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The Domar-Musgrave phenomenon and adverse selection
European
Journal
of Political
Economy
7 (1991) 41-53.
The Domar-Musgrave adverse selection
North-Holland
phenomenon
and
Kai A. Konrad* Seminar JGr Versicherungswissenschnft, 22, Germany Accepted
for publication
October
Universitiit
Miinchen.
Ludwigstraj?e
33. W-8000
kfiinchen
1990
In this paper, the incentive to increased risk taking caused by taxes on risky revenues (the Domar-Musgrave phenomenon) is revaluated. In a capital market equilibrium with adverse selection, the tax is not ineflective. An additional risk consolidation takes place within the collected tax proceeds. If owners of entrepreneurial tirms cannot react via a change of ownership structure, then the tax directly affects the investment decision. In other cases, the tax induces firms’ owners to cut back share issues.
1. Introduction
The taxation of risky profits induces an investor to increase his demand for the risky investment opportunity. This phenomenon was first studied by Domar and Musgrave (1944) and its robustness has been shown under quite general conditions.’ The intuitive explanation for this incentive to increase the risky fraction of the investor’s portfolio has often been described: government participates only in the revenues of risk-taking, but also assumes some of the risk burden. In a general equilibrium framework, the risk-taking incentive of a tax on risky revenues is less obvious. A risk-taking incentive may prevail if tax proceeds are used in a ‘neutral’ way, so that the investor’s behavior is not affected by the riskiness of the expenditures financed by these tax proceeds. If tax proceeds are lump-sum redistributed as risky transfers and capital markets are perfect, the incentive effect does not appear [cf. Atkinson and Stiglitz (1980), Bulow and Summers (1984), Gordon (1985)]. However, there are capital market imperfections. Entrepreneurs who found
*Paper presented at the Winter Symposium of the Econometric Society, January 16-19, 1990 in Nadarzyn. Comments by Roger Guesnerie, Manfred Holler, Tore Nilssen, Wolfram F. Richter, Hans-Werner Sinn and an anonymous referee are gratefully acknowledged. ‘Cf. Ashan (1974). Allingham (1972). Atkinson and Stiglitz (1980). Buchholz (1987). Mintz (1981). Mossin (1968), Sandmo (1977). Stiglitz (1969) for some central contributions and further references. 01762680/91/503.50
#Q 1991-Elsevier
Science Publishers
B.V. (North-Holland)
42
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and manage a firm generally know the true profitability of their firm better than potential buyers [cf. Myers and Mayluf (19&X4)].If buyers cannot discriminate between different firms, their willingness to pay will be the same for each. At best, the market price of a firm will be a weighted average of the true values of firms in the case of perfect information. Assume that entrepreneurs who own exceptionally profitable firms are unable to signal the exceptionally high quality of their firms. They choose to accept the market price for their firm below the true market value (related to perfect information), or they keep the firm for themselves. In the latter case they gain the higher expected revenues, but also they bear some of the non-systematic risks involved that could be diversified in the capital market. This paper considered the effect of a tax on the revenues of risk taking in a capital market equilibrium with adverse selection. Entrepreneurs decide endogenously about the size of their firm and about the fraction of their firm that they are willing to sell in the capital market. An equilibrium is considered where all firms of the same size are pooled, but where firm size can be seen by buyers and serves as a signalling variable. In this equilibrium, a tax on revenues from risk taking induces exactly the investor reactions known from the Domar-Musgrave literature if the entrepreneur cannot react via a change in the fraction of retention. The reaction of entrepreneurs is quite different if they can react to the tax-rate change by changing the fraction of retention of their firms. The tax reinforces the adverse-selection problem, driving an even larger proportion of high quality firms from the capital market and, thereby, increasing the amount of deliberately not diversified risks that could be diversified via the capital market.
2. A capital market model Consider a two-period one-good economy. There is a set of firms f E F, each firm founded and managed by a single entrepreneur. Firms are characterized by their twice continuously differentiable stochastic production functions: zl(u/,s) is the stochastic output in period 1, depending on capital input a/h0 in period 0 and the state of nature SEX It is possible not to produce, i.e., z/(0, s) = 0 for all firms. Buyers of shares in a firm are not able to observe the risk or return properties of the true production plan of a specific firm. The only tirmspecific observation they can make is firm size, i.e., the amount of capital u’ used in a particular firm. Only the entrepreneur who founded and manages a firm knows the true stochastic properties of ‘his’ ftrm. However, the types of firms that exist and their relative proportions are common knowledge. To make the analysis easier, it is assumed that risks of any firm are not
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43
systematic risks. A share of the market portfolio of traded shares is perfectly risk-free (no-systematic-risk assumption) and yields a safe rate of return.’ An entrepreneur who wants to found a firm in the first period t = 0 has to make three decisions. He determines a/, the size of his firm, he chooses the fraction (1 -&E [0, l] of his firm that he wants to sell in the capital market, and he decides how to invest his possible remaining resources (including the proceeds from selling a fraction of his firm) in the credit market and the capital market. Both credit contracts in the credit market and shares of the market portfolio in the capital market are equally safe. Buying shares of the market portfolio must yield the same, safe rate of interest i of the credit market. This makes the investor indifferent between investing in the credit market or buying shares of the market portfolio in the capital market. For the entrepreneurs’ choice of firm size and his selling decision the selling price of his firm is of major importance. Given that the amount of investment is observable while the fraction of retention is not, the former may serve as a signalling device while the latter cannot. In an equilibrium, therefore, all firms with the same amount of investment must be pooled, regardless of the choice of qf, while there might be a separating equilibrium with respect to firm size al. The situation is not so very different from models of capital market equilibrium with hidden knowledge, exogenous project size and endogenous and observable retentions. For the latter case, there is theoretical evidence that both pooling and separating equilibria can exist [DeMeza and Webb (1990)]. Moreover, given a smoothly distributed continuum of different firm types, the existence of a strongly separating signalling equilibrium can be shown using tools of Engers and Fernandez (1987). There the equilibrium outcome is characterized by a pricing function which assigns an equilibrium price of a firm to each possible amount of retention.3 Similar results can be expected to hold if firm size instead of retentions is the signalling variable, if firm types are smoothly distributed over the range of possible firm types and it is cheaper and more advantageous for good firms to be bigger than for bad ones. A proof is not given here. Instead, it will be assumed that a separating equilibrium exists where firms whose owners have chosen the same firm size are pooled and where the owners get the same price for their firm, while prices differ for firms of different sizes, so that
UJ= u(aJ)
(1)
‘The ‘no-systematic-risk assumption’ is quite usual in the adverse selection literature. Cf., for example, Prescott and Townsend (1984, p. 24) and Rothschild and Stiglitz (1976) for insurance examples and DeMeza and Webb (1990) for a capital market example. ‘Leland and Pyle (1977) considered the properties of a signalling equilibrium in capital markets where retentions act as a costly signal of true firm quality. The empirical evidence is inconclusive. See, for example, Downes and Heinkel (1982) and Krinsky and Rote&erg (1989) for opposite conclusions.
44
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describes the equilibrium valuation of firms, with v’ continuous and twice differentiable. The pricing function can be calculated to be the equilibrium pricing function by each entrepreneur, but it can also be observed in the equilibrium outcome if a sufficient number of firms exists. Let F, be the class of firms in equilibrium choosing a particular size a. Given the retentions qf chosen by the entrepreneurs f EF,, the value of shares sold in the capital market must have the property
(l+i)=
I (1-qr)z/df JEFO
44.I-U-q%!f 17
(2)
JEF.
with u(a) being the market price of a firm belonging to class F,. The reason is simple arbitrage. Only if the sum of the shares issued by firms of a firm class have the same return as alternative investment opportunities are portfolio investors indifferent between buying shares of these firms and buying other investment opportunities. Implicitly, eq. (2) determines the equilibrium value of firms of a particular class for all classes. The firm class F, and its price, however, appear simultaneously in equilibrium, as each entrepreneur can choose which firm class he would like to belong to.
3. Taxation of risk-taking revenues A tax on revenues from risk taking4 is given by T=r.q[z(a,s)-(l+i)o(u)].
(3)
The superscript will be omitted here and below. A tax is levied on the difference between what the entrepreneur actually gets by choosing a particular amount of retention q, and what he was sure to get, if he decided to sell his whole firm. The income difference between a complete buyout and retaining q>O is the revenue he receives for not selling the fraction q. However, to receive this uncertain additional revenue he has to bear fraction q of the risks implied by the production activity of his firm. The uncertain additional revenue is therefore called risk remuneration or revenue of risk taking. The tax base can also be written as (1 +i)(l -q)u(u)+q*z(u,s)-(1
+i)t’(u).
(4)
4F~r simplicity it is assumed here that tax proceeds are used in a way that doe-s not aNect the decisions of entrepreneurs. Unlike in the situation with perfect capital markets, this is a weak assumption as, here, lump-sum transfers of tax proceeds are not risky. For a discussion, see section 4.
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45
phenomenon
The tax differs from a profit tax. Instead of interest paid investment costs (1 +i)a being deductible, it is the interest paid equilibrium selling price of the firm which can be deducted. The final wealth V, of the entrepreneur, dependent on a, q and t, is therefore determined by
~,=q.z(a,s)+(l +i)[(l
-q)u(a)-a+
V,]-t.q(z(a,s)-(1
+i)c(a)),
(5)
or, equivalently, V,=q(l-t)[z(a,s)-(l+i)v(a)]+(l+i)[I/,-a+o(a)].
(6)
I/, in eq. (5) consists of the fraction q of output of his firm, in conformity with the retained fraction q, plus the returns of his investment in the credit and/or capital market, minus the tax T. V,, is his initial wealth which is exogenous. An expected-utility maximizing entrepreneur solves the problem maxEW(Vr) (4..)
(74
subject to eqs. (1) and (5), and
42.0,
I-qzo,
ahO.
Ub, c, 4
W(V,) is the Neumann-Morgenstern utility function of the entrepreneur’s final wealth I/,. Solving eq. (7) yields the Kuhn-Tucker conditions aL/dalO,
ahO and aaL/aa=O,
(84
aL/dq 2 0,
a 2 0 and q a Lpq = 0,
W
aLlan 2 0,
I >=0 and i. ayan= 0,
(84
These are necessary conditions for a maximum of problem (7) because the non-negativity constraints (7b), (7c), and (7d) are linear [cf. Chiang (1974, p. 713n)J. The taxation problem is uninteresting if u=O, because ~(0, s) =O. The taxation problem disappears with the source of taxes, i.e. da/dt =dq/dt =0 for a=O.
Similar reasoning applies to the case q = 0. Eq. (3) reveals that TsO holds in this case. An entrepreneur has no incentive to react to a tax change that does not affect him. He chooses
46
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da/dt = dq/dt = 0.
(9)
Entrepreneurial investors with q=O are founders of firms with expected revenue that at best is as high as the revenue of the market portfolio, i.e. they are initial owners of ‘relatively bad’ firms. Only if E z(a,s) s( 1 +i)u(a), does it not pay the initial owner to retain at least a small positive fraction. Eq. (9) shows that the supply of relatively bad firms will not be altered by a change in the tax rate. Consider now the cases with 420 not binding. From eq. (Sa), c,sE[W’(V,)[q(l-t)z,+(l+i)[(l-q(l-t))o’-l]]]=O,
(10)
with v’- dvlda, z, = &(a, s)/& (and z,, z S’z/S’a in what follows). First the case is treated where the optimal choice of the entrepreneur’s retention is O< 1 -q< 1. From (8b) and (8c), czzE[W’(V,)[z-(l+i)u]]=O.
(11)
The case of an entrepreneur who retains a positive fraction of his firm but also sells some fraction qE(0, l), seems to be very plausible for many entrepreneurial firms. Households are risk neutral with regard to small risks [cf. Merton (1982, p. 609)]. Therefore, an entrepreneurial investor who owns a firm that produces an above average expected revenue, i.e., Ez> v( 1 +i) prefers to retain at least some positive fraction of his firm. Positive retentions q>O imply a welfare loss compared to the case of a perfect capital market. The entrepreneur who deliberately decides to retain q>O bears the burden of risks that could be reduced via the capital market. The activity level a that is described by eq. (10) deviates from the activity level that is chosen under conditions of perfect information. Given perfect capital markets and complete risk diversification, the marginal condition for the optimal amount of investment is determined by E z,= 1+ i. In general, it might differ considerably from eq. (10). The reaction of an entrepreneur to a change of the tax rate t, given the case 1 > q>O and a >O, is obtained by differentiating eqs. (10) and (11) with regard to t with cij-dc,/LJj for i= 1,2 and j=q,a,t, (q>O),
c[:;]=[::. f::] [:;I=[ I::;]&.
(12)
Using Cramer’s rule yields
Wdt = Cl- C&Z~ -( -c,,)c,,]/det
C=O
(13)
K.A.
Konrad,
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41
and
Wdc=C(-c,,k,, -(-c,,)c,,]/det
C=q/(l-c).
(14)
This result can be summarized as follows. Proposition 1. Given a capital market equilibrium chat is described by eq. (1), consider an entrepreneur who sells some fraction O
An entrepreneurial investor has two ways of reacting to a change of the tax rate. He can change the investment activity a or the amount of retention q. Define y = q( 1 -c) and substitute in (6) and the optimality conditions (10) and (11). All equations only depend on a and y. Changing q appropriately to keep y constant if c is changed, the entrepreneur can restore the original optimum. As with the original Domar-Musgrave phenomenon, the entrepreneur’s possibility set does not change in a small neighbourhood of the old optimum. Some general equilibrium repercussions of this reaction might change this effect. Initial owners of relatively bad firms sell their entire firms and they do not react to a tax rate change, i.e., they do not increase their retentions if the tax rate increases. Only those entrepreneurs who own ‘more profitable’ or ‘better’ firms increase their retentions. As dq/dc>O is true only for these better firms, an even larger fraction of better firms’ shares are no longer supplied in the capital market, while the supply of ‘lower-quality’ firms remains unchanged. The profitability of capital market investments should fall tendentially if v(a) does not change. To equate the profitability of the credit and capital markets, market prices of firms must fall, and this again may lead to a reduction of the supply of ‘high-quality’ firms in the capital market. A welfare analysis would have to take all full general equilibrium repercussions of the tax-rate change into account. General equilibrium price changes, for example, might induce some entrepreneurs to change investment in order to enter another firm class. These aspects are outside the scope of this paper. The tax proceeds of a tax on risk-taking revenues have quite different stochastic properties than in the case of perfect capital markets considered by Gordon (1985). Asymmetric information leads to equilibria with some diversifiable, but deliberately not diversified, risk. Taxing the revenue of this non-diversified risk allows some of this risk to be diversified within the tax proceeds. These tax proceeds therefore do not have a market value of zero as in the case of perfect capital markets.
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Consider now an entrepreneur who, in equilibrium, chooses q= 1. The tax cannot induce him to increase the fraction of retention. Instead, he might be willing to change the size of his firm. The tax in this case has an immediate impact on the real investment choice. Differentiation of (10) with regard to t yields +i)v’)]/c,,
s=E[W’(z,-(1
+E[W”(z,(l
-t)-(1
+i)(l-tv’))(z-(1
+i)~)]/c,~.
(15)
c,. is negative if the second-order conditions for a maximum are fulfilled. The sign of the numerator determines whether an increase of the tax may increase or decrease the activity level of the firm. Assume that the market equilibrium fulfills the property v’(a) E du/da 2 1.
(16)
Larger firms are the better ones, i.e., their market value is higher by at least the additional investment. Proposition 2. Given a capital market equilibrium that fulfills eqs. (1) and (16), consider an increase of the tax rate t E [O, 1). Ifthere is an ordering of the states of nature such that z(a,s) and z,(a,s) are monotonically increasing in s, and the owner of the Jrm has constant or decreasing absolute and constant or increasing relative risk acersion, then da/dtzO holds for firms that are entirely retained by their entrepreneurial investor (q= l), i.e., the tax increasesjrm size.
The proof of this proposition follows lines drawn by Kihlstrom and Laffont (1983, p. 166). From (15) and using c,,
(17)
and E[fV(V,)[z,(l
-t)-(1
+i)(l-tfi’)][z-(1
+i)v]J
Then, (17) Q E[W’(V,)[z,-(l+i)u’](l-t)]S@ 0
E[W’(V,)[z,(l-t)+(l+i)tt”-_(l+i)u’]]$O
as t
(18)
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49
but
(18) 0
-(l/(l-t))E[W’(~~)(-W”/W’)[z(l-t)-(l+i)(l-t)c]d]~O
with A = z,( 1 -t) -( 1 + i)( 1 - tu’). The factor l/( 1 - t) does not change the sign and can therefore be neglected. Now add
to z(l-_t)-(l+i)(l-_t)u (18) 0
and re-order to get
-EIW’(VI)(-W”/W’)[z(l-t)+(l+i)[-a+V,]+t(l+i)u]A]
-EIW’(V1)(-W”/W’)[-(l+i)[-a+1/,] -t(l+i)u-(l+i)(l-Qu]A]50.
Note that V, = z( 1- t) + (1 + i) [ - a + V,] + t( 1+ i)u and substitute in the first expectations term and sum up the terms in the second expected value term to get
(18) -
-E[W’(V,)A(-W”/W’)V,] +(l+i)(Vo+u-a)E[W’(V,)A(-W”/W’)]sO.
By eq. (lo), E[W’( V,)A] =O. Therefore, the first term is negative under increasing relative risk aversion if z, and z monotonically increase in s, as this implies that V, and therefore A and (- W/W’) V, also monotonically increase in s. If relative risk aversion (- W”IW’)V’, is constant, then, because of (lo), the first term becomes zero. A similar reasoning applies to the second term. E[ W’( V,) A] = 0 by (10). If z and z, monotonically increase in s, then A and VI also monotonically increase in s. For decreasing absolute risk aversion in the second term negative values of W’A are multiplied with large values of (- W”/W’) and positive values of W’A are multiplied with small values of (- W”/W’), such that E[ W’( V,) A( - W”/W’)] becomes negative. (V, + u-a) is positive, V, > 0 is assumed, and, as u(O)>=0 and u’2 1, u 2 a. If absolute risk aversion (- W”/W’) is constant, the second expected-value term
50
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becomes zero. Relative and absolute risk aversion cannot be constant at the same time. Therefore, for at least one of the expressions in (18) strict inequality holds. Proposition 2 covers many questions that have been treated in the literature. For o(u) = a, condition (15) simplifies to E[W”(V,)Cz,-(l
+i)][z-(I
+i)u]]
d”~dt~E[W’(V,)~,,]+E[W”(~~)[z~-(*+i)]2(1-t)]7
z. ’
(19)
(i) Expression (19) is identical with the condition derived in the DomarMusgrave literature, where the reaction of an investor to a change in the rate of a tax on risk remuneration is determined. In this literature, an investor, who allocates an initial amount of wealth between a safe asset and a risky investment opportunity that has a stochastic return z(a,s), is considered. The stochastic return depends on the amount of investment a and the state of nature. Let a denote the amount of initial wealth an investor uses to invest in the risky investment opportunity. Then the maximization calculus of this investor yields precisely condition (19). In the case of constant returns to scale, i.e., if z(a,s) =a X(s) with X a random variable, condition (19) reduces to the ordinary Domar-Musgrave effect [cf., e.g., Mossin (1968)-J, da/dt = a/( 1 - t) > 0.
(20)
Mintz (1981) derived condition (19) within the Domar-Musgrave portfolio model and argued that du/dt cannot be signed without further assumptions concerning z(u,s). He provides a condition that implies an increase in risk taking. If z(u,s)-uz,(u,s)rconstant with regard to s for given a and the entrepreneur has constant or increasing absolute risk aversion, then du/dr>O. Mintz’s result (1981) draws on the property of the tax (in the case of decreasing returns to scale) to make the taxed entrepreneur tendentially less wealthy. This property is not ensured when the use of tax proceeds is taken into account. Proposition 2 shows that weaker and more plausible conditions exist which ensure that the tax increases the incentive for risk taking. (ii) The conditions of the stochastic production function cover, as a special case, the stochastic production function
(21) with X a random variable with finite moments, and h and g real-valued differentiable functions with h’>O. Re-ordering X(s) so that X(s) is mono-
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51
tonically increasing in s shows that the stochastic production function (21) fulfills the requirement made in Proposition 2. Proposition 2 therefore also generalizes Sandmo’s (1971) proposition concerning the effect of a profit tax for a generalized stochastic production function, without having to meet Katz’s (1983) criticism because here the measure of risk aversion in Proposition 2 is concerned with the final wealth of the entrepreneur and not with profits of a firm. 4. Discussion For completely credit-financed firms with exogeneous given project size, the welfare enhancing effect of a lOOaLprofit tax has been shown by DeMeza and Webb (1990). The tax resolves the adverse selection problem by imposing a 100% compulsory insurance. It might be difficult to implement a 100% tax for political reasons. Moreover, a 100% tax may have some adverse incentive effects which make it undesirable. Here, the impact of a small tax rate change is considered. Attention is drawn to an additional dimension of entrepreneural decision making. There is both a project size and the retention decision. These decisions are made simultaneously and are interdependent. For the impact of taxes on revenues that accrue from risk-taking, the endogeneity of the percentage of retentions turns out to be decisive. Given that the fraction of retentions is endogenous, the entrepreneur is forced to react to the tax in a way that amplifies the process of adverse selection. Entrepreneurs do not increase their risky production activity, but they may increase the fraction of retentions. The additional risk diversification within tax proceeds tends to be compensated by a reduced amount of risk diversification via the private capital market. If an entrepreneur cannot react to a tax-rate change via increasing his retentions because he holds all the shares of his firm, then the impact of an increase of the tax is more like that in the classical risk-taking and taxation model. An increase of the rate of this tax has an impact on the demand for risky activity. Under fairly general conditions, an increase in the tax rate increases this activity level, and a considerable amount of additional riskdiversification might take place within the tax proceeds. In this analysis it has been assumed that tax proceeds are used in a way that does not affect entrepreneurs’ decisions. This assumption is much weaker in a model with endogenously explained capital market incompleteness than in a model with perfect capital markets. It has been mentioned that the Domar-Musgrave phenomenon does not appear in a model with perfect capital markets. On the contrary, a lump-sum redistributed risk-takingrevenue tax is perfectly ineffective. The reason is that with perfect capital markets the risk-taking-revenue-tax proceeds are risky. No risk consolidation takes place within the tax revenue. All risk is transferred back to the private
52
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sector when the tax proceeds are redistributed. The only consistent way to use risky tax proceeds neutrally in the case of perfect capital markets is to spend it on public goods, assuming that these public goods affect the utility function of households in an additively separable way [Ahsan (1989), for example, makes this assumption]. The welfare impact of the use of tax proceeds in this case.depends largely on households’ attitudes towards the risky supply of these public goods. For a detailed consideration of these issues, see, for example, Allingham (1972). In the present model, a much easier way to use the tax proceeds neutrally is possible. Moreover, this way can be judged much more clearly from a welfare perspective. With the no-systematic-risk assumption all risks could be perfectly consolidated via the capital market. The consolidation does not happen because the information structure leads to adverse selection in the capital market. In this case a substantial risk consolidation can take place within the tax revenues and tax proceeds are expected to be practically riskless. If these proceeds are lump-sum redistributed, then practically no risk is transferred to the private sector. Households which receive the redistributions are simply made richer. If all tax proceeds are redistributed to households that are not entrepreneurs and if this does not affect the supply of these households to the credit and capita1 markets, then this is a perfectly neutral and clearly beneficial way to spend the tax revenue. If entrepreneurial households do receive some redistributions, things are more complicated. If decreasing or constant absolute risk aversion is assumed, however, it can be expected that the reaction of the entrepreneurs to a tax increase remains qualitatively the same. As they are richer now, entrepreneurs’ willingness to bear risk increases or remains constant. In the case of incomplete retention (q < l), therefore, the adverse selection problem can be expected to be reinforced and in the case of 100% retention (q= 1) the reaction of risky investment to a change in the tax rate can be expected to be reinforced. References Ahsan, S.M., 1974, Progression and risk-taking, Oxford Economic Papers 26, 318-328. Ahsan, SM., 1989, Choice of tax base under uncertainty: Consumption or income?, Journal of Public Economics 40,99-134. Allingham, M.G., 1972, Risk-taking and taxation, Zeitschrift fur Nationaliikonomie 32, 203-224. Atkinson, A.B. and J.E. Stiglitx, 1980, Lectures on public economics (McGraw-Hill, London). Buchholz, W., 1987, Risikoeffekte der Besteuerung, (unpublished) Habilitation thesis, Universitat Tiibingen. Bulow, J.I. and L.H. Summers, 1984, The taxation of risky assets, Journal of Political Economy 9220-39. Chiang, A.C., 1974, Fundamental methods of mathematical economics, 2nd ed. (McGraw-Hill, London). DeMeza, D. and D. Webb, 1990, Risk, adverse selection and capital market failure, Economic Journal 100, 206-214.
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Domar, E.D. and R.A. Musgrave, 1944, Proportional income taxation and risk-taking, Quarterly Journal of Economics 58, 388-422. Downes, D.H. and R. Heinkel, 1982, Signalling and the valuation of unseasoned new issues, Journal of Finance 37, l-10. Engers, M. and L. Fernandez_ 1987, Market equilibrium with hidden knowledge and selfselection. Econometrica 55, 425-439. Gordon, R.H., 1985, Taxation of corporate capital income: Tax revenues versus tax distortions, Quarterly Journal of Economics 100, l-27. Katz, E., 1983, Relative risk aversion in comparative statics, American Economic Review 73, 452453. Kihlstrom. R.E. and J.-J. Laffont, 1983, Taxation and risk taking in general equilibrium models with free entry, Journal of Public Economics 21, 159-181. Krinsky, I. and W. Rotenberg, 1989, Signalling and the valuation of unseasoned new issues revisited, Journal of Financial and Quantitative Analysis 24, 257-266. Leland, H. and D. Pyle, 1977, Informational asymmetries, financial structure and financial intermediation, Journal of Finance 32, 371-388. Merton, R.C., 1982, On the microeconomic theory of investment under uncertainty, in: K.J. Arrow and M.D. Intrilligator, eds., Handbook of Mathematical Economics, Vol. 2 (NorthHolland, Amsterdam), 601-669. Mintz, J.M., 1981, Some additional results on investment, risk taking and full loss offset corporate taxation with interest deductibility, Quarterly Journal of Economics 96, 631-642. Mossin, J., 1968, Taxation and risk-taking: An expected utility approach, Economica 25, 74-82. Myers, S. and N. Mayluf, 1984, Corporate Rnancing and investment decisions when firms have information that investors do not have, Journal of Financial Economics 13, 187-221. Prescott, E.C. and R.M. Townsend, 1984, Pareto optima and competitive equilibria with adverse selection and moral hazard, Econometrica 52, 21-45. Rothschild, M. and J.E. Stiglitz, 1976, Equilibrium in competitive insurance markets: An essay on the economics of imperfect information, Quarterly Journal of Economics 91, 629-649. Sandmo, A., 1971, On the theory of the competitive firm under price uncertainty, American Economic Review 61.65-73. Sandmo, A., 1977, Portfolio theory, asset demand and taxation: Comparative statics with many assets, Review of Economic Studies 44, 369-379. Stiglitz, J.E., 1969, The effects of income, wealth, and capital gains taxation on risk-taking, Quarterly Journal of Economics 83, 263-283.
Journal
of Political
Economy
7 (1991) 41-53.
The Domar-Musgrave adverse selection
North-Holland
phenomenon
and
Kai A. Konrad* Seminar JGr Versicherungswissenschnft, 22, Germany Accepted
for publication
October
Universitiit
Miinchen.
Ludwigstraj?e
33. W-8000
kfiinchen
1990
In this paper, the incentive to increased risk taking caused by taxes on risky revenues (the Domar-Musgrave phenomenon) is revaluated. In a capital market equilibrium with adverse selection, the tax is not ineflective. An additional risk consolidation takes place within the collected tax proceeds. If owners of entrepreneurial tirms cannot react via a change of ownership structure, then the tax directly affects the investment decision. In other cases, the tax induces firms’ owners to cut back share issues.
1. Introduction
The taxation of risky profits induces an investor to increase his demand for the risky investment opportunity. This phenomenon was first studied by Domar and Musgrave (1944) and its robustness has been shown under quite general conditions.’ The intuitive explanation for this incentive to increase the risky fraction of the investor’s portfolio has often been described: government participates only in the revenues of risk-taking, but also assumes some of the risk burden. In a general equilibrium framework, the risk-taking incentive of a tax on risky revenues is less obvious. A risk-taking incentive may prevail if tax proceeds are used in a ‘neutral’ way, so that the investor’s behavior is not affected by the riskiness of the expenditures financed by these tax proceeds. If tax proceeds are lump-sum redistributed as risky transfers and capital markets are perfect, the incentive effect does not appear [cf. Atkinson and Stiglitz (1980), Bulow and Summers (1984), Gordon (1985)]. However, there are capital market imperfections. Entrepreneurs who found
*Paper presented at the Winter Symposium of the Econometric Society, January 16-19, 1990 in Nadarzyn. Comments by Roger Guesnerie, Manfred Holler, Tore Nilssen, Wolfram F. Richter, Hans-Werner Sinn and an anonymous referee are gratefully acknowledged. ‘Cf. Ashan (1974). Allingham (1972). Atkinson and Stiglitz (1980). Buchholz (1987). Mintz (1981). Mossin (1968), Sandmo (1977). Stiglitz (1969) for some central contributions and further references. 01762680/91/503.50
#Q 1991-Elsevier
Science Publishers
B.V. (North-Holland)
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and manage a firm generally know the true profitability of their firm better than potential buyers [cf. Myers and Mayluf (19&X4)].If buyers cannot discriminate between different firms, their willingness to pay will be the same for each. At best, the market price of a firm will be a weighted average of the true values of firms in the case of perfect information. Assume that entrepreneurs who own exceptionally profitable firms are unable to signal the exceptionally high quality of their firms. They choose to accept the market price for their firm below the true market value (related to perfect information), or they keep the firm for themselves. In the latter case they gain the higher expected revenues, but also they bear some of the non-systematic risks involved that could be diversified in the capital market. This paper considered the effect of a tax on the revenues of risk taking in a capital market equilibrium with adverse selection. Entrepreneurs decide endogenously about the size of their firm and about the fraction of their firm that they are willing to sell in the capital market. An equilibrium is considered where all firms of the same size are pooled, but where firm size can be seen by buyers and serves as a signalling variable. In this equilibrium, a tax on revenues from risk taking induces exactly the investor reactions known from the Domar-Musgrave literature if the entrepreneur cannot react via a change in the fraction of retention. The reaction of entrepreneurs is quite different if they can react to the tax-rate change by changing the fraction of retention of their firms. The tax reinforces the adverse-selection problem, driving an even larger proportion of high quality firms from the capital market and, thereby, increasing the amount of deliberately not diversified risks that could be diversified via the capital market.
2. A capital market model Consider a two-period one-good economy. There is a set of firms f E F, each firm founded and managed by a single entrepreneur. Firms are characterized by their twice continuously differentiable stochastic production functions: zl(u/,s) is the stochastic output in period 1, depending on capital input a/h0 in period 0 and the state of nature SEX It is possible not to produce, i.e., z/(0, s) = 0 for all firms. Buyers of shares in a firm are not able to observe the risk or return properties of the true production plan of a specific firm. The only tirmspecific observation they can make is firm size, i.e., the amount of capital u’ used in a particular firm. Only the entrepreneur who founded and manages a firm knows the true stochastic properties of ‘his’ ftrm. However, the types of firms that exist and their relative proportions are common knowledge. To make the analysis easier, it is assumed that risks of any firm are not
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systematic risks. A share of the market portfolio of traded shares is perfectly risk-free (no-systematic-risk assumption) and yields a safe rate of return.’ An entrepreneur who wants to found a firm in the first period t = 0 has to make three decisions. He determines a/, the size of his firm, he chooses the fraction (1 -&E [0, l] of his firm that he wants to sell in the capital market, and he decides how to invest his possible remaining resources (including the proceeds from selling a fraction of his firm) in the credit market and the capital market. Both credit contracts in the credit market and shares of the market portfolio in the capital market are equally safe. Buying shares of the market portfolio must yield the same, safe rate of interest i of the credit market. This makes the investor indifferent between investing in the credit market or buying shares of the market portfolio in the capital market. For the entrepreneurs’ choice of firm size and his selling decision the selling price of his firm is of major importance. Given that the amount of investment is observable while the fraction of retention is not, the former may serve as a signalling device while the latter cannot. In an equilibrium, therefore, all firms with the same amount of investment must be pooled, regardless of the choice of qf, while there might be a separating equilibrium with respect to firm size al. The situation is not so very different from models of capital market equilibrium with hidden knowledge, exogenous project size and endogenous and observable retentions. For the latter case, there is theoretical evidence that both pooling and separating equilibria can exist [DeMeza and Webb (1990)]. Moreover, given a smoothly distributed continuum of different firm types, the existence of a strongly separating signalling equilibrium can be shown using tools of Engers and Fernandez (1987). There the equilibrium outcome is characterized by a pricing function which assigns an equilibrium price of a firm to each possible amount of retention.3 Similar results can be expected to hold if firm size instead of retentions is the signalling variable, if firm types are smoothly distributed over the range of possible firm types and it is cheaper and more advantageous for good firms to be bigger than for bad ones. A proof is not given here. Instead, it will be assumed that a separating equilibrium exists where firms whose owners have chosen the same firm size are pooled and where the owners get the same price for their firm, while prices differ for firms of different sizes, so that
UJ= u(aJ)
(1)
‘The ‘no-systematic-risk assumption’ is quite usual in the adverse selection literature. Cf., for example, Prescott and Townsend (1984, p. 24) and Rothschild and Stiglitz (1976) for insurance examples and DeMeza and Webb (1990) for a capital market example. ‘Leland and Pyle (1977) considered the properties of a signalling equilibrium in capital markets where retentions act as a costly signal of true firm quality. The empirical evidence is inconclusive. See, for example, Downes and Heinkel (1982) and Krinsky and Rote&erg (1989) for opposite conclusions.
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describes the equilibrium valuation of firms, with v’ continuous and twice differentiable. The pricing function can be calculated to be the equilibrium pricing function by each entrepreneur, but it can also be observed in the equilibrium outcome if a sufficient number of firms exists. Let F, be the class of firms in equilibrium choosing a particular size a. Given the retentions qf chosen by the entrepreneurs f EF,, the value of shares sold in the capital market must have the property
(l+i)=
I (1-qr)z/df JEFO
44.I-U-q%!f 17
(2)
JEF.
with u(a) being the market price of a firm belonging to class F,. The reason is simple arbitrage. Only if the sum of the shares issued by firms of a firm class have the same return as alternative investment opportunities are portfolio investors indifferent between buying shares of these firms and buying other investment opportunities. Implicitly, eq. (2) determines the equilibrium value of firms of a particular class for all classes. The firm class F, and its price, however, appear simultaneously in equilibrium, as each entrepreneur can choose which firm class he would like to belong to.
3. Taxation of risk-taking revenues A tax on revenues from risk taking4 is given by T=r.q[z(a,s)-(l+i)o(u)].
(3)
The superscript will be omitted here and below. A tax is levied on the difference between what the entrepreneur actually gets by choosing a particular amount of retention q, and what he was sure to get, if he decided to sell his whole firm. The income difference between a complete buyout and retaining q>O is the revenue he receives for not selling the fraction q. However, to receive this uncertain additional revenue he has to bear fraction q of the risks implied by the production activity of his firm. The uncertain additional revenue is therefore called risk remuneration or revenue of risk taking. The tax base can also be written as (1 +i)(l -q)u(u)+q*z(u,s)-(1
+i)t’(u).
(4)
4F~r simplicity it is assumed here that tax proceeds are used in a way that doe-s not aNect the decisions of entrepreneurs. Unlike in the situation with perfect capital markets, this is a weak assumption as, here, lump-sum transfers of tax proceeds are not risky. For a discussion, see section 4.
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The tax differs from a profit tax. Instead of interest paid investment costs (1 +i)a being deductible, it is the interest paid equilibrium selling price of the firm which can be deducted. The final wealth V, of the entrepreneur, dependent on a, q and t, is therefore determined by
~,=q.z(a,s)+(l +i)[(l
-q)u(a)-a+
V,]-t.q(z(a,s)-(1
+i)c(a)),
(5)
or, equivalently, V,=q(l-t)[z(a,s)-(l+i)v(a)]+(l+i)[I/,-a+o(a)].
(6)
I/, in eq. (5) consists of the fraction q of output of his firm, in conformity with the retained fraction q, plus the returns of his investment in the credit and/or capital market, minus the tax T. V,, is his initial wealth which is exogenous. An expected-utility maximizing entrepreneur solves the problem maxEW(Vr) (4..)
(74
subject to eqs. (1) and (5), and
42.0,
I-qzo,
ahO.
Ub, c, 4
W(V,) is the Neumann-Morgenstern utility function of the entrepreneur’s final wealth I/,. Solving eq. (7) yields the Kuhn-Tucker conditions aL/dalO,
ahO and aaL/aa=O,
(84
aL/dq 2 0,
a 2 0 and q a Lpq = 0,
W
aLlan 2 0,
I >=0 and i. ayan= 0,
(84
These are necessary conditions for a maximum of problem (7) because the non-negativity constraints (7b), (7c), and (7d) are linear [cf. Chiang (1974, p. 713n)J. The taxation problem is uninteresting if u=O, because ~(0, s) =O. The taxation problem disappears with the source of taxes, i.e. da/dt =dq/dt =0 for a=O.
Similar reasoning applies to the case q = 0. Eq. (3) reveals that TsO holds in this case. An entrepreneur has no incentive to react to a tax change that does not affect him. He chooses
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da/dt = dq/dt = 0.
(9)
Entrepreneurial investors with q=O are founders of firms with expected revenue that at best is as high as the revenue of the market portfolio, i.e. they are initial owners of ‘relatively bad’ firms. Only if E z(a,s) s( 1 +i)u(a), does it not pay the initial owner to retain at least a small positive fraction. Eq. (9) shows that the supply of relatively bad firms will not be altered by a change in the tax rate. Consider now the cases with 420 not binding. From eq. (Sa), c,sE[W’(V,)[q(l-t)z,+(l+i)[(l-q(l-t))o’-l]]]=O,
(10)
with v’- dvlda, z, = &(a, s)/& (and z,, z S’z/S’a in what follows). First the case is treated where the optimal choice of the entrepreneur’s retention is O< 1 -q< 1. From (8b) and (8c), czzE[W’(V,)[z-(l+i)u]]=O.
(11)
The case of an entrepreneur who retains a positive fraction of his firm but also sells some fraction qE(0, l), seems to be very plausible for many entrepreneurial firms. Households are risk neutral with regard to small risks [cf. Merton (1982, p. 609)]. Therefore, an entrepreneurial investor who owns a firm that produces an above average expected revenue, i.e., Ez> v( 1 +i) prefers to retain at least some positive fraction of his firm. Positive retentions q>O imply a welfare loss compared to the case of a perfect capital market. The entrepreneur who deliberately decides to retain q>O bears the burden of risks that could be reduced via the capital market. The activity level a that is described by eq. (10) deviates from the activity level that is chosen under conditions of perfect information. Given perfect capital markets and complete risk diversification, the marginal condition for the optimal amount of investment is determined by E z,= 1+ i. In general, it might differ considerably from eq. (10). The reaction of an entrepreneur to a change of the tax rate t, given the case 1 > q>O and a >O, is obtained by differentiating eqs. (10) and (11) with regard to t with cij-dc,/LJj for i= 1,2 and j=q,a,t, (q>O),
c[:;]=[::. f::] [:;I=[ I::;]&.
(12)
Using Cramer’s rule yields
Wdt = Cl- C&Z~ -( -c,,)c,,]/det
C=O
(13)
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and
Wdc=C(-c,,k,, -(-c,,)c,,]/det
C=q/(l-c).
(14)
This result can be summarized as follows. Proposition 1. Given a capital market equilibrium chat is described by eq. (1), consider an entrepreneur who sells some fraction O
An entrepreneurial investor has two ways of reacting to a change of the tax rate. He can change the investment activity a or the amount of retention q. Define y = q( 1 -c) and substitute in (6) and the optimality conditions (10) and (11). All equations only depend on a and y. Changing q appropriately to keep y constant if c is changed, the entrepreneur can restore the original optimum. As with the original Domar-Musgrave phenomenon, the entrepreneur’s possibility set does not change in a small neighbourhood of the old optimum. Some general equilibrium repercussions of this reaction might change this effect. Initial owners of relatively bad firms sell their entire firms and they do not react to a tax rate change, i.e., they do not increase their retentions if the tax rate increases. Only those entrepreneurs who own ‘more profitable’ or ‘better’ firms increase their retentions. As dq/dc>O is true only for these better firms, an even larger fraction of better firms’ shares are no longer supplied in the capital market, while the supply of ‘lower-quality’ firms remains unchanged. The profitability of capital market investments should fall tendentially if v(a) does not change. To equate the profitability of the credit and capital markets, market prices of firms must fall, and this again may lead to a reduction of the supply of ‘high-quality’ firms in the capital market. A welfare analysis would have to take all full general equilibrium repercussions of the tax-rate change into account. General equilibrium price changes, for example, might induce some entrepreneurs to change investment in order to enter another firm class. These aspects are outside the scope of this paper. The tax proceeds of a tax on risk-taking revenues have quite different stochastic properties than in the case of perfect capital markets considered by Gordon (1985). Asymmetric information leads to equilibria with some diversifiable, but deliberately not diversified, risk. Taxing the revenue of this non-diversified risk allows some of this risk to be diversified within the tax proceeds. These tax proceeds therefore do not have a market value of zero as in the case of perfect capital markets.
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Consider now an entrepreneur who, in equilibrium, chooses q= 1. The tax cannot induce him to increase the fraction of retention. Instead, he might be willing to change the size of his firm. The tax in this case has an immediate impact on the real investment choice. Differentiation of (10) with regard to t yields +i)v’)]/c,,
s=E[W’(z,-(1
+E[W”(z,(l
-t)-(1
+i)(l-tv’))(z-(1
+i)~)]/c,~.
(15)
c,. is negative if the second-order conditions for a maximum are fulfilled. The sign of the numerator determines whether an increase of the tax may increase or decrease the activity level of the firm. Assume that the market equilibrium fulfills the property v’(a) E du/da 2 1.
(16)
Larger firms are the better ones, i.e., their market value is higher by at least the additional investment. Proposition 2. Given a capital market equilibrium that fulfills eqs. (1) and (16), consider an increase of the tax rate t E [O, 1). Ifthere is an ordering of the states of nature such that z(a,s) and z,(a,s) are monotonically increasing in s, and the owner of the Jrm has constant or decreasing absolute and constant or increasing relative risk acersion, then da/dtzO holds for firms that are entirely retained by their entrepreneurial investor (q= l), i.e., the tax increasesjrm size.
The proof of this proposition follows lines drawn by Kihlstrom and Laffont (1983, p. 166). From (15) and using c,,
(17)
and E[fV(V,)[z,(l
-t)-(1
+i)(l-tfi’)][z-(1
+i)v]J
Then, (17) Q E[W’(V,)[z,-(l+i)u’](l-t)]S@ 0
E[W’(V,)[z,(l-t)+(l+i)tt”-_(l+i)u’]]$O
as t
(18)
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but
(18) 0
-(l/(l-t))E[W’(~~)(-W”/W’)[z(l-t)-(l+i)(l-t)c]d]~O
with A = z,( 1 -t) -( 1 + i)( 1 - tu’). The factor l/( 1 - t) does not change the sign and can therefore be neglected. Now add
to z(l-_t)-(l+i)(l-_t)u (18) 0
and re-order to get
-EIW’(VI)(-W”/W’)[z(l-t)+(l+i)[-a+V,]+t(l+i)u]A]
-EIW’(V1)(-W”/W’)[-(l+i)[-a+1/,] -t(l+i)u-(l+i)(l-Qu]A]50.
Note that V, = z( 1- t) + (1 + i) [ - a + V,] + t( 1+ i)u and substitute in the first expectations term and sum up the terms in the second expected value term to get
(18) -
-E[W’(V,)A(-W”/W’)V,] +(l+i)(Vo+u-a)E[W’(V,)A(-W”/W’)]sO.
By eq. (lo), E[W’( V,)A] =O. Therefore, the first term is negative under increasing relative risk aversion if z, and z monotonically increase in s, as this implies that V, and therefore A and (- W/W’) V, also monotonically increase in s. If relative risk aversion (- W”IW’)V’, is constant, then, because of (lo), the first term becomes zero. A similar reasoning applies to the second term. E[ W’( V,) A] = 0 by (10). If z and z, monotonically increase in s, then A and VI also monotonically increase in s. For decreasing absolute risk aversion in the second term negative values of W’A are multiplied with large values of (- W”/W’) and positive values of W’A are multiplied with small values of (- W”/W’), such that E[ W’( V,) A( - W”/W’)] becomes negative. (V, + u-a) is positive, V, > 0 is assumed, and, as u(O)>=0 and u’2 1, u 2 a. If absolute risk aversion (- W”/W’) is constant, the second expected-value term
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becomes zero. Relative and absolute risk aversion cannot be constant at the same time. Therefore, for at least one of the expressions in (18) strict inequality holds. Proposition 2 covers many questions that have been treated in the literature. For o(u) = a, condition (15) simplifies to E[W”(V,)Cz,-(l
+i)][z-(I
+i)u]]
d”~dt~E[W’(V,)~,,]+E[W”(~~)[z~-(*+i)]2(1-t)]7
z. ’
(19)
(i) Expression (19) is identical with the condition derived in the DomarMusgrave literature, where the reaction of an investor to a change in the rate of a tax on risk remuneration is determined. In this literature, an investor, who allocates an initial amount of wealth between a safe asset and a risky investment opportunity that has a stochastic return z(a,s), is considered. The stochastic return depends on the amount of investment a and the state of nature. Let a denote the amount of initial wealth an investor uses to invest in the risky investment opportunity. Then the maximization calculus of this investor yields precisely condition (19). In the case of constant returns to scale, i.e., if z(a,s) =a X(s) with X a random variable, condition (19) reduces to the ordinary Domar-Musgrave effect [cf., e.g., Mossin (1968)-J, da/dt = a/( 1 - t) > 0.
(20)
Mintz (1981) derived condition (19) within the Domar-Musgrave portfolio model and argued that du/dt cannot be signed without further assumptions concerning z(u,s). He provides a condition that implies an increase in risk taking. If z(u,s)-uz,(u,s)rconstant with regard to s for given a and the entrepreneur has constant or increasing absolute risk aversion, then du/dr>O. Mintz’s result (1981) draws on the property of the tax (in the case of decreasing returns to scale) to make the taxed entrepreneur tendentially less wealthy. This property is not ensured when the use of tax proceeds is taken into account. Proposition 2 shows that weaker and more plausible conditions exist which ensure that the tax increases the incentive for risk taking. (ii) The conditions of the stochastic production function cover, as a special case, the stochastic production function
(21) with X a random variable with finite moments, and h and g real-valued differentiable functions with h’>O. Re-ordering X(s) so that X(s) is mono-
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tonically increasing in s shows that the stochastic production function (21) fulfills the requirement made in Proposition 2. Proposition 2 therefore also generalizes Sandmo’s (1971) proposition concerning the effect of a profit tax for a generalized stochastic production function, without having to meet Katz’s (1983) criticism because here the measure of risk aversion in Proposition 2 is concerned with the final wealth of the entrepreneur and not with profits of a firm. 4. Discussion For completely credit-financed firms with exogeneous given project size, the welfare enhancing effect of a lOOaLprofit tax has been shown by DeMeza and Webb (1990). The tax resolves the adverse selection problem by imposing a 100% compulsory insurance. It might be difficult to implement a 100% tax for political reasons. Moreover, a 100% tax may have some adverse incentive effects which make it undesirable. Here, the impact of a small tax rate change is considered. Attention is drawn to an additional dimension of entrepreneural decision making. There is both a project size and the retention decision. These decisions are made simultaneously and are interdependent. For the impact of taxes on revenues that accrue from risk-taking, the endogeneity of the percentage of retentions turns out to be decisive. Given that the fraction of retentions is endogenous, the entrepreneur is forced to react to the tax in a way that amplifies the process of adverse selection. Entrepreneurs do not increase their risky production activity, but they may increase the fraction of retentions. The additional risk diversification within tax proceeds tends to be compensated by a reduced amount of risk diversification via the private capital market. If an entrepreneur cannot react to a tax-rate change via increasing his retentions because he holds all the shares of his firm, then the impact of an increase of the tax is more like that in the classical risk-taking and taxation model. An increase of the rate of this tax has an impact on the demand for risky activity. Under fairly general conditions, an increase in the tax rate increases this activity level, and a considerable amount of additional riskdiversification might take place within the tax proceeds. In this analysis it has been assumed that tax proceeds are used in a way that does not affect entrepreneurs’ decisions. This assumption is much weaker in a model with endogenously explained capital market incompleteness than in a model with perfect capital markets. It has been mentioned that the Domar-Musgrave phenomenon does not appear in a model with perfect capital markets. On the contrary, a lump-sum redistributed risk-takingrevenue tax is perfectly ineffective. The reason is that with perfect capital markets the risk-taking-revenue-tax proceeds are risky. No risk consolidation takes place within the tax revenue. All risk is transferred back to the private
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sector when the tax proceeds are redistributed. The only consistent way to use risky tax proceeds neutrally in the case of perfect capital markets is to spend it on public goods, assuming that these public goods affect the utility function of households in an additively separable way [Ahsan (1989), for example, makes this assumption]. The welfare impact of the use of tax proceeds in this case.depends largely on households’ attitudes towards the risky supply of these public goods. For a detailed consideration of these issues, see, for example, Allingham (1972). In the present model, a much easier way to use the tax proceeds neutrally is possible. Moreover, this way can be judged much more clearly from a welfare perspective. With the no-systematic-risk assumption all risks could be perfectly consolidated via the capital market. The consolidation does not happen because the information structure leads to adverse selection in the capital market. In this case a substantial risk consolidation can take place within the tax revenues and tax proceeds are expected to be practically riskless. If these proceeds are lump-sum redistributed, then practically no risk is transferred to the private sector. Households which receive the redistributions are simply made richer. If all tax proceeds are redistributed to households that are not entrepreneurs and if this does not affect the supply of these households to the credit and capita1 markets, then this is a perfectly neutral and clearly beneficial way to spend the tax revenue. If entrepreneurial households do receive some redistributions, things are more complicated. If decreasing or constant absolute risk aversion is assumed, however, it can be expected that the reaction of the entrepreneurs to a tax increase remains qualitatively the same. As they are richer now, entrepreneurs’ willingness to bear risk increases or remains constant. In the case of incomplete retention (q < l), therefore, the adverse selection problem can be expected to be reinforced and in the case of 100% retention (q= 1) the reaction of risky investment to a change in the tax rate can be expected to be reinforced. References Ahsan, S.M., 1974, Progression and risk-taking, Oxford Economic Papers 26, 318-328. Ahsan, SM., 1989, Choice of tax base under uncertainty: Consumption or income?, Journal of Public Economics 40,99-134. Allingham, M.G., 1972, Risk-taking and taxation, Zeitschrift fur Nationaliikonomie 32, 203-224. Atkinson, A.B. and J.E. Stiglitx, 1980, Lectures on public economics (McGraw-Hill, London). Buchholz, W., 1987, Risikoeffekte der Besteuerung, (unpublished) Habilitation thesis, Universitat Tiibingen. Bulow, J.I. and L.H. Summers, 1984, The taxation of risky assets, Journal of Political Economy 9220-39. Chiang, A.C., 1974, Fundamental methods of mathematical economics, 2nd ed. (McGraw-Hill, London). DeMeza, D. and D. Webb, 1990, Risk, adverse selection and capital market failure, Economic Journal 100, 206-214.
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Domar, E.D. and R.A. Musgrave, 1944, Proportional income taxation and risk-taking, Quarterly Journal of Economics 58, 388-422. Downes, D.H. and R. Heinkel, 1982, Signalling and the valuation of unseasoned new issues, Journal of Finance 37, l-10. Engers, M. and L. Fernandez_ 1987, Market equilibrium with hidden knowledge and selfselection. Econometrica 55, 425-439. Gordon, R.H., 1985, Taxation of corporate capital income: Tax revenues versus tax distortions, Quarterly Journal of Economics 100, l-27. Katz, E., 1983, Relative risk aversion in comparative statics, American Economic Review 73, 452453. Kihlstrom. R.E. and J.-J. Laffont, 1983, Taxation and risk taking in general equilibrium models with free entry, Journal of Public Economics 21, 159-181. Krinsky, I. and W. Rotenberg, 1989, Signalling and the valuation of unseasoned new issues revisited, Journal of Financial and Quantitative Analysis 24, 257-266. Leland, H. and D. Pyle, 1977, Informational asymmetries, financial structure and financial intermediation, Journal of Finance 32, 371-388. Merton, R.C., 1982, On the microeconomic theory of investment under uncertainty, in: K.J. Arrow and M.D. Intrilligator, eds., Handbook of Mathematical Economics, Vol. 2 (NorthHolland, Amsterdam), 601-669. Mintz, J.M., 1981, Some additional results on investment, risk taking and full loss offset corporate taxation with interest deductibility, Quarterly Journal of Economics 96, 631-642. Mossin, J., 1968, Taxation and risk-taking: An expected utility approach, Economica 25, 74-82. Myers, S. and N. Mayluf, 1984, Corporate Rnancing and investment decisions when firms have information that investors do not have, Journal of Financial Economics 13, 187-221. Prescott, E.C. and R.M. Townsend, 1984, Pareto optima and competitive equilibria with adverse selection and moral hazard, Econometrica 52, 21-45. Rothschild, M. and J.E. Stiglitz, 1976, Equilibrium in competitive insurance markets: An essay on the economics of imperfect information, Quarterly Journal of Economics 91, 629-649. Sandmo, A., 1971, On the theory of the competitive firm under price uncertainty, American Economic Review 61.65-73. Sandmo, A., 1977, Portfolio theory, asset demand and taxation: Comparative statics with many assets, Review of Economic Studies 44, 369-379. Stiglitz, J.E., 1969, The effects of income, wealth, and capital gains taxation on risk-taking, Quarterly Journal of Economics 83, 263-283.