Non-computable rational expectations equilibria

Mathematical

Social Sciences

25 (1993) 133-142

133

North-Holland

Non-computable equilibria Jonathan

rational

expectations

Thomas

Center for Economic Research, Tilhurg University, Tilhurg, Netherlands and University of Warwick, Coventry. UK Communicated

by H. Gottinger

Received 15 November

1991

Revised 2 April 1992

Using techniques

drawn from recursive

bria in infinite horizon economies of an overlapping

generations

function

is investigated.

theory,

the computability

of rational

The concept of a recursive economy

expectations

equili-

is defined. An example

model is given which is recursive but possesses only monetary

equilibria

which

are non-computable. Key words: Recursive

function;

recursive economy

1. Introduction When a rational expectations equilibrium exists it is normal to assume that agents are able to find it and solve for it. In this paper it will be shown that this may not be possible. Our notion of what can be solved for will be the standard concept of a computable function in recursive function theory. Roughly speaking, a function is computable if some finite set of instructions - an ‘algorithm’ or ‘effective procedure’ - can be designed which computes the function at each point of its domain. It might be thought that if the description of the economy itself is in some sense computable or effective, then an equilibrium which is computable would be expected. Nevertheless an example will be given of an overlapping-generations model whose description is effective, in a sense to be made precise, but which has only non-computable monetary equilibria. The basic model will be a standard one with no pathological features. A monetary policy rule will be constructed which leads to the result (the rule constructed will, admittedly, be a strange one). In a different context Lewis (1985) and Gottinger (1987) have considered the

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134

J. Thomas

/ Non-computable

equilibria

computability of choice functions (see also Rustem and Velupillai, 1990), and Rabin (1957) has analysed a game in which the winning strategy is not computable. Binmore (1987) and Anderlini (1990) amongst others have analysed what happens when players in games are modelled as Turing machines. Anderlini (1988) has analysed the problem of trying to match a given price sequence when the costs of computation matter. Finally, Spear (1989) considers whether effective procedures exist which enable agents to learn a rational expectations equilibrium.’

2. Computable

functions

and recursive economies

Brief definitions of relevant concepts will be given. For more extensive discussion the reader should refer, for example, to Spear (1989), Minsky (1967, Ch.5) or Cutland (1980). We characterize computability in terms of Turing machines, where a Turing machine is one particular formalization of an algorithm. Church’s thesis claims that Turing computability corresponds to the intuitive notion of an effective procedure (see Anderlini, 1990). Using Church’s thesis we will restrict arguments to informal descriptions of algorithms as opposed to detailed descriptions of Turing machines. We shall consider functions defined on the natural numbers, IN, with range in N. Suppose we have a Turing machine that takes inputs from the natural numbers, and given an input, it runs its program until it eventually halts, producing a number as its output, or it never halts. Then we can think of this machine computing the function with domain equal to those inputs for which the computation halts, and which takes values equal to the outputs from the computation. Alternatively, if for a given functionfthere exists a Turing machine which computesf, then f is said to be computable, and if no such machine exists, f is non-computable. As mentioned above, the domain of a function need not be the whole of N, but when it is, we say the function is total. Computability over natural numbers can straightforwardly be extended to computability over Q, where CQis the set of rational numbers, by use of a coding device a : IN -+ Q, where a is an explicit and effective bijection (which exists). (The extension to computability over Qk, where k E N, is similarly straightforward.) Turning to the description of an infinite horizon economy, suppose without loss of generality that there is a single variable, for example the price level, q(t) E N, to be determined each period t, t = 0,1,2,. . . . It is normally the case that the equilibrium conditions can be expressed in the form of a sequence of conditions, one set of

t The approach taken differs from that taken in the current paper in a number of important respects. In particular Spear assumes that a computable equilibrium exists, whereas it is this very existence question which is analysed here. Also, he considers only stationary equilibria, though with a potentially uncountable equivalent the current

state space. While the concept of a rational expectations equilibrium is then mathematically to that given here, it is not possible to interpret the learning mechanisms which he studies in context

where attention

is not restricted

to stationary

equilibria.

J. Thomas

/ Non-computable

equilibria

135

for each date, involving current variables q(t) and variables at other let F be the set of all (not necessarily total) functions dates {q(t))l”_, . Formally from N to n\l. Then we could formalize, at the most abstract level, the economy as a mapping, r : F-t F.2 Thus l-allows the determination of prices at I, T(q)(t), from knowledge of an input function q. We want to express the idea of an economy which can be effectively described, though the meaning of the word effective here is not so clear since we are considering the manipulation of infinite objects (the price sequences). The usual idea of effectiveness in this context would require that prices in any period can be effectively calculated in a finite time using only a finite part of the input sequence. Formally we say that a function y is finite if its domain is finite, and that y is a finite part of q if they coincide in value at all points where y is defined, i.e. y(t) = q(t) for all t E domain (y). It is easy to show that any finite function can be coded by a natural number in a one-to-one, effective manner, and we denote this code by c(y). So given any finite function y the code can effectively be found, and given any value for c, y can be effectively derived. The notation = means ‘equal when defined’. Then conditions

Definition (Cutland, 1980, p. 183). r : F+ F is a recursive operator if there is a computable function @(x, t) such that for all q E F and t E N, y E N, T(q)(t) = y if and only if there exists a finite part of q, say y, such that @(c(y), t)ly. We shall say that an economy is recursive if there exists a recursive operator r such that a total q( . ) is a rational expectations equilibrium if and only if r(q) = q.3 Notice that the requirement is only that some representation of the equilibrium conditions exists in the form of a recursive operator r. It need not hold for every representation. The extension to prices belonging to Q or CQk is achieved by interpreting y, @Jand O(t) as natural numbers which code for points in Q or Qk. For an extensive discussion of recursive operators the reader is referred to Rogers (1967).4 The following characterization of recursive operators will prove useful below: 2 This is analogous via excess demand mapping

to a static n-good functions,

as providing

the model correspond

general

a mapping

from

all we need to know about

by Spear (1989) if one restricts

he does not explicitly of the same forecast

function

value for q(t), given other prices,

of recursiveness

would

usually of this

in that equilibria

of

here we shall look for fixed points of I‘.

to be exactly equivalent

its violation

to define,

into itself. We can think

of the economy

g in Assumption

could lead to two different

is against the spirit of his analysis. 4 It may be thought that the definition a unique

Likewise

Theorem

his mapping

make this restriction,

model where it is possible

the description

to fixed points of this mapping.

3 This turns out by the Myhill-Sheperdson studied

equilibrium

the set of price vectors

to the class of economies

2.3 to extensional

functions.

imply that two different

temporary is unduly

equilibrium restrictive

while there may be multiple

price functions, as it appears

(temporary)

While

representations which

to require

equilibria.

This is

not the case, however. If at time t there is a condition of the form fr(q(0), q( l), , q(t), . ) = 0 this can always be expressed as T(q)(t) =f,(q(O), q(l), , q(t), ) + q(t). Recursiveness can therefore be interpreted as saying that given knowledge of a proposed t if the necessary equilibrium condition holds.

price sequence

q, it is possible

to check at each time

136

J. Thomas / Non-computable equilibria

Theorem (Cutland, 1980, p. 185). I- is a recursive operator if and only if (a) r is continuous: for any q E F and all t E tN, y E N, T(q)(t) = y if and only if there is a finite part of q, say y, with r(y)(t) = y; (b) the function @(x,t) given by @(c(y), t) = r(y)(t) for y E F (undefined for other

x) is computable. We give two examples of recursive economies, where q(t) E Q. First, at each time t the price level in equilibrium is related to the previous period’s price level and the following period’s price level in a simple fashion. Example

1.

T(q)(t) = aq(t - 1) + pq(t + l), r(4)(0)

t 2 1,

where a, p E Q,

= P.

Given any input sequence, q, T(q)(t) is found simply from q(t - 1) and q(t + 1). Thisisclearlycontinuous, and @(c(v),t)=(~y(t-l)+/?y(t+ I), trl, with@(c(y),O)= ,u, is a computable function. In the next example the price level at t depends upon all previous values. Example

2. 1-l

T(q)(t) = c s’q(t-r),

tzl,

where ~EQ,

T=O r(4)(0)

= P.

Here T(q)(t) is found from the first t values of q, and is therefore continuous while @(c(y), t)= c:zb d’y(tT), 62 1, with @(c(y),O)=p, is a computable function.

3. A non-computable

equilibrium

Even the restriction to a recursive economy does not guarantee that the equilibrium will be computable if T(q)(t) requires knowledge of future variables, q(r), for r>t.’ (If, on the other hand, the determination of prices at t involves only backward looking variables, then a trivial recursive procedure will produce the equilibrium, which is therefore computable.) To show that all equilibria may be non-computable, the concept of a simple set will be used. Two preliminary ideas are needed. First, let A be a subset of N. Then A is recursively enumerable (henceforth r.e.) if the function f, given by

5 If we drop computable

the requirement

solution

that the equilibrium

can be proved

price

using the recursion

function

theorem.

must be total, See McAfee

then existence

(1984).

of a

J. Thomas

“f(x) =

/ Non-compulable

equilibria

137

:ndefined, ;;=:A”’ -c 9

is computable. Secondly, it is possible to effectively number all Turing machines (such an enumeration, not uniquely defined, is a ‘Godel numbering’). Let @, be the function computed by a Turing machine with number x. A set A is simple if (a) A is recursively enumerable, (b) A (the complement of A in TV) is infinite, and (c) A contains no infinite recursively enumerable subset. The following simple set will be used below. First define a (partial) computable function a: a(x) is computed by computing in turn G,(O), G,(l), Q,(2), . . . (do not proceed to the computation of&(r) unless and until @Jr- 1) has been successfully computed); stop as soon as (and only if) an r is found such that o,(r) > 2x and put a(x) = &(r). Clearly a(x) need not be defined. Then the range of a(. ) is a simple set (this is the construction used in Rogers, 1967, Theorem II, Section 8.1). Call this set A. Moreover, because it is r.e. it is also the range of some total computable function g (Cutland, 1980, Theorem 7.2.7). The main idea used below is very general and could be applied to show the possibility of non-computable equilibria in other areas of economics such as noncooperative game theory.6 The model used is no more than an example. The critical idea is to construct an economy in which an equilibrium can only exist if and only if the price sequence generates an increasing sequence of numbers (there will be an effective rule which associates with each price sequence another sequence of numbers) which belongs entirely to A. By definition of A being a simple set, the set of numbers contained in this latter sequence, which has an infinite number of distinct elements, is not r.e. and consequently cannot be generated by a computable function. This implies in turn that any equilibrium price sequence is not computable, despite the fact that equilibria exist (because A is infinite). To be able to construct such an economy, there must first be sufficient indeterminacy in equilibrium so that an arbitrary increasing sequence of natural numbers can be generated by the price sequence. In our example it will be the case that prices can take on one of two types of values an infinite number of times and this will generate a binary sequence which will be ‘converted’ into a sequence of natural numbers. The second major difficulty is to ensure that the process of checking that the numbers generated belong to A can be done effectively. Consider the following simple stationary overlapping-generations model. Such a model has been extensively studied in recent years (see, for example, Cass, Okuno and Zilcha, 1979; Grandmont, 1985; Azariadis and Guesnerie, 1986). Individuals live for two periods, the same number being born each period. An individual born at time t is young at time t when he consumes c,, of the single perishable good, and old at time t + 1, consuming c2[+, , and has utility function u(cII, czI+ t). Each individual has endowment et when young and e2 when old. The exception is the old 6 I wish to thank

Joseph

Greenberg

for alerting

me to this possibility.

J. Thomas / Non-computable

138

equilibria

at t = 0, who live for one period and possess one unit of money. Holding money is the only way of saving. Let pI be the price of the good in terms of money at I and define zI = l/p,. Government monetary policy involves at each date t the determination of the money stock MI (per capita). It is assumed that changes are brought about by making transfers to the current young (only increases in Mt will be considered). Let (1 + 6,) be the proportional change in M,, where 0, E [0, oo), so Mt = (1 + B,)M,_ 1 and each young individual at t receives a transfer e,M,_ 1. We assume M,, = 1. The money supply rule is a function of previous prices: 0, = The rate of return on savings is R =zt+ ,/zt. Let s(R,co)= UZorZl, **., .z_i). arg maxoSSSe, +,u(e, + o -.s, e2 + Rs) be the real savings of the representative household (assuming a unique maximum exists), where o = e,M,- lzt is the real money transfer received at t. A rational expectations monetary equilibrium must satisfy s

(Zt+l,

e,iw-

zt

= (I+

,zt >

e,)bf_

lzt,

tro.

This says that the real savings of the young at time t must equal the real money supply and implies of course that the goods market clears in the sense that the excess demand from the old at t, M,_,z,, plus the excess demand from the young, B,A4_ izt-S, sum to zero. It will be assumed that s is defined and computable on Additionally we shall require that s has three special properties. First, note Q?+. that {(s(R, 0), Rs(R, 0)) : R ?O} is the usual ‘reflected offer curve’. We require this to have a certain shape: first it must initially be below the 45” line, be backward bending and cross the 45” line with slope smaller in absolute value than unity. Consider, for any value of ZL 0, the following set: {Rs(R, 0) : there exists R L 0 such that s(R, 0) =z}. This has at most two values which are just the intercepts of the offer curve with a vertical line originating at z, and define E(z) and h(z) to be the larger and smaller of the two (equal if there is a single value), respectively. Then in the usually considered case where the money stock is constant (M,= 1 for all t) the h(z,)} for all t 2 0. Consider ‘forward’ solution’ to the model satisfies zI+ 1E (ii( the (unique) stationary monetary equilibrium in this case: real money z* is given by the solution to z* = fi(z*). The second special property is that there exists an interval D around z* such that for ZE Q, an n’ exists such that h”@(z)) E 52 for n 2n’ (if necessary choose D small enough so that E(z) has slope smaller than one in absolute value for z E Q). A reflected offer curve with these two properties is drawn in fig. 1. The essential idea here is that if we are following some forward solution originating in 52, then using the top part of the offer curve to generate the next value of z1 keeps the sequence in Q because the slope is flat enough; but even if the lower part of the offer curve is used once, reverting to using the top part again for long enough will bring the sequence back into Q: see the dotted line in the diagram. These

7 This simply says given a value for z,, we look for a value for zr+ 1 such that (1) holds, value of zl+ I, we look for a value for .Q+Z such that (1) holds, and so on.

and with this

J. Thomas / Non-computable equilibria

Fig.

139

1.

two properties will be satisfied, for example, if the offer curve has a slope less than unity in absolute value for a sufficiently wide range of values either side of z* and there exists a cycle of period 3 in the case where the money stock is held constant. (Grandmont, 1986, has shown that a sufficient condition for this latter characteristic is that the utility function be additively separable into two constant relative riskaversion functions and the coefficient of risk aversion in the second period be sufficiently high.) The third property is that there exists a ~5 such that s(R, o) I o for R 10, WL ti.This merely says that for sufficiently large real transfers the individual will consume at least his endowment in his first period, no matter what the return on savings is. Notice that if such a transfer were to take place, (1) could not hold. None of these conditions requires a pathological utility function. The construction works briefly as follows. Take a candidate monetary equilibrium sequence {z,}. Suppose initially z. E Q, so z1 can be chosen either equal to h(z,-J or fi(zo). Let this choice generate a binary digit, 0 or 1, respectively. If /i(zo) was chosen, z1 EQ and the process can be repeated, generating the next digit. Otherwise either z,+, = h(z,) for tz 1 until z,+ I ~52 again, when the next binary digit is generated as before, or this is not the case, in which case set the money stock too large to be consistent with equilibrium. So if the money stock is to stay constant, an infinite sequence of binary digits will be generated. This is converted into a sequence of natural numbers by a given way of ‘cutting up’ the binary sequence, and the money rule will effectively specify that if ever one of these numbers belongs to A the money supply is again set too large to be compatible with equilibrium. So only a non-computable sequence can be a monetary equilibrium.

140

J. Thomas / Non-computable

equilibria

More precisely, the money supply rule will be specified as follows. A finite sequence up to T, (zo,zl, . . . . zT), is defined as an information sequence if it satisfies z,,eQ, ~,+~~{fi(z~),h(z~)}, and z,$Q implies z,+~=~(z,), for tsT-1. If, at any T, {z,}: fails this condition, Br+ 1= h/_h(zT). Any infinite information sequence produces a sequence of binary digits {pun},“=, : suppose at t, zI E Q for the nth time; then at t + 1, ,D,~, is produced:

1, if zt+ I k-1

=

-t 0,

=

&z,),

if z(+i = _h(z,).

Notice that the maximum number of periods elapsing between ,L+, and ,uu, is 1 +min{r:~‘(_h(max{z~52}))~52}, which is finite by definition of 52, so {p,} is an infinite sequence. This sequence of binary numbers is next cut up to convert it into a sequence of natural numbers {a(u)),“,,, where o(u) is the number represented by a block of binary numbers of length N satisfying 2r> 2~22~~~ with N= 1 when ..-,LI~+~), where ,u~ was the last digit used in computing u=O (so do)=(Pq+lPq+2 a(o- 1)). If at time T, (u+ 1) such numbers, {o(O), . . . , o(u)}, have been produced from {zc, . . . . zr }, define the set G(u) = {g(O), g(l), . . . , g(u)}. Then 8, = 0, ifa(i)$G(u),a(i)~2ianda(i+l)>a(i),i=O,...,u, or if o(O) has not yet been produced,

0, I3T+1=

ci,

otherwise, h(ZT)

’

for T>O. This fully defines the money supply rule. Notice that it is entirely backward looking and is computed by an effective procedure using only previous values of zt. Define r(z)(t)

=s

(%eJ4_1zI Zt

(l+B,)M,_,,

tro,

>I

r is a recursive where 8, and M1_ 1 are functions of zo, zl, . . . , z,_~ as described. operator: because the calculation of r(z)(t) only involves {z,}~” it is continuous; the function @(c(y), t)=T(y)(t) for yeF is computable since given any finite function y with domain {0, 1, . . . , t + l} we have a finite algorithm given by the money supply rule to calculate 8, and M1_ 1, and by assumption s( . , . ) is computable; consequently by Church’s thesis $J is computable (@ is undefined for y with different domain). Now for a monetary equilibrium to exist we must have zt+ 1E {_h(z,), fi(z,)} and 8,= 0 for all t 10. Thus (z,} must be of the type which generates an information sequence. However, it cannot be computable. To see this, suppose first that a(.) is computable. Since a(o) > a(u - l), a( a) enumerates an infinite r.e. set, which can-

J. Thomas / Non-computable

equilibria

141

not be a subset of A; then there exists u^such that ~(6) EA. Since g( . ) enumerates A, there exists some u’ such that g(o’) = a(6), hence o(6) E G(u) for u I u’. So m will be non-zero (at the latest) from the period after a(max(0, o’}) is produced. Now if {z,} is a computable sequence, we have an effective procedure for producing a(. ), which is therefore computable, implying a monetary equilibrium does not exist. On the other hand, there exist sequences which work. By construction of A a total a( .) can be found satisfying o(u)< a(u + l), a(o)l2v and a(o) EA for all ~20. Notice that a(o) can be expressed in binary using no more than 2N digits, where N is defined as before, so there exists an information sequence which will produce a(o), and any information sequence with 8,=0 for all t is a monetary equilibrium. This completes the argument. Even if, as has been shown, no computable function exists which is an exact equilibrium, it may be thought that this is unimportant in the sense that some computable price sequence can be found which is ‘close’ enough to an equilibrium so as not to matter. This is not true, however, at least in this example. Whatever computable sequence is chosen, there will come a time when 8, is set positive and at this point any short-sighted monetary equilibrium will break down dramatically; presumably before this happens money will have zero value (the barter equilibrium being the only equilibrium from this point on). The usual backward induction arguments then imply that money will have zero value at all dates. More precisely, it can be shown that with standard assumptions on the utility function, for E small enough, there is no strictly positive computable {z,} sequence and market-clearing trades such that each generation’s utility function is within E of its maximum utility at the given prices. While the non-existence of a computable monetary equilibrium has been shown, there will of course exist a non-monetary, in this case autarkic, equilibrium which is trivially computable (z, = 0 for all t). It would be straightforward but tedious to extend the model to rule out the existence of a non-monetary equilibrium: see Cass, Okuno and Zilcha (1979), who give an example with a monetary equilibrium but no non-monetary equilibrium. In this case there would exist no computable equilibrium.

Acknowledgement I wish to thank participants at seminars at Warwick and the Center for Economic Research (CentER) for helpful comments. The financial support and hospitality of CentER and the financial support of the Netherlands Organisation for Scientific Research (NWO) is greatly appreciated.

J. Thomas / Non-computable equilibria

142

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