UNlVfcKSin Of M STACKS » person ctor ? AS* s or *• ££, £ s rvS4° '»'"?- SS.^S«d for g (l) where y is aggregate net productive supply, x is aggregate net consump- tive demand, g is this vector of administrative inputs, and the inequal- ity is not strict for any commodity which is a productive resource (non-zero productive price) . The remaining equations defining an equilibrium specify that the prices agents face support the allocations included in the materials balance equation. Government policies can specify wedges or distor- tions by which the price vector any agent faces differs from that faced by the base agent (say, firm 1) . For convenience, this paper will follow notationally the simplification of Diamond-Mirrlees, and collapse these policies into the equation q - p = T (2) -4- where q is the price vector facing all consumers, p the productive sec- tor prices, and t a vector of taxes (including subsidies). An extensive set of policies can be so modelled, including agri- cultural surplus and price support programs. If, as below, a monetary equilibrium is being modelled, the monetary components of (1) allows specification of currency printing and open market policies, and the monetary components of (2) allows differential taxation of bonds. The fundamental complexity of equilibrium theory is that any (p,q) satisfying (2) and supporting allocations satisfying (1) is an equilibrium for an economy where the government has selected policies (g,x). Equilibrium is not unique, except under unusually heroic as- sumptions, even given a specification of government policies which is as complete as consistency allows. Government in competitive equili- brium does not have specifiable policies powerful enough to enable the direct selection of prices or allocations. Recognition of this fact complicates the formulation of the pro- ductive efficiency question. Diamond-Mirrlees and Heller-Shell char- acterized optimal policies as exhibiting productive efficiency, in that public sector activity is efficient when evaluated at producer prices. This will not in general be true for any government policy package, that prespecified policies will be efficient at every set of prices which are equilibrium prices. The only known exception is laissez-faire policies ((g» T ) " 0) i which will not be feasible if, for example, contract en- forcement is a costly activity. Full equilibrium analogues to the Diamond-Mirrlees and Heller-Shell theorems are well-defined, however, in the sense of attainable welfare, -5- which is the highest level of social welfare attainable in any equilibrium associated with a given policy package. The Diamond-Mirrlees and Heller- Shell assumptions apply directly in full equilibrium models, and in each, their theorem implies: Equilibrium Productive Efficiency: Limiting attention to policies for which there exists a productively ef- ficient equilibrium does not reduce attainable welfare. 3. The Model Outlined 3.1. Markets and prices The economy to be analyzed has open spot markets for j = 1,...,J commodities in each time period t = l,....,ft. Futures markets, exten- sively analyzed elsewhere, are eliminated here for simplicity. On all markets, goods exchange solely for fiat money, identified as a quantity of greenbacks, one unit of which has value p_ (to producers) or q~ (to consumers) in period t = l,....,ft. This notational convention is maintained throughout: for t = l,...,ft, any market statistic v is (J + 1) -dimensional, with a zero component, v fi , relating to money, v' is v absent v_ , v = (v.,.... t v _) has dimen- sionality 8 = ft (J + 1) » and v 1 = (v. 1 vA) is ftj-dimensional. Where reference is made to an arbitrary component v , the possibility of refer- ence to v„ is disallowed unless explicit. All agents are presumed to know with certainty prices for all periods at the beginning of the economy, and to show no concern for occurrences after period ft. Uncertainty is simply beyond the scope of the present analysis. -6- Firms face producer prices p = (p, , . . . . ,P t , P ) , and households face consumer prices q = (q, » . . . . ,q n ) , with q = p + t, where t ■ (x.. , t ) is the vector of quantity taxes. Producer prices are chosen from B J L p = { P 6 r° | I p - i, t- i ,n}\ (3) j-0 JC and the candidate space for consumer prices is Q = {q 6 P | q 0t 6 (0,1), t = 1, ,«}. (4) Reference will frequently be made to prices associated with an arbi- trary terminal horizon price for money, which is labelled u. Define P(-) : (0,1) + o(P) 5 by P(y) = {p 6 P | p Qfi = u}. (5) As p and q are identically normed, t lies in T = {t GIR 9 | I t,. - 0, -1 < T. < 1, j - 0...J, t - 1,..,R} . (6) Let e , (•), e TT (*) : (0,1) ->-]R be defined arbitrarily, subject to, Li U for all u e (0,1), < e D (w) < 1 - U < 1 + e L (y) < 1. (7) Define !(•) : (0,1) ■*■ o(T) by T(u) = {t G T 1 e L (u) <_ T Qfi < Eu (u)}. (8) When p G P(p), choosing t G T(p) will satisfy p + t ■ q G Q. -7- Placing p in the simplex by (3) is familiar. Much less common is the simplicial restriction of q . As modelled below, firm supply corre- spondences are homogeneous of degree zero in p , and household demand correspondences exhibit zero homogeneity in q . Thus, if an equilibrium J J existed where £ p. = 1 and £ ? 1f , ■ 3| then p and q /3 would yield j=0 Jt j=0 Jt the same allocations, thus still being an equilibrium and allowing J I t = 0. So the norm restrictions in (4) and (6) constitute a per- j=0 Jt missible numeraire choice. With this numeraire convention, t may be interpreted as the "average" excise tax level on goods in period t, and t as the com- modity-specific deviation from this average. 3.2. Households A superscript h represents any of the H households, which differ substantively from usual equilibrium treatments only in the requirement that purchases be covered by beginning-of-period money inventories, and in the monetary tax liability at the end of the economy. Notation: for h = 1,...,H, j=l,...,J, t ■ l,...,ft: x : net purchase by household h of good j in period t. T-T5 \-l x. = max(0,x. ): gross purchase by h of j in t. c S c , (9) where C C IR, is the feasible consumption set. This is a standard con- straint: nonnegative, physically consumable consumption out of stocks. Fidelity: q x£ <_ 0. (10) This is merely a definition of x„ , satisfied whenever the value of goods received does not exceed the value of cash dispersed. Solvency: <_ m <_ m + x . (11) -9- A household raising or lowering its money holdings as purchases and sales are (potentially) d is synchronized is constrained to be solvent, that is, to maintain at all times a nonnegative money inventory. m Q >_ is perhaps the closest analogue in this model to a budget constraint. The right-hand inequality is merely an accounting balance: closing inventory cannot exceed opening inventory plus net receipts. Liquidity: q t x t - ° ^ 12 ^ This constraint codifies the timing convention chosen as perhaps the simplest modelling of a critical role for money in economizing on trans- actions. During each period, purchases must be covered with cash before compensation for sales during the period is realized. Thus period t pur- ■u chases are limited in value by (12) to the value of m . , the previous period's closing money inventory. Period t sales will raise m and cover purchases in t + 1. Denote the household's feasible or budget correspondence X (•) : a Q ■+ oOR ), defined by X^q) = {x h eiR 9 | x h satisfies (9)-(12)}. (13) Household behavior is then specified by max U (c ) subject to x 6 X (q) , m_ <_ m fl . (14) The convention specified by the final inequality in (14) is separated from the constraints of (13) to draw attention to the treatment of ter- minal money stock. In positing a terminal horizon of the economy known with certainty from the beginning by all agents, an anomaly not character- istic of real-world economies is introduced. As money is inherently -10- valueless upon termination of the economy, all agents will attempt to rid themselves of it for the sake of greater last-minute consumption, making a positive price of money impossible in period ft. But if valueless then, money must also have a zero price in ft - 1, and by induction, in all periods. Following Heller [1974] and Okuno [1976] in adopting a suggestion of Lerner [1947], an artificial contrivance is used to respond to this artificially created anomaly. The government is presumed to collect in cash a tax liability from each household, upon the close of period ft trading, in nominal amount equal to the initial cash endowment of the household, ny.. Contrained by (14) to have at least this much money on hand, the household is willing to supply commodities for money in the final period. The possibility of a positive terminal money price being substainable is ensured. The (net) demand correspondence x (•) : Q ■* o(jR ) is defined by: x h (q) = {x h € X h (q) | x h is a solution to (14)}. (15) Maintained assumptions, for h = 1,...,H: h.l. C is closed, convex, bounded below, and contains u in its interior, for t = l,...,ft, m n > 0. Clearly intX (q) ^ by these assumptions, which then have the familiar purpose of ensuring that X (•) is continuous, compact- and convex-valued on Q (Heller [1974], Lemma 2, p. 101). m > guarantees the possibility of nonautarkic behavior, allowing some credulity for the presumed inefficiency of barter transactions. -11- 1_ h.2. u (•) is a continuous, nondecreasing, real-valued function which exhibits convex epigraphs and local nonsatiation. Presumably familiar, this guarantees that x (•) is upper hemi- continuous (uhc) and compact- and convex-valued on Q (Berge [1963], p. 116). k. 3. (Diamond-Mirrlees) For all x e x(q) for any q e Q, either (1) x. < for some k e {1,2, . .. ,Jfi}, or (2) x, > for some k e (1,2, .. ., Jfl} for which q > 0, k = j + J(t-l). 9 3.3. Firms Extensive exploration of firm behavior is well outside the focus of this study. Consequently, modelling of firms has been simplified as much as possible, even at the expense of treatment asymmetric with household modelling. Any of the F firms is represented by superscript f. Net supply by firm f of good j in period t is y. , f = 1,...,F, j = 1,...,J, t ■ l,...,ft. y n designates net units of money dispersed on markets in t = !,...,&. Thus, for t = l,...,ft, if p Q > 0, y ot - ~ p t y t' /p 0f (16) f £' f f By rearrangement, p fc y t = 0. As before, y fc = (y lt ,...,y Jt ) , f , £ f' f t f f\ p f «v _ f y t = (y ot' y t )j y = Wi*"**^* y = Z f y » y = x f y • f « A production plan y may be interpreted as the production of out- f ' f ' puts max (0, y ) from inputs max (0, -y ) . The set of feasible produc- tion plans is Y C]R . -12- Futures markets have been excluded as cuaberscne, and analysis of their import can be found elsewhere. Production is modelled as intra- period with possibilities unchanged over the course of the economy. Firm behavior, presumed to be price-taking profit maximization, can thus be described intraperiodically, as the distinction between f i nominal profits in t (.pi y r ) and the discounted value of profits in t (-y n ) has no behavioral significance. ' f 6 The (net) supply correspondence y (•) : P -*• o(B. ), f = 1,...,F, is defined by: y f (?) = {y f 6 1R 9 | y*' 6 Y* , P V f t ' > p' J» for all j • € Y f , P t yJ = 0, p Qt = implies | y^ | t = 1 Q}, (17) where k is an arbitrary, large bound. Assumptions for firms: £.1. For all f = 1,...,F : Y is a closed, convex cone containing Convexity is a standard assumption, and Cass [3] and McKenzie [17] establish the use of a fictitious factor of production to represent decreasing returns production, with profits distribution to share- 12 holders, as choice over a cone technology. *.2. Production is irreversible. For this modelling of firm behavior, y (•) is uhc on P (Earstad [1977], Lemma 2.1). -13- 3.4. Governmental activity The government selects a vector of quantity taxes x £ T, and a vector g of inputs to administration, constrained by (g.x) 6 G C {(g,t) 6 8 29 |t6I, g ;>0, | g 0t | < k g , t = l,... f fl}. (18) where G is the administrative feasibility set, and k is some arbitrary, © large bound. g It is useful to define the feasible correspondence G(*) : T •*■ oOR ), by G(t) = {g€ R 6 | (g, T ) e G}. (19) The following assumption is maintained: g.l. G is closed and exhibits limited free disposal ((g,x) e G and g' >_ g* imply (g,x) € G) . G(x) * for all x 6 T. This ensures the feasibility of an active commodity policy which could purchase more than the minimal requirement of any administrative input. Additionally, the assumption opens a full role for monetary policy, as only physical commodities are required for administration — any inflow or outflow of money is administratively feasible. So the stream (g ni ,g n „,...,g n ) will represent the governmentally selected monetary policy, the net inflows of money to the treasury. 3.2. (Heller-Shell) G( # ) is a continuous correspondence on T. 3.5. Equilibrium for given policies •\* 'v 48 For ease of notation, let a = (x, y, p, q) £ R , and A = [x(Q) x y(P) x P x Q] C fc 46 . -14- For given (g,T) G G, an equilibrium is a vector a e A satisfying q - P = t (20) x - y + g < 0, p(x - y + g) =0, (21) x h e x h (q) for all h = 1.....H, (22) and y f 6 y f (p) for all f = 1,...,F. 13 (23) Monetary equilibrium models generally are ensured of the existence of equilibrium, as setting all spot market money prices at zero leaves autarky the only possibility for firms and households. Such a position, however, constitutes excessive reliance on the assumption that barter transactions are unused since less efficient than monetary transactions. This assumption is quite tenable when money has a positive price, but barter transactions, not necessarily in balancing amounts, would un- doubtedly be desired in the event of monetary collapse. Autarky has also been ignored in the mechanisms used to model ad- ministrative costs, which are deemed to arise from the need to monitor economic activity. This is in the spirit of the endogenous policy ap- proach of Heller and Shell [1974], for upon finding desired redistribu- tion impossible or prohibitively costly in a barter economy, a government could be predicted to encourage and support the development and mainten- ance of monetary modes of exchange. For these reasons, equilibrium has been defined so as to include the requirement (q £ Q) that monetary methods of exchange are being used by consumers. The possibility that P~ = in equilibrium is included -15- only due to the simplified representation of firm behavior which has made this variable insignificant, as a component of a. The (possibly empty) equilibrium correspondence E(») : G ■*■ o(A) is defined by: E(g,x) = {a e A | a satisfies (20)-(23)} (24) Let D = {(g,i) £ G | E(g,t) ± 0}. (25) D is the domain for policy selection in an equilibrium model of endogenous policy theory. It would be convenient to have a guarantee that D is nonempty. Only minor adaptations are needed to apply the equilibrium theorem of Heller [1974] to show that an equilibrium exists for (g,x) = (0,0). However, the administrative feasibility set may not include the zero vector. Mantel [1975], in a barter model, demonstrates the existence of equilibrium for any tax package satisfying a complex nonexcessive sub- sidies assumption. He does not show that the set of taxes meeting this restriction is nonempty. As nonexcessive subsidies roughly requires that the government satisfy its budget constraint at any price vector (not just at equilibrium prices) , I have not succeeded in constructing an argument that any nontrivial (g,x) meets this condition. The non- emptiness of D remains a technical difficulty. 3.6. The equilibrium-policy set It is convenient to designate z = (a, g, t) = (x, y, p, q, g, x), and to define the graph of the correspondence E(«) as the equilibrium- policy set: -16- Z = {z e a x D J a£ E(g,x)}. (26) Let the correspondence Z( # ) : (0,1) ♦ o(Z) be defined by: Z(u) = {z e z J x 6 10,), p Qfi = „}. (27) Then Z is bounded, and Z(y) is compact for y € (0,1) (Harstad [1977], 14 Lemmata 2.2, 2.3, Theorem 2.4). 3.7. Optimal policies Let DC') : (0,1) ■»■ o(D) be defined by DGO = C(g,T) 6 D | t € T(y)}. (28) The relationships among terminal price levels, policy packages, and attainable consumption sector allocations are represented by e(») : (0,1) x D + o[x(Q)], defined by e(u,g,x) = {x|(x,y,p,q) 6 E(g,x), for some q,y,p S P(u)}, t £ T(u) (29) 0, otherwise. H8 Let the welfare index w(») : JR. ■* IR be a continuous function of alloca- tions, and Paretian in the sense that it is derivable from a welfare function which is an increasing function of utility levels. Designate the attainable welfare function, W(») : (0,1) x D -+TR, defined by r [max w(x) subject to x G e(y,g,x)], (g,x) e D(y) W(y,g,x) = < (30) 0, otherwise. -17- Existence of this maximum is guaranteed by compactness of Z(y). Theorem 1. For any y £ (0,1), attainable welfare W(y,«) achieves a maximum on D(y). Proof: Pick arbitrary y e (0,1). Define b(«) : Z(y) ■*■ JR by b(z) = w(x) . As Z(y) is compact, and b(») is continuous by construction, there exists z* £ Z(y) satisfying b(z*) >_ b(z) for all z e Z(y). By construction, W(y,g*,T*) >_W(y,g,x) for all (g,t) € D(y). The existence of optimal policies in the face of price indeterminacy cannot be guaranteed. 4. Productive Efficiency For convenience, represent g as (g ,g'). where g Q = (g Q1 , . . . ,g QJ p , g* - (4»....gy. and define r(-) : (0,1) ■* o(IR fi ) by r(l») = {g Q e a | (g ,g',T) 6 D(y) for some g' > 0, t e T(y)} (31) where D(y) is D(y) adjusted for component reordering. Let T = {g Q e R " | g Q € r(y) for some y € (0,1)}. (32) Define D°(«) : (0,1) x T ■*■ o(D) by {(g,x) e D(y) | g Q = y}, Y e r(y) D°(y,Y) = w(x) for all x G e°(y,Y)}, Y 6 Uy) Z^y.Y) = > (38) L 0, otherwise, and Z 2 («) : (0,1) x T -»- o(Z) by r {z G Z°(y,Y) | y' - g' « n' € bdy N(t)}, y G r(y) Z 2 (y,Y) = I (39) 0, otherwise. Given y,Y> Z (u,y) is the set of equilibrium-policy relations for which 2 the maximum level of attainable welfare is reached, and Z (p,y) is the productively efficient subset of Z. 1 2 Theorem 2. Given any y G (0,1), any y e T(y), Z (y,Y) n Z (y,Y) f 0. That is, for arbitrary monetary policy, optimal tax and administrative policy responses are productively efficient. The proof of the theorem, which is presented in section 5, handles several complications which do not arise in the Diamond-Mirrlees model, but essentially the same logic is used. Diamond-Mirrlees presume an in- efficient optimum. Given their demand assumption, there exists a Pareto- improving price change. If the presumed optimal allocation is interior to the aggregate net production possibilities set, there exists a nearby feasible allocation supported by the Pareto-superior price vector. This contradicts the original presumption. Notice that their argument held net tax receipts constant at zero while showing feasibility of a dominat- ing allocation. In this model, zero monetary inflows have no special role, -20- so the Diamond-Mirrlees argument can be replicated holding monetary in- flows at an arbitrary level. 5. Proof of Theorem 2 1 2 1) Suppose the contrary: Z (u,y) n Z (y,y) = 0. For any z* € Z (u,y), n* 6 int N(t*). 2) By /x.3., there exists q G 3R , q ^ 0, so that q = (q* + Aq) 6 Q is Pareto-superior to q* for any A G (0,1], •v 'v % . 17 For arbitrary e.. > 0, suppose x(q) O n (x*) = for every A e (0,1]. e l , Oj As Z (h,y) is compact by construction (Karstad [1977], Thro. 2.4),andx(*) is uhc on Q, there exists z** £ Z (y,y) such that either: 2 a) z** e Z (u,y)> as desired, or b) there exists q - , Pareto-superior to q** and arbitrarily close, so that x(§) n n (x**) ± for any e 1 > 0. £ 1 3) Assume, then, without loss of generality that for any A (e-, £„) >> 0, there exists (x,§) satisfying: A a) q £ Q is Pareto-superior to q*, A b) q e n (q*), and G 2 ' „ c) x e x(q), x e n (x*). e l 4) As n*' S int N(t*) and G'(») is continuous on T by Q.2., A A A A n** S N(t) for any x 6 w. (x*). This in turn implies n' e N(t) for £ 3 any n' £ it (n*'), and e„ > 0, any e, > for suitably small (e^e,^.). 4 A 5) Therefore, there exists (g,y) satisfying: AAA a) y* - g« = n«, A A b) g£ G(t), as G(») is continuous, A c) g Q = Y, by (18), and -21- A d) y e yCp) for some p £ n (p*) , for any e s > 0, for £ 5 3 suitably small (£■.,£„,£,). A A AAA 6) As q S n (q*) and p e n (p*) , (q - p) = t S n (t*) for £ 2 e 5 £ 3 any e„ > for suitably small (e„,e»). A 7) If for any component jt, x* < n* , then p* = 0, and p could A A have been chosen above with p. = 0, while satisfying p G n (p*) , J e 5 n' £ n (n*) , guaranteeing x.^ < n. - e 4 Jt - jt 8) For any component jt with x* = n* , x. S fl(n* ) . Thus, A AAA for suitably small (e 1 e,) , z can be chosen to guarantee a G ECg»T) As q was chosen Pareto-superior to q*, w(x) > w(x) , contradicting supposition that z* S z (y,y)^ -22- Footnotes 1. Assumption k.3. below, used to guarantee the existence of Pareto- improving price changes. 2. Assumption Q.2. below, used to guarantee that small changes in tax rates can be feasibly administered given small changes in adminis- trative resources. Heller-Shell present a realistic example of an administrative technology which does not satisfy this assumption. 3. Diamond-Mirrlees model the government as selecting consumer prices and firm allocations, subject to being on the boundary of the private sector possibilities set. Heller-Shell have the govern- ment choosing firm and household allocations, subject to the existence of a vector of buying and selling prices, each for pro- ducers and consumers, which supports the allocations, given house- hold-specific lump-sum taxes and transfers, and firm-specific profits taxes and license fees. 4. TR is the 9-dimensional real space. Throughout, E. = {v 6E | v >_ 0}, 6 8 JR, , = int JR . Vector inequalities: v > implies v. > 0, all j in- ""■ + — j — - eluding 0; v > implies v >_ ^ v; v >> implies v. > all j including . J 5. o(S) is read "the set of subsets of S" and is the power set for S. A mapping : S -*• S is point-valued, : S ■*■ o(S) is set-valued. 6. An obvious generalization, avoided for simplicity, is to require nonnegative money holdings in a model with bond markets. If the sale of bonds includes the incurrence of transactions costs, possibly through a timing mechanism similar to (12) (some feature is needed to make bonds and money distinct assets in equilibrium), earlier drafts and various futures markets literatures convince me that the qualitative results below are maintained. 7. More complicated modellings of transactions costs may well leave households with nonconvex budget sets, which would create substan- tial mathematical difficulties best avoided here. If impact of transactions costs is restricted to maintain convexity, and money still plays a critical role, the structure of household behavior should be qualitatively unaltered, and the analysis to follow should apply. -23- 8. ul 6 int C h implies uP » 0. Heller [1974], p. 103: "This is a familiar if absurd assumption. I believe it can be replaced in the usual manner at the usual cost of a substantial increase in the complexity of the proofs." Heller does not vary underlying para- meters and examine the resultant variation in equilibria. I cannot be certain the assumption is not crucial here. Weymark [1978], p. 4: "In the context of the optimal taxation literature, where no lump-sum taxation is possible, this is a highly restrictive assumption." On the one hand, with administrative costs incumbent upon all tax instruments, lump-sum taxation no longer enjoys such a unique position. On the other hand, (12) is more readily de- fended when time periods are interpreted as short, making u> >> 0, all t, even less tenable. 9. This assumes the existence of either a common demand or a common supply good. Weymark [1978] shows in essence that this can be weakened to being able to find a basis for the commodity space such that some composite good is in common demand or common supply. No primitive assumptions (that is, no assumptions about individual characteristics of agents, rather than combined char- acteristics of sets of agents) suffice to guarantee the existence of common demand or supply goods. 10. Note that if p n = 1 for any t e {l,...,fi}, any feasible y is profit-maximal, and y n =0. If p» = 0, any t, nominal profit maximization is well-defined, and the supply correspondence defined in (17) includes any y which is optimal in other periods, and for f which y maximizes nominal profits (that is, it includes all c f bounded values of y_ ) • 11. Readers may be interested in whether the analysis to follow extends to a model where firms are treated symmetrically to households, and must pay cash for period t purchases before receiving payment for period t sales. First, the model would have to allow for firms having initial endowments of cash, and a terminal tax would be re- quired to provide an incentive for firms to supply goods for money in period ft. Second, in such a model, the distinction between nominal and discounted profits would be critical. Without addi- tional features, the objective of price- taking firms would be un- clear. If a full set of future markets were added, and provision was made for an equilibrium distinction between bonds and money as assets, and money still economized on transactions, firms would maximize the first-period discounted value of profits streams. Analysis virtually identical to that presented below could be applied by similar argument to such a model, with the candidate set for producer prices restricted to Q. A significantly dis- similar argument would be required at one step, noted in footnote 14, below. -24- 12. Household behavior has been modelled consistent with the possibility of a commodity in which utility is degenerate. The strict positivity of endowments, discussed in footnote 8, implies that each household holds a positive ownership share of each firm. R. ={v e JR | v < 0}. 13. A direct implication of the equilibrium conditions is the exact balance of the government budget at the level of the selected mone- tary policy in each period where p fi > (if p- = 0, y_ is in- determinate — footnote 10) . From (21) , s ot " *ot - x 0t = q t x [ ' q ot - p t y t ' Pot , <;*- T t *' t * ,jg - y') — - — + , so T 0t P 0t q 'x 1 t'x 1 p 'g' Vt , t t "t 5 t 8ot = T 0t + ^t ^t * The last line states that the net inflow of money to the government in period t equals the net receipts from the average tax rate plus the net receipts from the commodity-specific deviations from this average rate (either of these two terms may be negative) minus the money spent to purchase administrative inputs. 14. Lemma 2.2: Z is bounded. Lemma 2.3: Given any y G (0,1), any sequence {q } -»■ q with q G Q for all n, and q G [P(y)/Q]; for any {x } with x 6 x (q ) , {x } is unbounded, for any h = 1,...,H. Theorem 2.4: For any u G (0,1), Z(y) is compact. To extend the present analysis to the model discussed in footnote 11, a lemma similar to Lemma 2.3 would be required, show- ing that firm supplies explode when {p } -*• p with p € Q for all n, p° 6 [P(u)/Q]. 15. Harstad [1977] presents an example. It assumes a convergent se- quence of policies increasing in attainable welfare. The asso- ciated welfare-attaining sequence of allocations converges to an allocation which is not feasible, because the corresponding se- quence of equilibrium prices converges to monetary collapse. 16. Resolving price level indeterminacy does not directly enter the efficiency argument, but is needed to ensure the existence of an optimum, that is, to ensure Z (y,y) f 4 0. A A 17. n (v) = {v G s| | |v-v| | < e} for any v G s. -25- References 1. Kenneth J. Arrow and Gerard Debreu, Existence of equilibrium for a competitive economy, Econometrica 22 (1954), 265-290. 2. Claude Berge, "Topological Spaces," Macmillan, New York, 1963. 3. David Cass, Duality: a symmetric approach from the economist's vantage point, J. Econ. Theory 7 (1974), 272-295. 4. Gerard Debreu, Economies with a finite set of equilibria, Econometrica 38 (1970), 387-392. 5. Peter A. Diamond and James A. Mirrlees, Optimal taxation and public production, Amer. Econ. Rev . 61 (1971) , 8-27 and 261-278. 6. Ronald Morris Harstad, "Elements of an Equilibrium Theory of Tax- ation," Ph.D. dissertation, University of Pennsylvania, 1977. 7. Walter Perrin Heller, The holding of money balances in general equilibrium, J. Econ. Theory 7 (1974) , 93-108. 8. Heller and Karl Shell, On optimal taxation with costly adminis- tration, Amer. Econ. Rev . 64 (1974), 338-345. 9. Werner Hildenbrand, "Core and Equilibria of a Large Economy," Princeton University Press, 1974. 10. Tjalling C. Koopmans, "Three Essays on the State of Economic Science," McGraw-Hill, New York, 1957. 11. Abba P. Lerner, Money as a creature of the state, Proc. Amer . Econ. Assoc . 37 (1947), 2. 12. Rolf R. Mantel, General equilibrium and optimal taxes, J. Math . Econ . 2 C1975), 187-200. 13. Lionel McKenzie, On the existence of a general equilibrium for a competitive market, Econometrica 27 (1959) , 54-71. 14. Masahiro Okuno, General equilibrium with money: indeterminacy of price level and efficiency, J. Econ. Theory 12 (1976), 402-415. 15. John A. Weymark, On Pareto-improving price changes, J. Econ. Theory 19 C1978), 338-346. M/D/232 t\y