDP (see Hawrylyshyn 1976).

We use the framework of “temporary equilibrium with rationing” (or non-Walrasian equilibrium) as formulated, for example, by Barro and Grossman (1971, 1976), Benassy (1975), Malinvaud (1977), and Muellbauer and Portes (1978) (see the survey by Grandmont 1988). The model we develop has two production sectors, each with

*This paper was written while the author was visiting the Department of Eco- nomics, Queen’s University at Kingston, Ontario. The constructive comments of two anonymous referees are gratefully acknowledged.

Journal of Macroeconomics, Spring 1990, Vol. 12, No. 2, pp. 289405 289 Copyright 8 1990 by Louisiana State University Press 0164-0704/90/$1.50

John Bennett

its own homogeneous output. In the formal sector, firms operate within a regime of Keynesian unemployment. With the output price and wage rate determined exogenously, there is an excess supply of labor to the sector and an excess supply of output by it. In the second sector there is shadow economy household production activity. Each individual household may produce shadow economy output, but its consumption of such output may differ from its production because it may also buy or sell on the black market. It is assumed that black market trade occurs at a market-clearing price. Households that sell on the black market are regarded as being “self-employed’ in this respect (there is no explicit black market wage in the model). An individual household may earn income in both sectors at once, for it may have separate members working in each sector or it may have a member or members employed in the formal sector but “moonlighting” in black market activity.

Perhaps the most important limitation of our analysis is that we assume, for simplicity, that all households are rationed in the formal sector labor market. This condition is assumed to hold both before and after each of the parameter changes to be considered. In a more general model there would also be some households unrationed in the formal sector labor market. The main technical dif- ficulty then is that it would have to be taken into account that, when a parameter is altered in value, the proportion of households unrationed will generally change.

The present formulation does, however, cover the possibility that some households will have no formal sector employment at all. Also, it is general enough to allow for households in a variety of states in the shadow economy, any given household possibly being (a) both producer and consumer, (b) only a producer, (c) only a consumer, or (d) neither producer nor consumer.

A shadow economy production sector is also inserted into a model of temporary equilibrium with rationing by Ginsburgh et al. (1985) and Adam and Ginsburgh (1985). Unlike in our model, it is assumed in these papers that there is a (market-clearing) shadow economy wage rate. But the inclusion of this additional endogenous variable in the model complicates the analysis considerably, and so it is found necessary to assume specific functional forms. We shall not make such restrictions below.’

‘Ginsburgh et al. consider various combinations of Keynesian unemployment, classical unemployment, and full employment in the two sectors, while Adam and Ginsburgh use a simplified version of the same approach to obtain numerical es-

Keynesian Unemployment and the Shadow Economy

As in the formulations by Ginsburgh et al. and Adam and Ginsburgh, we assume that there is no uncertainty facing agents. Thus, we do not allow for the probability that the authorities will detect black market sales (with a subsequent imposition of penal- ties). Our main concern is with how policy changes directed at the formal sector have general equilibrium interactions with the shadow economy (especially with black market activity) that lead to repercussions on the formal sector. In Section 2 we set up the model, and in Section 3 we examine some comparative static results. In Section 4 some concluding comments are made, while in the Ap- pendix the effects of parameter changes on shadow economy activity are briefly derived.

2. The Model It is assumed that two commodities, each of which is homo-

geneous, are produced in the economy. Commodity 1 is produced in the “formal” sector, and commodity 2 in the “shadow” sector. The formal sector is modeled as a representative firm whose output of 1 is denoted by xi. The employment of any household, h, in the formal sector is denoted by e:. Assuming that all the labor performed in this sector is homogeneous, we write &et = e,, where the summation, as all the summations in this paper, is over all households, h = 1, 2, . . . , H. Production of 1 is assumed to be given by

x1 = X1(4 , x;>o, x:50.

In the shadow sector, all H households are potential producers of commodity 2. The production of 2 by household h is denoted by x;. It is assumed that

xi = Xk(e!J , x;’ > 0 ) xyso, (h = 1, 2, . . . , H) ; (2)

where ek is the labor performed by h in producing 2. The produc-

timates for the Belgian economy. A shadow economy production sector is incor- porated into a model of repressed inflation (representing a centrally planned economy) by Hare (1987) and Bennett and Phelps (1989). Like Ginsburgh et al. and Adam and Ginsburgh, Hare assumes that there is wage employment in the shadow economy, but Bennett and Phelps assume self-employment, as in the present paper. For general two-sector models of temporary equilibrium with rationing (not tailored to take into account the specific features of the shadow economy) see Mal- invaud (1980) and Fourgeaud, Lenclud, and Michel (1981).

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tion function Xt is possibly different for each h. The reason for dis- tinguishing the behavior of separate households in the shadow sector, rather than proceeding in terms of a representative household, is that the model will be based on the heterogeneity of household behavior with respect to commodity 2. (If households all behaved identically there would be no trading of 2.)

The quantity of 1 purchased by any household h is denoted by c:. Also, an amount, g,, is purchased by the government. Cur- rent production of 1 is assumed to equal current uses (no inventories being held between periods):

x1 = lx 4 + g1 . (3) h

It is supposed that the price of 1, p,, is fixed and that the firm would always like to supply more of 1 than households and the government wish to buy. Assuming the “short-side” rule to hold, the quantity of 1 actually traded is demand-determined.

The quantity of 2 consumed by h is denoted by ci. In general, c; # z$; if (c; - r!J is positive (negative) h buys (sells) some of good 2. It is assumed that the government does not buy any of 2. It seems realistic to suppose that the market for 2 is highly compet- itive, with no institutional constraints on the price charged. It is therefore assumed that the price of 2, p,, is flexible and market- clearing. Given that no inventories are held between periods, we have

h h

The budget constraint facing any household h is

riih + wlet + p2(i4 ‘- $) = p,c: + mh , (h = 1, 2, . . . , H) , (5)

where #ih and mh are the nominal money balances with which h respectively begins and ends the period, while wi is the net-of-tax wage rate earned in the formal sector. It is assumed that wi is (parametrically) fixed. Household h begins the period with the balance eh, earns wie: in the formal sector, and obtains a net revenue (which may be negative) of p2(xk - c,“) from shadow sector activity. It spends p,ct on I, and its terminal money balance is mh. (It is assumed that any current period formal sector profit is undistri- buted. If X’; = 0, such profit is in any case zero). We assume that h possesses the quasi-concave utility function.

292


uh = ti(ct, ck, ef, ei, mh) , (h = 1,2, . . . , H) . 63

The variable Uh is increasing in c:, $, and mh and decreasing in et and et. It is supposed that h’s employment in the two sectors together never approaches the physical maximum.

We wish to consider the implications of there being an excess supply in the formal sector labor market. For simplicity, we assume that every household, h, would like to obtain more formal sector employment than it actually can; this assumption is maintained throughout the comparative statics. With the short-side rule apply- ing, et is therefore an exogenous parameter for each h. Thus, we suppose that h chooses its demands for commodities 1 and 2, cFd (i = 1, 2), its supply of labor to the shadow sector, ep, and its demand for terminal money balances, mhd, to maximize uh subject to (2), (5), and the given value of e:. (For more detailed analysis of such household decision problems see Shishko and Rostker 1976, and Gronau 1977.) As there are no further quantity constraints on h’s behavior, these demands and supplies are the amounts actually transacted:

h Ci = cfd(pl, P,, wl, d, rfLh) , (i = 1, 2) ,

d = &h, ~2, WI, et, fib> ,

(h = 1, 2, . . . , H) . (7)

We do not list mM separately because, once cp (i = 1, 2) and ek are chosen, mhd is determined. From the first-order conditions for the households utility-maximization problem in the presence of the constraint on et, it is easily shown that

a2" h azh a2ih azh -= awl

-=w1--, e1 s’ ae; afih (zh = cY, c?, eF> . (8)

These relationships will prove useful in investigating the comparative statics of the model.

The representative formal sector firm employs as much labor as it wishes. Its demand for labor depends purely on the amount of commodity 1 it can sell. Output, x1, is set equal to the demand for 1, and the labor required to produce this output is given by Equation (1). To complete the model it must also be specified how this total amount of employment is allocated between households. But, as our main concern is with comparative statics, we shall only specify how changes in formal sector employment are allocated. We return to this point below.

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To summarize, we have a conventional representation of a Keynesian unemployment regime in the formal sector, while at the same time having a shadow sector in which there is market-clear- ance for both labor and the commodity produced. The model allows for each household to produce this commodity both for its own consumption and for selling to other households. The rationale for (black market) trade between households in the model is that digerent households have different tastes and/or face different conditions. That is, as the model is formulated, such trade occurs because the utility function, Uh, the initial money endowment, riih, the formal sector employment ration, e:, and the shadow sector production function, Xi, may differ over h.

We now turn to comparative statics. From (l)-(4),

dx, = Xidel ,

dxl = r, dc: + dgl , h

c dx; = 2 dc,h . h h

Using (8), the comparative static form of (7) becomes

. -

add apl

+ ac,hd

+ aCFd ap2

- aC,hd

apl

? aek

ap2

? aei

apl

(h = 1, 2, .

a2

. > HI ’

+ - acy aeih

- af$ atih

&I

efdwl + wide: + drii h

02)

(13)

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The signs above the partial derivatives in (13) are those which, to simplify the discussion, we shall assume to hold below. Com- modity 1 is assumed normal and a gross substitute for 2, horn which the signs in the first row of the matrix follow. The second row of the matrix relates to commodity 2, for which our basic assumption is also normality. However, the likelihood of inferiority seems greater for commodity 2 than for commodity I-if only because a household whose real income rises may choose to switch its expenditure away from purchases that contain an element of illegality, for which there may be no warrantee attached, and so on. Thus, although we attach a positive sign to a#/amh, we also note the alternative case in which it is negative. Nonetheless, we exclude the possibility that 2 is Giffen; that is, we assume that &$/8pz < 0. It is also assumed in (13) that &?/a~, > 0 (gross substitutability); this is consistent with 2 being either normal or inferior.

The partial derivatives of et appear in the third row of the matrix. In making the sign assumptions shown there we are supposing that h would always rather supply a given increment of labor to the formal sector than to the informal sector. This preference would result from consideration by h of the combination of monetary and non-monetary rewards from each type of work. Given that h is always rationed in the formal sector labor market, any change in its demand for leisure (resulting from the variation of some parameter value) can therefore be regarded as a change, of opposite sign, in its supply of labor to the shadow sector. (Here and throughout the paper we are referring to effective, not to no- tional, labor supply.)

We shall assume that leisure is normal, and so at$/afih < 0. However, we do not assume specific signs for a$/ap, and a$/ap2. There are two reasons for this. First, recent empirical work suggests that it is unclear whether leisure and commodities are gross substitutes or gross complements in consumption.’ Second, in trying to determine the sign of a&lap,, there is the added complication that, as h is a (potential) supplier of 2, a higher value of p, causes the implicit marginal wage rate in shadow sector production, p,X!$, to be greater. (If h is a buyer of 2, p,Xi’ is an implicit mar-

‘See Browning, Deaton, and Irish (1985), who find in a life-cycle model for UK households that an increase in commodity prices causes an increase in the supply of labor (commodities and leisure being gross complements in this sense) but that an increase in the wage rate causes an increase in commodity demands (commodities and leisure being gross substitutes in this sense).

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ginal wage rate in an opportunity cost sense.) As we would treat the rise of a wage rate in a clearing labor market as having an indeterminate effect on the labor supply to that market, it seems ap- propriate to regard the labor supply response to a change in p,Xk’ in the present model as indeterminate.

For simplicity, it is assumed that a change in e, is distributed between households according to

h (14)

This means that a change in the aggregate level of formal sector employment never leads to a change in the opposite direction of the formal sector employment obtained by any household, h. A parallel assumption will be made for the total initial money balances of households, which we denote by rir. (= &,cz~):

dcxh = phdfi, 05ph51, c ph = 1. (15) h

Substituting into (11) and (12) from (9)-(15) and also using (8), it is found that

2" xp z > (

dpl + 5 - Xk’ 2 >

(efdwl + phd&) 1 + k! (efdwl + fShdriz) 1 + dgl (16)

To avoid using unwieldy expressions in some parts of Section 3 we shall rewrite the determinant of the matrix on the left-hand side of (16) as

A= (17)

296


It will be assumed that A # 0. Note that if the shadow economy is insignificant in size, al2 and a21 are small. It is easily checked that if, as an approximation, aI2 and a21 are set equal to zero, Equa- tions (18)-(23) below reduce to the results that would be obtained in a “one-sector model” in which there were no shadow economy.

3. Discussion In this section we shall consider the effects of changes in the

parameters g,, wi, fi, and p,. Since the intuitive explanation of these effects is similar in each case we focus on the example of a change in g, and deal with the other changes more briefly.

(a) dg,: The Multiplier Using Cramer’s rule, it is found from (16) that

6 1 hs hd

Zi = A h C( x,“r de2 ac2

ap, 1 ap2 ’

We refer to dxl/dgl as the multiplier throughout. Using (17), Equa- tion (18) can be written

Al 1 -= da a22 - a12a2Jh ’

09)

Now l/a% is what the multiplier would be in a one-sector model (with no shadow economy); it is the same in form as the multiplier found for the Keynesian unemployment regime by, for example, Muellbauer and Portes. As is assumed in the one-sector model, we assume that l/a, > 1 (that is, 0 < a% < 1).3 It is worth noting here that if the sale of commodity 2 were excluded from the model, each household consuming its own production, the multiplier would be l/a, rather than being of the form shown in (19). It is only when there is exchange between households that the parameter p, has a role to play, and the multiplier is that shown.

In Equation (19) all the terms on the right-hand side of (17) appear. By referring to these terms we can explain intuitively how

3Note that, although partial derivatives such as Xi and Jcy/Jfi” would appear in both a one-sector and a two-sector model, they would in general differ in value across the models. In assuming that l/a, > 1, we are in effect supposing that the differences are not “too” great.

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the existence of black market exchange affects the multiplier. Con- sider a one-unit increase in g,, the government’s purchases of 1. This causes an increase in the demand for labor, and therefore employment, in the formal sector. One effect is qualitatively the same as in the one-sector model; this is indicated by the term uz2 in (19). However, in our model the rise in formal sector employment has two types of effect on activity in the shadow sector, and these have repercussions on the formal sector.

First, the rise in el will generally affect the households’ will- ingness to work in the shadow sector. Given the sign assumption in (13) the supply of labor in the shadow sector will be reduced,4 and therefore the supply of 2 will fall. Second, the rise in el will generally affect households’ demand for 2. Given that 2 is normal for all households, the demand for 2 will rise. The two effects men- tioned in this paragraph mean that the excess supply of 2 falls at the rate ui2. In turn, p, is affected. The extent to which p, must change to restore market equilibrium for 2 is indicated by l/~,~, the reciprocal of the rate of change of the excess supply of 2 with respect to p,. It is assumed that all > 0,5 in which case there is an increase in p,. But, as represented by the term u2i, the increase in p, affects the demand for 1, the sign assumption made in (13) being that this demand rises. Thus, we have identified an effect on the demand for 1 which, by definition, does not exist in the one- sector model. Due to this effect on demand, there are further repercussions on formal sector output.

The assumptions made in (13) and in this section together mean that ali > 0, 1 > uz2 > 0, and u12, u2i < 0. With these assumptions alone it is possible that A (s u11u22 - ur2u2J, and therefore dx,/dgi will be negative; that is, although we have assumed I/uz2, the “one- sector model multiplier,” to be greater than unity, it is possible that

41t is assumed in (13) that a&/atih < 0 for all h; using (S), it follows that a&/&~ < 0. A parallel argument will apply to each of the partial derivatives of household behavior with respect to e, or w, discussed in the text.

5u,l = Z,(Xz a&/ap, - ac~d/dp,). In (13) we assume that acid/ap2 -C 0, but that the sign of a&apz is uncertain (h = 1, 2, . . , H). In assuming that a,, > 0 we are supposing that there is not a preponderance of numerically large negative x;‘a&/ap,s. (Viewing commodity 2 purely as a consumption good, the evidence of Browning, Deaton, and Irish quoted in note 4 above is consistent with a&lap, being positive. Viewing 2 as household production, a higher value of p, means that h’s implicit shadow sector wage rate is higher, ceteris pa&us, and so a positive value of a&/apa would be equivalent to having an upward-sloping labor supply curve.)

298


the existence of the shadow sector will cause general equilibrium interactions that result in a negative multiplier. Note that this result does not depend on the existence of unusual signs for the partial derivatives in (13). For the rest of the paper, however, we shall suppose, as seems likely to be the case in practice, that A > 0. It then follows from (19) that, under the sign assumptions we have made on the aii-terms, dx,/dg, > 1.6 Furthermore, it is found in the Appendix that under the same sign assumptions dx,/dgl > 0; greater government expenditure on commodity 1 is also associated with a greater output of 2.

These conclusions must be modified if we make the alternative assumption indicated in Equation (I3)-that ~c~~/c&’ < 0 (2 being inferior) for some households, h. Referring to (16) and (I7), this can be seen as opening the possibility that al2 > 0. Now from (19)

dx ‘Sl dg, = ’

as a12a21 Z da22 - 1) . (20)

It is easily checked that if aI2 > 0 each of the alternatives shown in (20) is possible. (Notice that a positive value of aI2 does not in- validate the assumption that A > 0; in fact it ensures that A > 0.) Thus, the existence of market interactions between the formal and shadow sectors may, when the shadow sector output is inferior for some households, cause the multiplier on formal sector output to be less than unity. Additionally, it is shown in the Appendix that with such inferiority dx2/dgl may be negative.

(b) The Effects of Other Parameter Changes We now consider how changes in the parameters wl, ti, and

p, affect rr. (Unlike the other parameter changes, a change in & is

‘Under our sign assumptions an increase in g, results in a feedback from the shadow sector to the demand for 1, which is positive, and this causes dx,/dg, to exceed l/h. However, due to the argument made in note 5, it does not follow that the existence. of the second economy necessarily causes the multiplier to be greater. In particular, we might expect &~/a& to take a smaller value than in the one-sector model because, in our model, a portion of any additional money balances is used to buy some of commodity 2. Thus, we might expect o22 to be greater in our model than in the one-sector model, and so it is unclear in which model the multiplier is greater. It is worth noting that (on the basis of a less general theo- retical model than the one we have presented) Adam and Ginsburgh find that for Belgium the existence of the shadow sector has caused the (formal sector) multiplier to be less than it would otherwise have been, though still greater than unity.

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not directed at the formal sector, but we include this case because it is closely related to a change in w1 and so can be dealt with briefly.) The effects of these parameter changes on ;r, are shown in the Appendix.

The change in dw, can be interpreted simply as a change in the wage rate paid by employers or as a change in the average tax rate on formal sector income. With the latter interpretation, our analysis is applicable provided the tax rate change is not accom- panied by a fully compensating change in the (gross-of-tax) wage rate paid by the firm.’ From (16) and (17),

dx, 1 dw, = 2 a11 >I . (21)

If 2 is normal for all h, it follows from the rest of our sign assumptions that dxl/dwl > 0, as would be found in a one-sector model. If, however, 2 is inferior for some households, it is possible that dx,/dw, < 0. If we interpret the increase in w1 as a reduction in the tax rate on formal sector income, this result may be restated: it is possible that a reduction in the tax rate on formal sector income will lead to a fall in formal sector output.

Intuitively, (21) can be explained as follows: Since there is an excess supply in the formal sector labor market, an increase in wi is equivalent to an increase in initial money holdings (see Equation [S]). As indicated by the summations in (21), this affects the households’ demand for commodities 1 and 2, and its supply of labor to produce 2. These changes in behavior by households feed into the same multiplier process as was discussed in Sub-section (a) for the policy change dg,. A major difference between the changes dg, and dwl, however, is that in the former case there is a direct effect on output, whereas in the latter there is not; that is, a one-unit increase in g, directly causes one more unit of 1 to be produced, and then has indirect effects, while an increase in wi only has indirect effects. This is the reason inferiority of 2 may cause dx,/dg, to be less than unity but dx,/dw, to be less than zero.

‘If there were such a fully compensating change, the net-of-tax wage rate re- ceived by households would remain constant, and so households’ behavior would

be unafTected. And, provided we assume that the formal sector remains in a Keyne- sian unemployment regime, a change in the real cost of labor would not affect firm

behavior.


Proceeding similarly for an increase in initial money holdings, dti, it is found that

. (2.9

This is the same as (21), except that ph has replaced e:. The qual- itative conclusions obtained from (21) therefore also apply to (22). If 2 is normal for all h, dxl/dti is certainly positive; otherwise, it may be negative.

Finally, for an increase in the formal sector commodity price, pl, we find

It can be seen that if an increase in pl had no direct effect on the excess supply of 2 (the second summation being zero), the demand for 1, and therefore the output of 1, would fall. Under the sign assumptions made in (13), however, the excess supply of 2 may therefore fall, and if it would fall, (23) would be indeterminate in sign.

Before concluding, it is worth commenting briefly on the roles played by the distributional parameters oh (h = 1, 2, . . . , H). (Similar comments would apply to ph.) For illustrative purposes, suppose that there is a group of relatively poor households for which the marginal propensities to consume both commodities 1 and 2, and leisure, out of lump-sum income are relatively large. Thus, compared to other households, any such household h has a large plac~/&iih, p,ac~d/&ih, and -XE’ pzaek/aGih." Assume that oh is relatively large for any such household h; that is, such households receive a more than proportionate share of any increase in el. In comparison with the alternative of giving every household an equal share of an increase in ei, it is then found from (16) that a,, is small and aI2 large. It follows from (19) that dx,/dg, is large. The explanation is that (a) such poor households spend a large proportion of any extra income on 1, and (b) they increase the demand for 2 and

‘The marginal propensity to consume leisure is defined by analogy with the other marginal propensities. Given that h is rationed in formal sector employment, X:’ p, is the implicit price of a unit of leisure to h, while -aef/aGih is the rate of change of h’s (effective) demand for leisure with respect to lump-sum income.

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John Bennett

decrease the supply of 2 by large amounts, and so the rise in p, is large, as is the resulting switch in demand towards 1. Because of (a) and (b) there is a large increase in the production of 1.

4. Conclusions In this paper a model of temporary equilibrium with rationing

is set up with two particular types of production sector. In the “formal” sector there is a fixed wage rate, a fixed output price, and Keynesian unemployment, while in the “shadow” sector there is self-employment by households, the commodity produced selling for a market-clearing price. The possibility is allowed that any household may work in both sectors at once and that a household may consume units of its own shadow sector production, rather than selling them.

It is investigated how policy changes that are directed at the formal sector have general equilibrium repercussions via the shadow sector that feed back on the formal sector. It is found that, in theory, government purchases in the formal sector might have a negative multiplier on formal sector output even though the equivalent one- (formal-)sector model would have a positive multiplier. When further sign assumptions in partial derivatives are made to eliminate this case, a sufficient condition for the multiplier to be greater than unity is found to be that the shadow sector output is normal for all households. If this output is inferior for some households, the multiplier may be less than unity.

An increase in the formal sector wage rate or a decrease in the tax rate on formal sector income has effects on formal sector output that are broadly similar to those of an increase in households’ initial money balances. If the shadow sector output is normal for all households, then the formal sector output increases; but if the shadow sector output is inferior for some households, formal sector output may fall. Finally, the effect on formal sector output of a change in its price is found to be indeterminate in sign.

Receioed: January 1989 Final oersion: August 1989

References Adam, M.G., and Victor A. Ginsburgh. “The Effects of Irregular

Markets on Macroeconomic Policy.” European Economic Review 29 (October 1985): 15-33.

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Barro, Robert J., and Herschel I. Grossman. “A General Disequi- librium Model of Income and Employment.” American Economic Review 61 (March 1971): 82-93.

-. Money, Employment and Inflation. London and New York: Cambridge University Press, 1976.

Benassy, Jean-Paul. “Neo-Keynesian Disequilibrium Theory in a Monetary Economy.” Review of Economic Studies 42 (October 1975): 503-23.

Bennett, John, and Michael Phelps. “The Supply Multiplier with a Self-Employed Private Sector.” Economics of Planning Forth- coming, 1989.

Browning, Martin, Angus Deaton, and Margaret Irish. “A Profitable Approach to Labour Supply and Commodity Demands Over the Life Cycle.” Econometrica 53 (May 1985): 503-43.

Fourgeaud, Claude, Bernard Lenclud, and Philhpe Michel. “A Two- Sector Model with Quantity Rationing.” Journal of Economic Theory 24 (June 1981): 413-36.

Ginsburgh, Victor A., Phillipe Michel, Fiorella Padoa Schioppa, and Pierre Pestieau. “Macroeconomic Policy in the Presence of an Irregular Sector.” In The Economics of the Shadow Economy, edited by Wulf Gaertner, and Alois Wenig. Berlin: Springer-Ver- lag, 1985.

Grandmont, Jean-Michel. “Temporary Equilibrium Theory: An Ad- dendum . ” In Temporary Equilibrium, edited by J.M. Grand- mont. San Diego: Academic Press, 1988.

Gronau, Reuben. “Leisure, Home Production, and Work-the Theory of the Allocation of Time Revisited.” Journal of Political Economy 85 (December 1977): 1099-1123.

Hare, Paul. ’ “Supply Multipliers in a Centrally Planned Economy with a Private Sector.” Economics of Planning 21, nos. 2-3 (1987): 53-61.

Hawrylyshyn, Oli. “The Value of Household Services: A Survey of Empirical Estimates.” Review of Income and Wealth 22, no. 2 (1976): 101-31.

Malinvaud, Edmond. The Theory of Unemployment Reconsidered. Oxford: Blackwell, 1977.

-. “Macroeconomic Rationing of Employment.” In Vnemploy- merit in Western Countries, edited by Edmond Malinvaud and Jean-Paul Fitoussi. London and Basingstoke: Macmillan, 1980.

Muellbauer, John, and Richard Portes. “Macroeconomic Models with Quantity Rationing.” Economic Journal 88 (December 1978): 788- 821.

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Shishko, Robert, and Bernard Rostker. “The Economics of Multiple Job Holding.” American Economic Review 66 (June 1976): 298- 308.

Smith, Stephen. Britain’s Shadow Economy. Oxford: Clarendon Press, 1986.

Appendix: Effects of Parameter Changes on x2 Multiplying the third line of (13) by Xi’, summing over h, and

using (8),

h.s

+ae,

hs

aeh wlahdel + ae, dp2 .

ap2 > (AU

When g,, wr, rot, or pl is altered there are in each case three (potential) effects on x2. First, there is the “direct” effect, as represented by the first three terms in parentheses (the direct effect for dgr is zero). Second and third, there are the “indirect” effects via the changes that occur in the endogenous parameters el and p,, as represented by the final two terms in parentheses.

For the determination of indirect effects, consider any policy change, dv (v = g,, wl, e, pJ. To find the indirect effect via en we first take the relationship between dv and dxl, as discussed in Section 3. Since dx, = Xi&,, we therefore have the relationship between de1 and dv. This is substituted for the term in de, in (Al). To find the indirect effect via p, we go back to (16) and solve for P2:

“g$) - a,,g](e:dw, + phdti)

X!‘g) - q2$]dpl - a12dgl}. (A2)

We then substitute from (A2) for dp2 in (Al). Given the sign assumptions made in the text we obtain the

results shown in the table, where the signs without parentheses

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TABLE 1.

Policy Change Direct Effect

on x2

Indirect Effects on x2

via e, via p,

43 0 + +(-I dwi - +(-I ? drii - +(--) ? dPl - ? 3-t)

apply when 2 is normal for all h, and the signs in parentheses show what may possibly (but not necessarily) happen when 2 is inferior for some h. It can be seen that only one definite result is obtained: if 2 is normal for all h, then dx,/dg, > 0.

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Keynesian unemployment and the shadow economy