Financial Control of a Competitive Economy with Public Goods but Without Randomness I. Karatzas, M. Shubik, and W.D. Sudderth December 28, 2010 Abstract The monetary and fiscal control of a simple economy without outside randomness is studied here from the micro-economic basis of a strategic market game. The government’s bureaucracy is treated as a public good that provides services at a cost. A conventional public good is also considered.
1 1.1
Introduction Context
When one considers the broad sweep of public goods and services from bridges and roads to defense and law enforcement, there are many distinctions to be made concerning both how to supply them and how to pay for them. There is a large literature on these matters (see Musgrave & Musgrave (1973), Samuelson (1954), Salani´e (2003), Shubik (1984), Bergstrom, Blume & Varian (1986), Diamond (2006)). There has also been a certain amount of experimental gaming concerning public goods (see Ledyard (1995)). Underlying all of this work has been a recognition of the enormous variety of institutional structures involving public goods and their financing. The gap between abstract mathematical economics and applied public finance is considerable, although works such as that of Salani´e (2003) are devoted to closing this gap. This paper is part of our attempt both to narrow this gap and to provide sufficiently well-defined models that could serve as a platform for experimental games. 1
It is clear that one of the critical problems (or class of problems) in public economics and in macroeconomics can be classified as the government control problem. A control problem in economics, unfortunately, is not like a control problem in mechanical engineering where the physical objects and the power sources can be easily defined and controlled. For the very simple models of this paper, it is, however, possible to formulate precise government control problems in terms of variables such as the rate of interest and the rate of taxation. The basis of our work is the classical article on a single pure public good by Samuelson (1954), but we note that the treatment in that article was completely static. Here we introduce a class of simple well-defined process models based on this highly simplified idealization of one type of public good. For these simple models it is possible to provide complete, mathematically precise descriptions of the dynamics. Although we do characterize equilibria for the models, they are also available for study out of equilibrium. This possibility of dynamic disequilibrium requires us to consider the nontrivial and economically relevant problems of default and cash flow constraints. These problems are not easily discerned when (as is often done in economic theory) attention is devoted heavily to equilibrium conditions, especially with extra assumptions such as rational expectations. To make possible the specifications for playable experimental games, more is required than a well-defined mathematical model. The experimental subjects must be briefed and debriefed. This involves providing a qualitative description of the context of the game and what it is meant to represent; see Huber et al. (2011) for details.
1.2
Approach
In the models of this paper we attempt to forge a link between full process models and rational expectation equilibria not because we believe that the rational expectations analysis provides more than some moderate insights into dynamic equilibrium, but because the labor of having to spell out any complete process opens up the underlying theory to a host of questions that can be easily ignored when one’s attention is given only to equilibrium. Details such as the importance of time lags and whether the timing and the actual accounting conventions of the economy match are brought out. We purposely restrict attention to economic models without exogeneous uncertainty, because there is so much modeling underbrush to be cleaned 2
up and organized before one can tackle exogenous uncertainty. Although the solutions for the non-stochastic models are relatively simple, the careful modeling of the laws of motion for the agents requires laying out virtually all of the details required to study the stochastic models. Another reason to study these relatively simple models is that they often have analytically tractable solutions, which is rarely the case for stochastic models with independent agents. An important feature of our non-stochastic models with simultaneous moves by small homogeneous agents is that the symmetric equilibria coincide with the equilibria of representative agent models. Of course, many of the most critical and vexing problems in the provision of public goods appear with the introduction of uncertainty. For example, the problem of how to fund smoothly the salaries of a bureaucracy appears. We plan to treat in a subsequent article the much more difficult case of independent agents with exogenous uncertainty. There are common words and phrases that we use daily and successfully in broad economic analysis such as money, credit, public goods, profits, bonds, and various forms of taxes. All of them appear to have multiple meanings and usually need not be defined too tightly for the purposes and questions at hand. But the difficulties encountered in making them operationally precise enable us to appreciate the delicacy of the link between careful formal mathematics and an application. We regard our models as caricatures of reality, but believe that they start to close the gap between the formal models of mathematical economics and their verbal descriptions, which are often far apart. In this and two companion essays, macro-economic problems of the monetary and fiscal control of an economy are viewed building up from the microeconomic basis of a dynamic strategic market game. In previous works (cf. Geanakoplos et al. (2000), Karatzas et al. (1994, 2006, 2006a), we have considered a highly simplified economy with either a representative agent or a continuum of small, strategically independent agents. We continue this approach here with a more explicit consideration of government and the role of bureaucracy in policing the economy. Our contribution is complementary with much of the literature on monetary and fiscal policy in the sense that we present in explicit detail closed strategic models for the microeconomic processes of the economy. It has been said that a good tax is one that is easily collected. This contains a mixture of considerations of both public approval of the purpose and the efficiency and honesty of the collection system. Collection costs are not considered in our discussion. 3
The models presented below illustrate at a high level of simplification some fundamental features of government control of taxation, bureaucratic enforcement of default conditions, and public debt creation. These features are treated with emphasis on information conditions, the equations of motion, and the logical necessity for default rules and their enforcement. The simple, low-dimensional models treated here, like many such lowdimensional models studied in the literature (cf. Lucas (1978, 1980), Lucas & Stokey (1983) among others), are highly restricted when compared with the large macro-economic models utilized by practitioners (cf. Fair (2010)). These simple models may provide some insight into qualitative properties of an economy and raise questions about the logical completeness of the large models. However, they are not substitutes.
1.3
Outline
We shall consider five basic models. Before each of the models we comment on the nature and relevance of the micro-economic details appearing in them. In each model there is a continuum of producer-consumer agents who seek to maximize their total discounted utility from consumption over a countably infinite sequence of time periods. In Models 1 and 2 the government taxes the agents and uses the revenue to support a bureaucracy that provides government services. In Models 3, 4, and 5 a conventional public good is supported. In the first three models a central bank sets an interest rate and stands ready to make loans or accept deposits. In the fourth model one-period bonds are introduced. Model 5 considers consols, which are also known as perpetuities. For each of the models we construct a type-symmetric stationary equlibrium. Some of the details of the proofs are in the Appendix. It may at first appear that, with no stochastic elements, the distinction between representative agent and independent agent models appears to be a distinction without an operational difference. Even at this level this is not quite true. It takes a fair amount of care to establish that there are no equilibria other than typesymmetric equilibria in a game that is structurally intrinsically symmetric. A simple example utilizing a 2 x 2 bi-matrix game is sufficient to illustrate this. 2, 1 0, 0 0, 0 1, 2 4
There are three equilibria in this symmetric game, two of which are nonsymmetric in payoffs. We restrict ourselves below to type-symmetric equilibria. However, as soon as uncertainty is introduced, symmetric equilibria generally do not exist. In particular this makes any taxation schemes far more difficult than if all agents were identical.
2
Model 1. An Economy with Taxation to support a Government Bureaucracy
Possibly one of the purest public goods is the presence of a system of law and order, under which the society and economy flourish. A bureaucracy is required to enforce the law and order and it requires resources, mainly in the form of wages to maintain it. In a separate paper Shubik and Smith (2005) consider the optimal size for the bureaucracy; here it is of a given size and its wages are determined by the flow through to the bureaucracy of the taxes collected. All of the models we consider will have a government and a continuum I = [0, 1] of producer-consumer agents. In each model there will also be a single private perishable good, which will serve as both a consumption and a production good. Model 1 has in addition another continuum J = [0, δ], δ > 0, of government bureaucrats. The size δ of the bureaucracy is exogenously given. The government taxes the producer-consumers in order to support the bureaucrats. The productivity of the bureaucracy is implicit in the model and can be interpreted as supplying the basic running of the government and enforcement of the laws. For simplicity we assume that the bureaucrats obtain a tax-free income. This saves on some accounting but yields substantially the same problem. We turn now to a detailed description of the model. • At the beginning of every time period n = 1, 2, . . ., every producerconsumer agent α ∈ I holds cash mαn ≥ 0 and a quantity qnα ≥ 0 of goods. Every bureaucrat γ ∈ J holds cash mγn ≥ 0 but no goods. (The bureaucrats neither hold nor produce the private good.) Every agent α sells her goods in a market so that the total amount of goods sold in period n is Z Qn = qnα dα. (1) I
5
In every period n , every agent α ∈ I bids cash bαn in the market in order to obtain goods for consumption and as input for production in the next period; likewise, every bureaucrat γ ∈ J bids cash bγn for goods. The total amount of cash bid is Z Z A B α Bn = Bn + Bn = bn dα + bγn , dγ (2) I
J
and the price of the good is formed endogenously as Bn . pn = Qn
(3)
Each agent α ∈ I receives in cash the revenue pn qnα from the sale of her goods. However, this income is taxed at a rate θ ∈ (0, 1) set by the government so that α’s net income is θ¯ pn qnα in period n , where θ¯ := 1 − θ . Prior to bidding in the goods market, each α ∈ I can borrow from or deposit in a central bank which sets the rate of interest ρ > 0, the same for borrowers and for lenders. The bank predicts the price pn to be pˆn and permits α to borrow up to (θ¯pˆn qnα )/(1 + ρ) which is the amount that α is expected to be able to repay with interest. Thus α is able to bid any amount θ¯pˆn qnα α α . (4) bn ∈ 0, mn + 1+ρ In a rational expectations equilibrium, we have pˆn = pn . The agent receives her bid’s worth in goods bαn /pn , and then selects a portion knα ∈ [0, bαn /pn ] to be put into production. She consumes the remaining goods xαn = bαn /pn −knα and receives in utility u(xαn ), where u : [0, ∞) → [0, ∞) is a concave, increasing, differentiable utility function such that u(0) = 0. Each agent α seeks to maximize the total discounted utility ∞ X
β n−1 u(xαn ),
n=1
where β ∈ (0, 1) is the discount factor. Agent α begins period n + 1 with cash ¯ nqα. mαn+1 = (1 + ρ)(mαn − bαn ) + θp (5) n The agents α ∈ I have a common production function f : [0, ∞) → [0, ∞) which is also assumed to be concave, increasing, and differentiable with f (0) = 0. We assume further that f 0 (0) = +∞, lim f 0 (k) = 0. k→∞
6
(6)
The input knα of agent α results in the output f (knα ). Agent α begins period n + 1 with goods α qn+1 = f (knα ) + y. (7) The quantity y ≥ 0 of goods can be viewed as an additional endowment that could be flowing in from a source external to the economy. (It is included with an eye toward generalizing to a random variable Y .) • The situation for a bureaucrat γ ∈ J is similar, except that the bureaucrats do not hold goods, do not produce them, and pay no income tax. The bureaucrats receive as income the tax revenues collected from the producerconsumer agents. The total revenue collected in period n is θpn Qn and this amount is divided equally among the bureaucrats. Since the size of the bureaucy is δ, each of them receives the cash amount θpn Qn /δ at the end of the period. Prior to bidding in period n the bureaucrat γ is allowed to borrow from the bank an amount based on his expected income and can thus bid any amount θpˆn Qn γ γ . (8) bn ∈ 0, mn + δ(1 + ρ) As already mentionned above, the bank’s predicted price pˆn will agree with the actual price pn in rational expectations equilibrium. The bureaucrat γ then receives xγn = bγn /pn in goods, consumes all of his goods, and thereby gets u(xγn ) in utility. Each bureaucrat γ seeks to maximize the total discounted utility ∞ X β n−1 u(xγn ). n=1
(Our analysis would be unchanged if the bureaucrats had a different utility function from that of the producer-consumer agents.) The bureaucrat γ begins period n + 1 with cash θpn Qn . (9) δ We are now ready to construct an equilibrium for this model. However, we pause first for a brief aside on solution concepts. mγn+1 = (1 + ρ)(mγn − bγn ) +
2.1
An aside on solution concepts
For each of our models, we shall construct a type-symmetric Nash equilibrium. That is, we shall find initial conditions and strategies for the agents 7
such that all agents of a given type use the same strategy and every agent is playing optimally given that all other agents follow the prescribed strategies. For these equilibria the phenomenon called by macro-economists “time inconsistency” (cf. Kydland and Prescott (1977), Chari and P.J. Kehoe (2006)) does not arise. From the viewpoint of formal game theory this notion raises problems in the definition of the extensive form of a game and whether nonbinding language statements are formally modeled as moves in the game. Many years ago Schelling (1960) launched a well-directed critique against formal game theory; but apart from several intuitively attractive observations on the problems of precommitment and threat he offered no formal solution. The work of Kydland and Prescott (1977) deals somewhat with this question. However there are many plausible models that depend delicately on the order of moves and disclosure of information between the government and the small agents. We do not pursue this further beyond observing that this provides an example of where the relationship between context and the model and the mathematical formulation is delicate.
2.2
An equilibrium for Model 1
To construct a type-symmetric equilibrium, we suppose that every producerconsumer agent α ∈ I begins play with the same cash mα = mA > 0 and goods q α = qA > 0, and that every bureaucrat γ ∈ J begins with cash mγ = mB > 0. Let M be the total cash holdings of both types. Thus Z Z α M = m dα + mγ dγ = mA + δ · mB . I
J
The total amount of goods available is Z Q = q α dα = qA . I
We assume and then prove that the following set of strategies are optimal. Suppose that every agent α ∈ I bids the same amount bα = bA = aA · M, and that every bureaucrat γ ∈ J bids bγ = bB = aB · M. 8
The total bid is then Z Z α B = b dα + bγ dγ = bA + δbB = (aA + δaB )M = aM , I
J
where we have set a = aA + δaB . The price of goods is p=
B aM = . Q Q
Suppose also that each agent α ∈ I inputs the same quantity k for production and that Q = qA = f (k) + y. (10) Thus the quantity of goods will again be Q in the next period. By the laws of motion (5) and (9), agents and bureaucrats, respectively, will in the next period have cash holdings ¯ = (1 + ρ)(mA − bA ) + θB ¯ m ˜ A = (1 + ρ)(mA − bA ) + θpQ
(11)
and
θB θpQ = (1 + ρ)(mB − bB ) + . δ δ The total cash in the next period will be m ˜ B = (1 + ρ)(mB − bB ) +
(12)
˜ =m M ˜ A +δ m ˜ B = (1+ρ)(M −B)+B = (1+ρ)(M −aM )+aM = τ M, (13) where τ := 1 + ρ − ρa
(14)
is the rate of inflation (or deflation). If the agents and bureaucrats continue to bid the same proportions of their money, then the price in the next period will be ˜ aM aτ M p˜ = = = τp . (15) Q Q This way, prices inflate at the same rate as the money supply. Theorem 1: Suppose that every producer-consumer agent α ∈ I begins with ¯ and goods q α = f (k1 ) + y and that every bureaucrat γ ∈ J cash mα = θM ¯ 1 begins with cash mγ = θM/δ, where M > 0 and f 0 (k1 ) = (1 + ρ)/β θ. 1
The existence of such a k1 follows from the condition (6). Indeed, our only use of this condition is to guarantee the existence of k1 .
9
Then there is an equilibrium in which every α ∈ I bids the constant proportion a of her cash in every period and inputs k1 for production, and every γ ∈ J bids the constant proportion a of his cash in every period where a=
(1 + ρ)(1 − β) . ρ
(16)
The proof of Theorem 1 is in the Appendix. ¯ aB = aθ/δ, and In the equilibrium of Theorem 1, we have aA = aθ, a = aA + δaB , where a is given by (16). Thus, by (14) and (16), money and prices inflate at the rate τ = (1 + ρ) − ρ ·
(1 + ρ)(1 − β) = β(1 + ρ) , ρ
(17)
in agreement with the classical Fisher equation. Also, the quantity of goods produced stays fixed at Z Q = q α dα = f (k1 ) + y. (18) I
The government influences inflation and production through its choice of the tax rate θ and the interest rate ρ. Clearly, there is no inflation if the central bank sets 1 1+ρ= . β However, production is an increasing function of 1+ρ 0 −1 k1 = (f ) . β θ¯ Since f 0 is a decreasing function, we see that lower interest rates increase production. The same is true of lower tax rates. However, a lower tax rate also results in less money to support the bureaucracy and the government services it provides. This trade-off between private and public production will be made explicit in Model 3 below. Out of equilibrium the government may not be able to balance the budget at the end of every period, hence a flexible system will be needed. One way of achieving the flexibility is to have bureaucratic wages be variable, following the tax collection. 10
A simple example is illustrative. Example 1: Consider the stationary equilibrium of Theorem√ 1 in the special case when the production function of the agents is f (k) = 2 k. and y = 0. Then 1+ρ 1 f 0 (k1 ) = √ = β θ¯ k1 so that k1 =
β θ¯ 1+ρ
2
and
β θ¯ . 1+ρ In a period when the money supply is M , the consumer-producer agents bid ¯ , the bureaucrats bid bB = aθM/δ, and the price of the good is bA = aθM p = aM/Q. Thus, in every period, the agents and bureaucrats consume, respectively, ¯2 β θ bA β ¯ − k1 = − k1 = θQ xA = 2− p 1+ρ 1+ρ Q = f (k1 ) = 2
and xB =
bB 2βθθ¯ = . p δ(1 + ρ)
Thus, their total discounted utilities are 1 · u(xA ) 1−β
and
1 · u(xB ). 1−β
It is easy to check that xA increases as the tax rate θ and the interest rate ρ decrease. On the other hand, xB increases as ρ decreases but has a maximum in θ at θ = 1/2. 4
2.3
A comment on Model 1
In this model we have provided for a closed system that pays for a bureaucracy via taxation. We have not yet considered the importance of its enforcement abilities. This is done in Model 2. Before proceeding to Model 2 we note that the reason why in Model 1 we were able to avoid the possibility 11
of bankruptcy, or agents being unable to repay their loans, is because we imposed the unrealistic condition that pˆ, the bank’s estimate of future price p, is completely accurate, thereby enabling it to limit precisely the level of permitted borrowing in equilibrium. If this condition is lifted then default or bankruptcy conditions must be specified as they are in Model 2.
3
Model 2: Unlimited lending with penalties for default
In Model 1, individuals were permitted to borrow up to a level limited by their expected income for the period. Instead of using this “secured lending” rule we now consider an economy with unsecured lending permitted. But with unsecured lending there is a need for default penalties to guard against failure to repay, and the penalties require enforcement. In this model we now view the contribution of the bureaucracy as being in charge of the enforcement of the laws, rules and instruments promoting efficient and honest trade (cf. Shubik and Smith (2005)). A natural extension is to consider the “production function” of the bureaucracy in being able to deter antisocial strategic behavior. We have not modeled this function explicitly. Instead Model 2 imposes a penalty in negative utility on agents who default on their debts. In reality, effective enforcement is a function of the size, honesty and efficiency of the bureaucracy. In most respects Model 2 is the same as Model 1. There is a continuum I = [0, 1] of producer-consumer agents and a continuum J = [0, δ] of bureaucrats. The producer-consumers hold cash and goods, and are taxed to pay the bureaucrats who hold cash but no goods. The critical difference from Model 1 is that the bank no longer imposes any bound on lending to either agents or bureaucrats. At the start of any period when loans are due, we assume in Model 2 that the bank can collect any liquid assets held by a debtor and will enforce a penalty in units of negative utility proportional to any debt that remains. The debt is then erased and the individual is permitted to borrow again. When the penalty for default is sufficiently severe, the agents and bureaucrats will not wish to incur large debts and will themselves limit their borrowing. To make these assumptions precise, consider first a producer-consumer agent α ∈ I who begins a period with cash mα ≥ 0 and goods q α ≥ 0.
12
The agent is permitted to bid any amount bα ≥ 0 for goods and input k α ∈ [0, bα /p] for production. The agent then consumes xα = bα /p − k α and receives u(xα ) in utility. Here p is the price formed in the market as in Model 1. (With no limit on bids, the bank does not need to estimate the price p as was necessary in Model 1.) Agent α begins the next period at (m ˜ α , q˜α ) where ¯ α m ˜ α = (1 + ρ)(mα − bα ) + θpq
and
q˜α = f (k α ) + y.
(19)
If m ˜ α ≥ 0, play continues as before. However, if m ˜ α < 0, the agent is in default and is punished in utility by the amount ζ·
m ˜α , p˜
(20)
where ζ is a positive parameter and p˜ is the price of the good in the period when the punishment takes place. The situation of a bureaucrat γ ∈ J with cash mγ ≥ 0 is analogous. The bureaucrat can bid any amount bγ ≥ 0 for goods, consume xγ = bγ /p, receive u(xγ ) in utility, and begin the next period in position m ˜ γ = (1 + ρ)(mγ − bγ ) +
θpQ . δ
If m ˜ γ < 0, the bureaucrat is punished by the amount ζ · m ˜ γ /p , and then plays from 0. If the punishment parameter ζ is sufficiently large, then there is an equilibrium in which no bankruptcy occurs. Indeed, the equilibrium is the same as that of Model 1. Theorem 2: If ζ ≥ u0 (0) , then the equilibrium of Theorem 1 is also an equilibrium for Model 2. The proof of Theorem 2 is in the Appendix. If the default penalty is sufficiently small, the equilibrium of Theorem 1 for Model 1 need not be an equilibrium for Model 2. Example 2: Let u(x) = x and ζ = 1/2. A producer-consumer α who bids ¯ α )/(1 + ρ) will have marginal utility b > mα + (θpq 1 1 1 0 b α u −k −ζ = 1− > 0, p p p 2 13
corresponding to the difference between the additional marginal utility from consumption and the disutility from the default penalty. Thus agent α will prefer to go bankrupt rather than follow the strategy of Theorem 1. 4 The penalty as defined in (20) is adjusted for inflation. In reality it may take decades to correct for inflation in many laws involving the worth of property. Thus the crime of stealing $20 in 1880 should be reclassified as a tort in 2000, but may remain on the books as a crime. Consider a variation of Model 2, call it Model 20 , in which the penalty for default is not indexed. Thus, if a player begins a period with a negative cash position m, ˜ the player is punished by the amount ζ · m ˜ regardless of 0 the price level. In all other respects, Model 2 is the same as Model 2. In the presence of inflation the effective penalty increases over time and, as in Model 2, the equilibrium for Model 1 is also an equilibrium for Model 20 . Theorem 3: Assume β(1+ρ) ≥ 1 and let p1 be the price for goods at the first stage in the equilibrium of Theorem 1. If ζ ≥ u0 (0)/p1 , then the equilibrium of Theorem 1 is also an equilibrium for Model 20 . The proof is similar to that for Theorem 2 as given in the Appendix. Note that a type-symmetric equilibrium with active bankruptcy cannot occur for the deterministic economy of Model 20 . If it were advantageous to default, agents would increase their bids to the point at which the advantage would disappear. Active bankruptcy can occur for a model with exogenous uncertainty (cf. Geanakoplos et al. (2000)).
3.1
A comment on Model 2
By linking the penalty ζ to the performance of the bureaucracy we close the model of a self-policing system or the games within the game. The bureaucratic control of government is overlaid on the economy linking the need not only for laws, but also for the ability to enforce them. In this context it becomes meaningful to ask questions about the optimal size of a bureaucracy, but to do so requires an ability to describe the production function that undoubtedly depends on many social and political factors not discussed here.
14
4
Model 3: An Economy with a generic Public Good
When public goods are discussed, the stress is usually not on bureaucracy as providing a self-policing system for the economy – even though this may be one of the most important of all public goods. Our third model is more conventional, it lays stress on a physical public good that may depreciate. As in Models 1 and 2 there is a continuum of producer-consumer agents α ∈ I who hold cash mαn ≥ 0 and goods qnα ≥ 0 at the start of each period n. However, we assume that the government now provides a quantity Gn ≥ 0 of a generic public good in every period n. For simplicity, Model 3 will not include the bureaucrats. Instead, we shall assume that the government spends all its tax revenues on the production and maintenance of the public good. Now for the details: As in Model 1, each agent α makes a cash bid bαn in period n where θ¯pˆn qnα α α . bn ∈ 0, mn + 1+ρ As in Model 1, pˆn is the bank’s estimate of the price pn of the private good, and, in rational expectations equilibrium, pˆn = pn . Although default is possible out of equilibrium, we do not model it explicitly. In period n each agent α sells her goods qnα in a market at price pn and pays taxes in the amount θpn qnα . The government spends its total tax revenues Z Z G α Bn = θpn qn dα = θpn qnα dα = θpn Qn I
I
in the private goods market. Thus the price pn is formed as pn =
Bn + BnG , Qn
where
Z Bn =
bαn dα
I
is the total bid by the agents. Each agent α purchases the quantity bαn /pn of goods and inputs knα ∈ [0, bαn /pn ] for production. Agent α’s cash mαn+1 and α goods qn+1 in the next period are given by (5) and (7) as in Model 1. 15
The government acquires the quantity knG =
BnG pn
of the private good, all of which is used for the production or maintenance of the public good. Suppose Gn ≥ 0 is the quantity of the public good available at the beginning of period n and that η ∈ (0, 1] is the depreciation rate. Then the amount of the public good in the next period is Gn+1 = (1 − η)Gn + F (knG ),
(21)
where F (·) is the government’s production function for the public good. The utility of an agent α in period n is u(xαn , Gn ), a concave increasing function of the agent’s private consumption xαn
bαn − knα = pn
and the public good Gn provided by the government. For fixed values of G, we assume that u(·, G) is concave, increasing, and differentiable. As before, each agent α seeks to maximize her total discounted utility ∞ X
β n−1 u(xαn , Gn ).
n=1
The government is regarded as a controller who moves first in the game. Its actions are specified by its selection of its control variables, the interest rate ρ and the tax rate θ. The tax rate determines the income level of the government and thereby also determines its production of the public good. Furthermore, the goal or the objective function of the government is assumed known to the agents. For example, the government may wish to supply some level of the public good subject to some condition on inflation.
16
4.1
An equilibrium for Model 3
We will construct a type-symmetric equilibrium in which money and prices may inflate, but consumption and production remain constant. Suppose that all agents begin with the same amount of cash mα = M and goods q α = Q = f (k) + y. Assume each agent makes the same bid bα = b = aM and inputs the same amount k α = k for production. Since the government bids its total income, namely θpQ, the price of the private good is given by R α b dα + θpQ aM = + θp p= I Q Q and hence aM p= ¯ . θQ The amount of private good consumed by each agent in the period is xα =
bα aM ¯ − k. − kα = − k = θQ p p
(22)
The government inputs kG =
θpQ = θQ p
for production of the public good and thus produces the quantity F (θQ). Assume that the government holds the quantity Gn of the public good in each period n equal to a constant G. Then, by (21), G = (1 − η)G + F (θQ), and thus
1 (23) G = F (θQ) η is the amount of the public good provided by the government in every period. By (22) and (23) the utility received by each agent in every period is 1 α ¯ u(x , G) = u θQ − k , F (θQ) . (24) η Theorem 4: Suppose that every agent α ∈ I begins with cash mα = M > 0 ¯ Assume also and goods q α = Q = f (k1 ) + y, where f 0 (k1 ) = (1 + ρ)/(β θ). 17
that the government initially provides the quantity G = F (θQ)/η of the public good. Then there is an equilibrium in which all agents bid the proportion a=
(1 + ρ)(1 − β) ρ
(25)
of their cash and input k1 for production in every period. In this equilibrium, the government inputs k G = θQ for production of the public good in every period, thereby holding Gn = G for all n. The proof is in the Appendix. In the equilibrium of Theorem 4, money and prices inflate at the rate τ = β(1 + ρ) as they did in the equilibrium of Theorem 1. To see this, suppose that all the agents begin at mα = M and q α = Q and play the strategy of the theorem. Then at the next stage, each agent has cash ¯ α ˜ =m M ˜ α = (1 + ρ)(mα − amα ) + θpq aM = (1 + ρ)(M − aM ) + θ¯ · ¯ · Q θQ = τ M. The price at the next stage is ˜ aM aM p˜ = ¯ = τ ¯ = τ p. θQ θQ
4.2
A Control Problem for the Government
A benevolent government will try to maximize the welfare of the agents through its choice of the values of its control variables, the interest rate ρ and the tax rate θ. Observe that, in the equilibrium of Theorem 4, the total discounted utility of every agent is, by (22) and (23), ∞ X 1 1 n−1 α ¯ − k1 , F (θQ) , β u(xn , Gn ) = · u θQ 1 − β η n=1 where 0 −1
k1 = (f )
1+ρ ¯ θβ
and 18
Q = f (k1 ) + y.
Thus the utility of an agent can be written as a function ϕ(θ, ρ) of the the government’s control variables θ and ρ. We illustrate the government’s optimization with a simple example. √ Example 3: Suppose f (k) = 2 k, F (k) = k, β = 1/2, y = 0, η = 1, u(x, G) = log(xG). Then 1 1+ρ 2(1 + ρ) f 0 (k1 ) = √ = = ¯ βθ θ¯ k1 so the equilibrium values are k1 =
θ¯2 , 4(1 + ρ)2
Q = f (k1 ) =
and G = F (k G ) = k G = θQ =
θ¯ , 1+ρ
θθ¯ . 1+ρ
The utility to maximize is ϕ(θ, ρ), where 1 ¯ − k1 , θQ) = 1 · u ϕ(θ, ρ) = · u(θQ 1−β 1−β
¯2 ¯ θ θ¯2 θθ − , . 1 + ρ 4(1 + ρ)2 1 + ρ
Since u(x, G) = log(xG), trivial algebra shows that ϕ(θ, ρ) = 3 log θ¯ + log θ + log(3 + 4ρ) − 3 log(1 + ρ) + C, where C does not depend on θ or ρ. So −3 1 ∂ϕ (θ, ρ) = + =0 ∂θ 1−θ θ if θ = 1/4. This is the optimal tax rate. It’s also easy to see that ∂ϕ (θ, ρ) < 0 ∂ρ for all ρ > 0. Hence small values of ρ are better for the agents.2
4
The Example 3 is fairly typical, in that there is an optimal tax rate that is an interior point of (0,1). This is intuitively obvious since a tax rate of 0 2
There are problems when ρ = 0, Theorem 4 does not apply.
19
results in no production of the public good and a rate of 1 means that the agents have no incentive to produce the private good. The example is also typical in that the agents’ utility is decreasing in the interest rate. However, this is misleading because the bank might (and usually does) have multiple goals. In particular, it may wish to maximize the welfare of the agents subject to a constraint on the rate of inflation. In the equilibrium of our deterministic model it is possible to reduce inflation to zero by setting 1 + ρ = 1/β. When there is exogenous uncertainty, this need not be the case (cf. Karatzas et al. (2006a)). Even without uncertainty this is not always true if some part of the population (such as pensioners) is living off its capital. (For a proof of this possibility, see Karatzas et al. (2006).)
4.3
A comment on Model 3
From the point of view of public finance there is a considerable difference between maintaining the upkeep of a public good and building a new one. Roads, dams and bridges provide examples. Each requires maintenance against very real physical deterioration. This model has dealt with this instance. It has not dealt with the basically different problem of raising large funds for the capital investment for a large dam, road or bridge, not because they are not important, but to sort out cases and to present the simpler case. Capital investment usually calls for long term financial instruments and the political-economy aspects of investment timing become important. The problems of long or short term are considered in the last two models. Another problem that merits observation is that there are several different ways to consider both the the government’s role and its goals in the production and maintenance of the public good. Here government production is considered together with the assumption of an altruistic goal for the government.
5
Model 4. Financing with National Debt: one-period Bonds
In Models 1, 2, and 3 above a public good or a bureaucracy was financed via an income tax. In Models 4 and 5 we consider financing the government through the introduction of public debt. To do so it is necessary to introduce the instrument of a government bond. A government debt for a period of a year or more can be considered to be a bond. In Model 4 we consider the 20
simplest possible case, that of a bond that has only a one period duration. Political reality calls for several alternative means of financing public goods that take different political pressures into account, but to make the situation even simpler, we assume that there is no income tax in Model 4. The government sets an interest rate ρ > 0 for its one-period bonds and sells them at face value. We can assume, with little loss of generality, that there is no bank available to accept deposits. If deposits paid a higher (respectively, a lower) rate of interest than bonds, then everyone (respectively, no one) would use the bank rather than buy bonds. We assume for this model that β(1+ρ) > 1. If the reverse inequality held, agents would have no incentive to buy bonds. (Notice that when β(1+ρ) < 1 in Model 3, the number a of (25) is greater than 1. This means that, in the equilibrium of Theorem 4, agents borrow from the bank and make no deposits. The income tax nevertheless allows the government to provide the public good.) As in Model 3 the government provides a quantity Gn ≥ 0 of a generic public good in each period n. As in all the previous models, there is a continuum I = [0, 1] of producer-consumer agents, and each agent α ∈ I holds cash mαn ≥ 0 and goods qnα ≥ 0 at the start of each period n. The goods are sold in a market, and each agent α bids bαn ∈ [0, mαn ] to purchase an amount of goods bαn /pn . Here pn is the price of goods formed in period n as explained below. (In this model agents cannot default, since their bids are limited to their cash holdings.) Any excess cash mαn − bαn is used to purchase bonds which mature at the end of the period. Thus agent α begins the next period with cash. mαn+1 = (1 + ρ)(mαn − bαn ) + pn qnα .
(26)
As in previous models, each agent α selects a quantity of goods knα ∈ [0, bαn /pn ] to input for production, consumes xαn =
bαn − knα , pn
(27)
and begins the next period with goods α qn+1 = f (knα ) + y.
(28)
The government bids in the private goods market all its income obtained 21
from the sale of bonds, namely Z G Bn := (mαn − bαn ) dα = Mn − Bn ,
(29)
R R where Bn = bαn dα is the total bid of the agents and Mn = mαn dα is their total cash holdings. The price of the private good in period n is Bn + BnG Mn pn = = , Qn Qn , where
Z Qn =
(30)
qnα dα
n
is the total amount of private goods sold in the market. The amount of private goods purchased by the government, namely knG = BnG /pn , is used as input to produce the quantity F (knG ) of the public good. If Gn is the quantity of the public good available at the beginning of period n and η ∈ (0, 1] is the rate of depreciation, then Gn+1 = (1 − η)Gn + F (knG ) .
(31)
As in Model 3, each agent α seeks to maximize ∞ X
β n−1 u(xαn , Gn ),
(32)
n=1
where the utility function u(·, ·) is concave, increasing and differentiable.
5.1
An equilibrium for Model 4
To construct a type-symmetric equilibrium, assume that every agent α ∈ I begins with cash mα = M > 0, goods q α = Q = f (k) + y, bids bα = b = aM for goods, and inputs k α = k for production. In the equilibrium constructed below, the proportion a will satisfy 0 < a < 1. After bidding, each agent spends her remaining cash M − aM = a ¯M to purchase bonds. The government thus receives an income of a ¯M , spends all
22
of it in the private goods market, and then inputs all of the private goods to produce the public good. The price of the private good is p=
aM + a ¯M M = Q Q
(33)
and
a ¯M =a ¯Q p is the government’s input for production of the public good. As in Model 3, we assume that the government holds the quantity of the public good Gn equal to a constant G so that kG =
G = (1 − η)G + F (¯ aQ) or, equivalently, 1 aQ). G = F (¯ η Each agent α purchases the quantity b aM = = aQ p p of the private good, consumes x = aQ − k, and derives the amount of utility aQ) in the period. u aQ − k , η1 F (¯ Theorem 5: Suppose every agent α ∈ I begins with cash mα = M > 0 and goods q α = Q = f (k2 ) + y where f 0 (k2 ) = (1 + ρ)/β, and that the government initially provides the quantity G = η1 F (¯ aQ) of the public good. Then there is an equilibrium in which all agents bid the proportion a = (1 − β) +
1 1+ρ
(34)
of their cash and input k2 for production in every period. In equilibrium the government inputs k G = a ¯Q for production of the public good in every period thereby holding Gn = G = η1 F (¯ aQ). The proof is in the Appendix. In the equilibrium of Theorem 5, money and prices inflate at the rate τ = β(1 + ρ). (Recall that the rate of inflation was the same for the previous 23
models in which ρ was the interest rate set by the central bank for deposits and loans.) To see this, suppose that all agents begin with cash M and goods Q, and follow the strategy of Theorem 5. Then, at the next stage, every agent will have cash ˜ = (1 + ρ)(M − aM ) + pQ = (1 + ρ)(M − aM ) + M M = [2 + ρ − (1 + ρ)a]M = τ M, where the last equality is by (34). The price at the next stage will be p˜ =
5.2
˜ aM aM =τ· = τ p. Q Q
Another Control Problem for the Government
The only control variable of the government in Model 4 is the interest rate ρ paid on the one-period bonds. To find an optimal value of ρ, we consider the total discounted utility of an agent α in the equilibrium of Theorem 5: ∞ X n=1
β
n−1
u(xαn , Gn )
1 1 · u aQ − k2 , F (¯ aQ) . = 1−β η
Since k2 = (f 0 )−1 ( 1+ρ ) and Q = f (k2 ), this utility is a function ϕ(ρ) of ρ. β To illustrate the optimization we take another look at Example 3 recast in the context of Model 4. √ Example 4: As in Example 3, let f (k) = 2 k, F (k) = k, β = 1/2, y = 0, η = 1, and u(x, G) = log(xG). Setting 1+ρ 1 f 0 (k2 ) = √ = = 2(1 + ρ), β k2 we see that k2 =
1 4(1 + ρ)2
and thus Q = f (k2 ) =
24
1 . 1+ρ
Also, by (31) and (34), we have a ¯ G = F (k G ) = k G = a ¯Q = 1+ρ 1 1 1 = · − , 1+ρ 2 1+ρ and 1 1 1 1 x = aQ − k2 = · + − 1+ρ 2 1+ρ 4(1 + ρ)2 3 1 1 + · . = 2(1 + ρ) 4 (1 + ρ)2 Since ϕ(ρ) = log(x) + log(G), an elementary calculation shows that ϕ(ρ) achieves its maximum at the value ρ = 1.85. 4 Unlike in Model 3, the agents’ utility is not now a monotonically decreasing function of the interest rate ρ. Indeed, ϕ(ρ) typically has an interior maximum as in the example. While the agents’ consumption of the private good is decreasing in ρ, this is not true of the public good since the government finances its production with income based on ρ.
5.3
A Remark on Government Debt
In Model 4 the government is explicitly financing the public good by selling bonds thereby creating public debt that grows geometrically. Indeed, if we measure government debt by the number of bonds outstanding, then, in the equilibrium of Theorem 5, the debt in period n is a ¯mn = a ¯τ n−1 m1 . Thus debt increases at the rate of inflation, and, if we correct for inflation, debt remains constant. There is debt implicit in the earlier models when there is inflation; that is, when β(1 + ρ) > 1. This is because agents are making deposits in the bank and the deposits earn interest. The amount of interest paid increases geometrically at the rate of inflation τ .
25
5.4
A Comment on Model 4
Short-term bond financing without taxes or other sources of revenue has to be inflationary. Even worse, given that we have limited ourselves to one period borrowing, the refinancing has to increase from period to period. We have not explicitly remodelled refinancing costs but our guess is that an order of magnitude or around 1% is probably not unreasonable. Essentially these considerations call for both taxation and longer term instruments. In essence Models 4 and 5 without taxation provide the essence of the infinite horizon ultimate Ponzi game. In Model 4 current bondholders are always paid off from the ever increasing borrowing from new bondholders. In Model 5 there is no need to refinance as the perpetuities, whose volume grows in order to be able cover the interest payments, need never be repaid. Nevertheless it can be regarded as a benign Ponzi as the combined scheme of issue and inflation have covered the maintenance of the public good.
6
Model 5. Financing with National Debt: perpetuities or consols
Although historically consols were invented after the existence of government debt of various lengths, they are by far the easiest long-term instrument to consider; thus for our final model, we assume that the government finances a public good through its sale of perpetuities or consols. These are bonds sold at face value that pay interest at a rate ρ > 0 in every future period. In a complex economy there is a time structure of interest rates stretching forward over many years and the shape of the yield curve is viewed by those setting policy. The flexibility of the economy is enhanced by this complex multidimensional control system. As a first step towards treating the multiperiod control possibilities, the perpetuity picks up long term financing but does not pick up the flexibility obtained with the full array of issues of all lengths. Model 5 differs from Model 4 in that the consols held by an agent do not mature at the end of each period but are held by the agent indefinitely. In a stochastic model we would expect that some agents would find it advantageous to sell consols to others in a private market. However, in a typesymmetric deterministic equilibrium, either all the agents would wish to sell or they would all wish to buy. Thus, in the equilibrium constructed below 26
there is no active private market for the consols. As in Model 4, we assume that β(1 + ρ) > 1 so that agents will have an incentive to buy bonds. Each agent α ∈ I holds cash mαn ≥ 0, goods qnα ≥ 0, and a quantity cαn ≥ 0 of consols at the beginning of each period n. Also, as in the previous model, the goods are sold in a market and each agent α bids bαn ∈ [0, mαn ] to purchase goods bαn /pn . We assume that each agent α spends her remaining cash mαn − bαn to purchase an equal quantity of consols. Thus α begins the next period with consols cαn+1 = cαn + mαn − bαn .
(35)
Agent α’s cash in the next period comes from the interest on the consols together with the profits from the sale of goods. Thus mαn+1 = ρcαn+1 + pn qnα .
(36)
As in all our models, each agent α chooses an amount of goods knα ∈ [0, bαn /pn ] as input for production, consumes xαn = bαn /pn − knα and begins the next period with goods α qn+1 = f (knα ) + y. The total bid Bn of the agents and the government’s bid BnG are defined exactly as in Model 4, and, consequently, the price pn is given by (30). Likewise, the quantity of the public good Gn provided by the government in period n is given by (31) and each agent α seeks to maximize her total discounted utility given in (32).
6.1
An equilibrium for Model 5
As in Model 4, we suppose that every agent α ∈ I begins play with cash mα = M > 0, goods q α = Q = f (k) + y, bids bα = B = aM , and inputs k α = k for production. For Model 5 we further assume that every agent starts with the same quantity cα = c = γM of consols in her portfolio. Here a and γ are positive constants. The price p of the private good is given by (33). At the next stage of play, every agent will have cash ˜ = ρ(M − aM + c) + pQ = [ρ(1 − a + γ) + 1] · M M and consols c˜ = c + M − aM = [γ + 1 − a] · M. 27
˜ = β(1 + ρ)M , to hold and In equilibrium we expect the Fisher equation, M ˜ . If this is so, then we have also that c˜ = γ M ρ(1 − a + γ) + 1 = β(1 + ρ) Solving for γ and a, we obtain 1 1 γ = · 1− ρ τ
and
and
γ + 1 − a = γβ(1 + ρ).
a=1−
2 1 · τ −1 , ρτ
(37)
where τ = β(1 + ρ) is the inflation rate. Recall that, by assumption, τ > 1. Thus γ > 0 and a < 1. Theorem 6: Let γ and a be given by (37) and assume that a > 0. Suppose that every agent α ∈ I begins with cash mα = M > 0, consols cα = γM , and goods q α = f (k2 ) + y where f 0 (k2 ) = (1 + ρ)/β. Then there is an equilibrium in which all agents bid the proportion a of their cash and input k2 for production in every period. In equilibrium the aQ) of the public good in every government provides the quantity G = η1 F (¯ period. The proof is in the Appendix.
6.2
The public debt
In the equilibrium of Theorem 6, we can measure government debt by the quantity of consols outstanding. If cn is the quantity in period n, then cn = γmn = γτ n−1 m1 . Thus, as in Model 4, the debt increases at the rate of inflation.
6.3
Non-inflationary finance
If conservation of the amount of money in the economy is desired then some device that removes the money injected by the growth of national debt payments is required. In essence the initial increase is ρc. This can be offset by a tax rate θ such that θpq = ρc. The introduction of taxes will, however, result in a change in the optimal strategy of the agents.
28
6.4
The Barro model and Ricardo equivalence
A stimulating article by Barro (1974) raised the question “Are government Bonds Net Wealth?”. The immediate answer from a game theoretic viewpoint is: Of course they are in a dynamically well-defined strategic market game, if the government is assumed to be serving the society. The question is how much net wealth do they represent. As was partially indicated by Barro’s article the increment of wealth depends on the details of the structure assumed. The introduction of government bonds and their acceptance in competitive markets represents an enlargement of the strategy sets available to society. They can be interpreted as a new production facility, much as a new way for making steel is an addition, so the bonds are a new way for facilitating finance. However, it is well known that with incomplete markets it is feasible that an enlargement of the strategy sets can make some individuals worse off 3 , thus one cannot give an unqualified comment without a full specification of the detailed assumptions. Two different but important questions concern the individual and societal implications on wealth of the differences between various combinations of national debt and taxation funding of the same economic program. In order to answer both of these questions, models involving both individual independent agents as well as a representative agent are required. When there is no exogenous uncertainty present, a model with independent but symmetric agents may yield the same results as a representative agent model. When uncertainty is present, the differences are considerable. We consider models with exogenous uncertainty in a subsequent paper. Here we can note that, for a stochastic model with a continuum of independent agents, a generically asymmetric wealth distribution is obtained. Although Barro appears to regard this as a second order effect, this does not appear to be so. The neutrality of the Ricardo effect that holds so well for the representative agent is challenged by the asymmetry of the independent agent equilibrium. 3
One need go no further than a game with a unique strategy for each individual with payoffs (10,10) then add a new strategy for each converting it into a prisoners’ dilemma game.
29
6.5
A comment on Model 5
The existence of stationary infinite-horizon nonstochastic equilibria are worth considering as a necessary first step in understanding economic process models. However they obliterate many of the timing differences that must be dealt with in public finance. Here Models 4 and 5 appear to yield substantially the same equilibria, whereas with uncertainty present, they will differ considerably.
7
Concluding Remarks
7.1
The Fisher equation
In all of our models the classic Fisher equation τ = β(1 + ρ) was seen to hold in equilibrium. (Indeed, this equation was key to our proofs.) We do not expect the equation to hold for stochastic models4 . Consider, for example, representative agent models in which the endowment y is replaced by a random variable Y which is the same for all agents. The rate of inflation τ will then be a random variable and we conjecture that in the study of these models of public finance the Fisher equation will be replaced by a “harmonic Fisher equation” of the form 1 1 = , E τ β(1 + ρ) as it was in Katatzas et al. (2006).
7.2
A comment on equilibrium and equations of motion
For models with no uncertainty, it is often easier to analyze the equilibrium state without paying close attention to the details of the laws of motion required for fully specified dynamic programs. This is not so when exogenous uncertainty and disequilibrium positions need to be considered. 4
We have not even commented on the modeling difficulties in matching ρ with β, where the latter is meant to be symbolic of both private and public discounting, and the former appears to be the outcome of a political process.
30
When a full dynamic model is specified, the distinction between building and maintaining a public good must be made. Here we are not examining the financing of construction of a public good but are concerned with financing by national debt, taxation, or other means for maintenance. Politics, or at least political-economy, not economics is the dominant force in selecting among the array of government control weapons to achieve the goals the political process selects. This can be seen when we consider yet another alternative means of financing, the use of the printing press and inflation. The government could finance its operations by paying for goods and services in newly printed bills. The constraints on this scheme lie with the psychology and political economy of acceptance not the economics and accounting.
7.3
Transactions costs and financing
Although the key stress in the selection of means of financing is often political, there is intertwined with this concern the transactions and bureaucratic costs. They require a level of microeconomic detail not considered here, but often critical in specific applications.
7.4
Public goods, political constraints and economic weapons
The subject of public goods is the subject of political economy. Economic weapons are pitched against political and social reality. The many weapons of finance are matched against the many constraints of the polity and society. Politics is the art of the societally feasible and acceptable. The weaponry of public finance must adjust in accordance with the political environment. In this work we have presented a m´elange of models without discussing directly the political environment. There were two reasons for doing so. The first was to stress the central role of bureaucracy as a key public good in the provision of a self-policing system. The success of trade occurs under the rule of law and its enforcement. The second reason was to show that many weapons can be used to achieve more or less the same outcome. When income taxes are politically unfeasible, other taxes, or short-term or longterm bonds can be used. In practice these choices require an understanding of micro-micro variables such as refinancing costs, collection problems and
31
accounting. But even without the uncertainties that are ever present, the outlines of many of the different cases can be seen. Many of the more realistic and relevant problems appear when uncertainty is added. In particular, even if we can establish that dynamic equilibria exist, 5 the importance of stochastic time lags qualitatively changes many of the conclusions noted above. A simple example is illustrative. In comparing stochastic versions of Models 1 and 2 with Model 3, one has to compare suffering a random shock in a system where the civil servants are looking for their monthly pay check with the situation where the call for funds is to pay for the maintenance and servicing of highways or national parks. With the former there is a pressure to serve the constituency, with the latter maintenance may be delayed until more prosperous times. When one views the physical properties of the many different public goods, the array of financial methods available and the political and social constraints present it appears that a considerable taxonomy of models needs to be constructed, and the models call for analysis. This article takes a step in this direction.
5
We conjecture this to be true, but it has not yet been established in any generality.
32
References [1] Barro, R.J. (1974). Are Government Bonds Net Wealth? Political Economy 82, 1095-1117.
Journal of
[2] Bergstrom, Th., L. Blume and H. Varian (1986). On the private provision of public goods. Journal of Public Economics, Elsevier, vol. 29(1), 25-49. [3] Chari, V.V. and P.J. Kehoe (2006). Modern Macroeconomics in Practice: How Theory is Shaping Policy. Journal of Economic Perspectives 20, 4:3-28. [4] Diamond, P. (2006). Optimal Tax Treatment of Private Contributions for Public Goods with and without Warm Glow Preferences. Journal of Public Economics, Vol. 90, No. 4-5, pp. 897-919. [5] Fair, R. (2010). Macroeconometrics. http://fairmodel.econ.yale. edu/main2.htm. [6] Geanakoplos, J., I. Karatzas, M. Shubik and W.D. Sudderth (2000). A strategic market game with active bankruptcy. Journal of Mathematical Economics 34, 359-356. [7] Huber, J., M. Shubik, W.D. Sudderth and S. Sunder (2011). An experiment on the optimal supply of a public good. Technical Report, Yale University and University of Innsbruck. [8] Karatzas, I., M. Shubik and W.D. Sudderth (1994). Construction of Stationary Markov Equilibria for a Strategic Market Game. Mathematics of Operations Research 19, 975-1006. [9] Karatzas, I., M. Shubik and W.D. Sudderth (2006). Production, interest, and saving in deterministic economies with additive endowments. Economic Theory 29, 525-548. [10] Karatzas, I., M. Shubik, W.D. Sudderth and J. Geanakoplos (2006a). The inflationary bias of real uncertainty and the harmonic Fisher equation. Economic Theory 28, 481-512.
33
[11] Kydland F.E. and E.C. Prescott (1977). Rules rather than Discretion: The Inconsistency of Optimal Plans. Journal of Political Economy 85, 473-91. [12] Ledyard, J. (1995). Public Goods: A Survey of Experimental Research. In Kagel and Roth, ed. Handbook of Experimental Economics, Princeton University Press. [13] Lucas, R. W. (1978). Asset prices in an exchange economy. Econometrica 46, 1429-1445. [14] Lucas, R. E. (1980). Equilibrium in a pure currency economy. Economic Enquiry 18, 203-220. [15] Lucas, R. E. and N.L. Stokey (1987). Money and interest in a cash-inadvance economy. Econometrica 55, 491-513. [16] Lucas, R. E. and N.L. Stokey (1983). Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12, 5593. [17] Musgrave, R. A., and P. Musgrave (1973). Public Finance in Theory and Practice. New York: McGraw-Hill. [18] Salani´e, B. (2003). The Economics of Taxation. Cambridge, MA: MIT Press. [19] Samuelson, P. A. (1954). The pure theory of public expenditure. Review of Economics and Statistics 36(4): 387–389. [20] Schelling,T.C. (1960). Strategy of Conflict. Harvard University Press. [21] Selten, R. (1965). Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit. Zeitschrift fur die gesamte Staatswissenschaft 12, 301-324. [22] Shubik, M. (1984). Game Theory in the Social Sciences: A Gametheoretic approach to Political Economy, Vol. 2. Cambridge: MIT Press. [23] Shubik, M. (1999). Theory of Money and Financial Institutions. Cambridge: MIT Press. 34
[24] Shubik, M and D.E. Smith (2005). Fiat Money and the Natural Scale of Government. Santa Fe Institute Working Paper Series 05-04-010.
8
Appendix
8.1
The proof of Theorem 1
We need to show that the strategies described in Theorem 1 are feasible and optimal for each agent and each bureaucrat, when all other agents and bureaucrats follow these strategies. Consider first a producer-consumer α ∈ I with cash mα and goods q α when the price is p. Then α faces a dynamic programming problem in which the optimal return V (mα , q α , p) satisfies the Bellman equation V (mα , q α , p) = b α α ¯ − k + βV (1 + ρ)(m − b) + θpq , f (k) + y, τ p u sup ¯ α p 0≤b≤mα + θpq 0≤k≤ b 1+ρ
p
Assuming an interior solution the Euler equations are: ! 1 0 b β(1 + ρ) 0 ˜b ˜ u −k = ·u −k p p p˜ p˜
(38)
and u0
b −k p
¯ θβ = f 0 (k) · u0 1+ρ
! ˜b − k˜ , p˜
(39)
where p˜, ˜b, and k˜ are the price, the agent’s bid, and the agent’s input for the next period. Under the hypothesized strategy for α , we have ˜b ¯ m ¯ aθ¯m ˜ aθτ aθm b = = = = , p˜ p˜ τp p p k˜ = k = k1, . Thus
β(1 + ρ) β(1 + ρ) 1 = = . p˜ τp p 35
and
˜b b − k = − k˜ p p˜
so that (38) and (39) are satisfied when k = k1 The appropriate transversality condition is trivial because the quantities consumed and input for production are the same in every period. We must also verify that the given solution is interior. That is, we need 0 < bα < mα + and 0 k1 . The proof of optimality for agent α ∈ I is now complete. Consider now a bureaucrat γ ∈ J. The bureaucrat γ has only one control variable, the bid for goods. Consequently, γ has the simpler Bellman equation: 36
b θpQ γ u + βV (1 + ρ)(m − b) + , τp (42) V (m , p) = max θpQ p δ 0≤b≤m+ δ(1+ρ) γ
and the Euler equation 1 0 u p where
b β(1 + ρ) 0 = u p p˜
˜b p˜
!
˜ aθM aθτ M ˜b b δ = = δ = p˜ p˜ τp p
and as before, β(1 + ρ)/˜ p = 1/p. Thus the Euler equation is satisfied. Again, the transversality condition is satisfied easily. Hence, the given strategy is optimal for γ ∈ J. 4
8.2
The proof of Theorem 2
Assume that all agents α ∈ I and bureaucrats γ ∈ J play their strategies described in the statement of Theorem 1. We must show that the given strategy is optimal for any single player. Consider a producer-consumer agent α ∈ I who begins a period with cash mα and goods q α , and suppose that the price in the period is p. If mα ≥ 0, then the Bellman equation for agent α takes the form α b α α α ¯ , f (k) + y, τ p . − k + βV (m − b)(1 + ρ) + θpq V (m , q , p) = sup u p 0≤bα α
α
α
0≤k≤ bp
However, if mα < 0 so that the agent has defaulted, then the Bellman equation is mα V (mα , q α , p) = ζ · + V (0, q α , p). (43) p The second form of the Bellman equation reflects our rule that the agent is punished in proportion to the default and then allowed to continue the game from a position of zero cash. 37
In the equilibrium of Theorem 1 every agent α ∈ I begins with cash ¯ m = θM and goods q α = f (k1 ) + y. They each bid bα = θaM where a is given by (16), and input k1 for production. We must show that their selections remain optimal in our present model. It suffices to show that it is never desirable for an agent α to choose a bid that exceeds ¯ α θpq mα + 1+ρ because, if the agent chooses a bid in the range ¯ α θpq α 0, m + 1+ρ α
she is, in effect, playing in Model 1. Let α b α ¯ α , f (k) + y, τ p . Ψ(b) = u − k + βV (mα − b)(1 + ρ) + θpq p If b > mα +
¯ α θpq 1+ρ
then m ˜ α < 0, where ¯ α m ˜ α = (1 + ρ)(mα − b) + θpq is the agent’s cash position in the next period as in (19). Hence, by (43), α ζm ˜ b −k +β + V ((0, f (k) + y, τ p) Ψ(b) = u p τp Recall that τ = β(1 + ρ). Hence, 0
Ψ (b) = = ≤ ≤
1 0 b βζ(1 + ρ) u −k − p p β(1 + ρ)p 1 0 b u −k −ζ p p 1 0 [u (0) − ζ] p 0.
Thus the optimal bid of a producer-consumer α in Model 1 remains optimal in Model 2. A similar argument shows that it also optimal for the bureaucrats to play their equilibrium strategy from Theorem 1. 4 38
8.3
The proof of Theorem 4
As explained in the paragraph after the statement of Theorem 4, money and prices inflate at the rate τ = 1 + ρ − ρa = β(1 + ρ) when agents play the given strategy. We must show that the strategy is optimal for a single agent α ∈ I when all others follow it. The Bellman equation for such an agent α is V (mα , q α , p) = b α α ¯ u − k, G + βV (m − b)(1 + ρ) + θpq , f (k) + y, τ p . sup ¯ α p 0≤b≤m+ θpq 0≤k≤ b 1+ρ
p
(44) Except for the dependence of the agent’s utility on the public good G, the Euler equations are the same as in Model 1 (see (38) and (39)) ! β(1 + ρ) 0 ˜b ˜ b − k,G = ·u − k,G , p p˜ p˜ ! ¯ ˜ b θβ b ˜ u0 − k, G = · f 0 (k) · u0 − k, G . p 1+ρ p˜
1 0 u p
Here u0 (·, ·) denotes differentiation with respect to the first coordinate. As in the proof of Theorem 1, the Euler equations are satisfied when the given strategy is followed; and it is not difficult to check that the solution is interior and satisfies a transversality condition. 4
8.4
The proof of Theorem 5
The Bellman equation for a single agent when others follow the proposed strategy is b α ∗ α ∗ − k, G + βV ((m − b)(1 + ρ ) + pq , f (k) + y, τ p) . V (m , q , p) = sup u p 0≤b≤mα α
α
0≤k≤ pb
The Euler equations are 39
! b β(1 + ρ∗ ) 0 ˜b ˜ − k,G = ·u − k,G , p τ ∗p p˜ ! 0 ˜b b βf (k) u0 − k,G = − k˜ , G · u0 p 1 + ρ∗ p˜
1 0 u p
It is straightforward to verify that these equations are satisfied when the agent follows the proposed strategy, and that the actions are interior. As before, the appropriate transversality condition also holds. 4
8.5
The proof of Theorem 6
Consider the situation of an agent α ∈ I with cash mα , goods q α , and consols cα when the price of the private good is p and all other agents follow the strategy described in Theorem 6. We must show that the same strategy is optimal for α when she begins in the same position as the others. The Bellman equation for agent α now takes the form V (mα , q α , cα , p) = b α α α α α − k, G + βV ρ(m − b + c ) + pq , f (k) + y, c + m − b, τ p . u = sup p 0≤b≤mα 0≤k≤ pb
Suppose agent α begins in the same position (m, q, c, p) as the other agents so that q = f (k2 ) + y, c = γm, and p = m/q. Suppose also that α, like the other agents, plays the strategy described in Theorem 6. Then α consumes the quantity am −k x= p of the private good in the first period of the game. Let (m, ˜ q˜, c˜, p˜) be the agent’s position in the next period. Thus the agent consumes am ˜ x˜ = − k˜ p˜ in the second period. Since m ˜ = τ m, k˜ = k = k2 , and p˜ = τ p, we see that x˜ = x. It follows that α consumes the same quantity x in every period. 40
For an interior solution, standard arguments yield the following envelope equations: 1 0 b 1 ∂V (m, q, c, p) = u − k2 , G = u0 (x, G), (45) ∂m p p p ∂V ∂V βp 0 ˜b 1 (m, q, c, p) = βp (m, ˜ q˜, c˜, p˜) = u − k2 , G = u0 (x, G), ∂q ∂m p˜ p˜ 1+ρ (46) ∂V ∂V ∂V (m, q, c, p) = βρ (m, ˜ q˜, c˜, p˜) + β (m, ˜ q˜, c˜, p˜) ∂c ∂m ∂c βρ 0 ∂V = u (x, G) + β (m, ˜ q˜, c˜, p˜). p˜ ∂c
(47)
In these equations and below we write u0 (x, G) for ∂u (x, G). ∂x The value function V for agent α’s dynamic programming problem has the homogeneity property: V (m, ˜ q˜, c˜, p˜) = V (τ m, q, τ c, τ p) = V (m, q, c, p). Hence, 1 ∂V ∂V (m, ˜ q˜, c˜, p˜) = (m, q, c, p). ∂c τ ∂c Now substitute for the left-hand-side in (47) and, after some algebra, we have 1 ∂V (m, q, c, p) = u0 (x, G). ∂c p
(48)
We can now differentiate the expression in brackets on the right side of the Bellman equation to get the Euler equations. Differentiating with respect to b and using (45) and (48), we have 1 0 b ∂V ∂V β(1 + ρ) 0 u − k, G = βρ (m, ˜ q˜, c˜, p˜) + β (m, ˜ q˜, c˜, p˜) = u (x, G). p p ∂m ∂c τp (49) This holds since, for α’s strategy, τ = β(1 + ρ) and x = am/p − k2 = b/p − k. Now differentiate with respect to k and use (46) to get b ∂V βf 0 (k) 0 0 u − k, G = βf 0 (k) (m, ˜ q˜, c˜, p˜) = u (x, G). (50) p ∂q 1+ρ 41
This also holds for α’s strategy since k = k2 = (f 0 )−1 ((1 + ρ)/β). To check that α’s strategy is interior, first notice that 0 < b = am < m since 0 < a < 1. We also require that 0 < k2 < b/p = aQ. The first inequality is clear. For the second, calculate as follows: aQ = a(f (k2 ) + y) ≥ af (k2 ) Z k2 1+ρ f 0 (t) dt ≥ ak2 f 0 (k2 ) = ak2 · =a . β 0 So k2 < aQ if a(1 + ρ) > 1. β Substitute the expression in (37) for a and use the equality β(1 + ρ) = τ to see that this inequality is equivalent to (1 + 3ρ + ρ2 )β 2 − (2 + 3ρ + ρ2 )β + 1 < 0. The expression on the left factors to give (β − 1)((ρ2 + 3ρ + 1)β − 1), which is negative since β < 1 and (ρ2 + 3ρ + 1)β > (ρ + 1)β > 1. Finally, the appropriate transversality condition holds because, as in the other models, consumption and input for production are the same in every period. The proof of Theorem 6 is now complete. 4
42
IOANNIS KARATZAS Department of Mathematics Columbia University, MailCode 4438 New York, NY 10027
[email protected]
and INTECH Investment Management One Palmer Square, Suite 441 Princeton, NJ 08542
[email protected] ——————————————————— MARTIN SHUBIK Cowles Foundation for Research in Economics Yale University, 30 Hillhouse Avenue New Haven, CT 06520
[email protected] ——————————————————— WILLIAM D. SUDDERTH School of Statistics, 313 Ford Hall University of Minnesota Minneapolis, MN 55455
[email protected]
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