}
Non-steady-state equilibrium solution of
a class of dynamic models
Jenny X. Li
Departments of Mathematics and Department of Economics, The Pennsylvania State University, University Park, PA, 16802, USA
Accepted12 June 2000
Abstract
We study a class of monetary and growth models by using both analytic and numerical tools. In particular, we consider a model that was studied by Rotemberg, (Journal of
Political Economy, 92 (1984) 40}58) andprovide a solution to the open problem
concerning the existence of the non-steady-state equilibrium of the model. We investigate the stable manifold solution to the underlying dynamical system and then use it to
generate the equilibrium path. We use a"xedpoint iteration to numerically evaluate the
stable manifoldsolution andeventually discover the exact solution. This investigation gives another illustration of the potential power of the general approach developed by the
author that combines mathematical analysis andnumerical simulations. 2001
Else-vier Science B.V. All rights reserved.
1. Introduction
In this paper, we shall present a mathematical methodandsome numerical algorithm for computing equilibria of a class of dynamic models. In particular, we are interestedin monetary andgrowth models. The equilibria of many of such models are often described by nonlinear di!erence equations of secondor higher order. One interesting phenomenon is that only one boundary condition (plus another condition at in"nity) can be imposedfor these equations.
E-mail address:[email protected] (J.X. Li).
Examples of such models include Baumol (1952), Grossman (1982), Grossman andWeiss (1983), Grossman (1985), Rotemberg (1984), Stockman (1980), Polemarchakis (1990) andJose-Victor andRios-Rull (1996). The usual treat-ments in the literature for this type of problems are: consider the steady-state equilibrium only, or linearize the nonlinear di!erence equation aroundthe steady-state and"nda local solution, or use the shooting methodto"ndthe numerical equilibrium.
For a dynamic model, however, a steady-state equilibrium does not provide complete information about the considered economy, especially when we are interestedin the e!ects of such non-steady-state events as open-market opera-tions. As for the local linearization method, we know that sometimes the higher-order terms matter so the local result cannot be extended. The shooting algorithm, on the other hand, has at least two major problems. First, there is little theoretical basis for the proposedterminal values andthe model can easily get pollutedby parameters that are arti"cially introduced. Second, the computa-tional e!ort associatedwith the arti"cial two-point boundary-value approach grows super-linearly with the distance of the terminating value from the initial value if one is trying to"nd the dynamics for an interval of initial conditions. In this paper, we shall propose a numerical methodthat is basedon our new mathematical understanding of the model. There are two important features of the proposed method. First, it provides the global closed-form solution of non-steady-state equilibrium sequences for many existing models. Second, the numerical approach has strong theoretical backgroundandvery e$cient. We shall describe our method using a particular model proposed by Rotemberg (1984).
The rest of the paper is organizedas follows. Section 2 brie#y presents the model and algorithm which we will apply for our method. Section 3 presents the perfect foresight equilibrium of the economy, where the equilibrium se-quence was governedby a thirddi!erence equation with only one initial value. Section 4 proposes the mathematical methodandthe numerical algorithm and establishes the existence of the dynamic equilibrium sequence. Section 5 gives some concluding remarks.
2. A special employed model
As mentioned earlier, we shall demonstrate our new approach by studying a particular model proposed by Rotemberg (1984). For clarity, we shall now give a brief description of this model. For details, the interested readers are referred to the original paper.
There are three actors in this model: "rms, consumers andgovernment. Competitive"rms in periodtproduce only one good that can be both consumed andinvested. The total outputQ
t andon K
R\ the amount of goodthat was producedbut not consumedat previous period. Such a dependence is described by constant-returns-to-scale production functionfas follows:
Q
R"¸(f(KR\/¸)).
We assume that the labor is suppliedinelastically¸"¸M, f'0 andf(0. To maximize the pro"t, the workers are assumedto be paidtheir marginal product. Therefore, the total return, denominated in period t goods, from forgoing the consumption of an additional good att!1 is given by 1#r
R\,
Consumers pick the path of consumption optimally. Because of the transac-tions cost, consumers visit their"nancial intermediary only occasionally. So, we assume that there are two types of consumers,nof them in each type andtotal 2nof them. Consumer type &a' engages in "nancial transactions in the even periods,t,t#2,t#4,2, type&b'engages in"nancial transactions in the odd periods, t#1,t#3,t#5,2, andconsumer i withdraws an amount MGO of money to support the consumption at periodand#1. Consumerifaces the optimal problem:
Ois the nominal price of the consumption andinvestment goodat.
The consumers have access to two assets, money andclaims on capital. But money is the only asset with which goods can be bought.
The lifetime budget constraint of the consumer attis given in terms of the consumers monetary withdrawals:
whereKGRare the claims on capital of consumeriatt,Bis the real cost of visiting the"nancial intermediary, and>GRis the non-investment income of the consumer
To"ndthe optimal path of consumption, one usually considers the auxiliary
Using the just establishedrelation above, we then have
lnCGO#lnCGO>"(1#) ln
MGSubstituting (4) into (2), we obtain
max subject to the lifetime budget constraint (3). This maximization yields
CGR>
The above relation states the rate of return on capital.
The government in this model does not have expenditures. It only levies taxes, issues money andholds capital. The evolution of the capital heldby the government is holding of capital att. An increase in money supply meansM
R>'MR. The equilibrium for the economy is a path of P
R,rR such that consumers maximize utility and"rms maximize pro"ts using these price sequenceP
R and satis"es the goods market and money market clearing conditions:
By (6) and(7) we then obtain
consumers that visit the"nancial intermediary atandK
O#KO"KO.
Using (8) and(5), andthe transversality condition lim
2(2KG2)/(MG2/P2)"0. We obtain the nonlinear third-order di!erence equation that governs the equi-librium level of capital sequence:
¸Mf
KO> quence of rates of return by (1), the total consumptions by (5), the individual consumption by (8) andthe equilibrium price sequence by (7).First, we consider the existence of the equilibrium sequence that converges to a steady state with positive consumption. In this case,M
R/MR> converges to a constant, so the steady state values of capitalKM then satis"es the following relation:
It follows that the unique steady state with positive consumption satis"es:
f
KM¸M
" 1. (9)
Now, we shall consider some speci"c production function, namely the Cobb}Douglas productionf(K
R)"K?R where 0((1. Without loss of gener-ality, we assume that the labor supply constant¸M"1. For this type of produc-tion funcproduc-tion the equilibrium level of capital sequence satis"es:
K?
In virtue of (9), for the Cobb}Douglas production function, the steady-state equilibrium is given by
Let us now begin to consider the non-steady-state equilibrium. We assume that no money stock increase will occur after time period t
51, namely
M
R>"MR fort5t. In this case, the"nite di!erence equation is reduced to
K?
R>!KR>"
1#(M
R>)/(MR>) 1#M
R/(MR>)
K?\
R>K?R>\(K?R!KR>),
04t4t
(12)
and
K?
R>!KR>"()K?R>\KR>?\(K?R!KR>), t5t. (13) The very basic question for this reduced model is, for a feasible initialK
, if there exists a pathK
Rthat satis"es (12) and(13) andconverges toKM , furthermore, if such a path is unique. These important questions were "rst addressed in the original work of Rotemberg (1984), but up to now, the answers to these questions still remain open.
Here we shall provide an a$rmative answer to the question on the existence. The answer to the uniqueness question is much more involvedandwe shall not get into the details here.
3. On the existence of non-steady-state equilibrium path
The main idea is that we shall look for equilibrium path in a special form, namely
K
R>"g(KR), t5t (14)
for some functiong. Then for such a special path, the"nite-di!erence equation (10) is satis"ed, ifgsatis"es the following functional equation:
g(g(x))?!g(g(g(x)))"g(x)?\g(g(x))?\[x?!g(x)] (15)
together with the following two conditions:
C1: g(KM )"KM ; C2: lim
R g(KR)"KM for anyKifKR is given by recursion:KR>"g(KR) for
t5t
;
We note that the trajectory given by the aforementionedfunctiongcorresponds to a stable manifoldof the dynamical system. For convenience, we shall call this function to be the equilibrium function.
equivalent form:
F(g(x))"g(x),
where
F(g(x))"x?!()\(g(g(x))?!g(g(g(x))))g(g(x))\?g(x)\?. (16)
The existence ofgof (15) is then equivalent to the existence of the"xedpoint forF. Among many mathematical techniques for establishing the existence of "xedpoints, the contraction mapping theorem oftentimes proves to be e!ective andfurthermore this technique also gives rise to a simple numerical algorithm to approximately compute the "xedpoint. We have successfully appliedthis technique before, see, for example, Li (1998).
The idea is that we can show that the nonlinear operatorFis contractive with respect to certain measurement, then the "xed-pointed theorem claims that a unique "xedpoint g exist for F andfurthermore this "xedpoint can be computedas a limit of the sequence that are recursively de"ned
gI>"F(gI), k50. (17)
The nonlinear functionalFgiven by (16) does indeed appear to be contractive in a neighborhoodofx"KM (notice thatF(KM )"KM ). We have carriedout some extensive numerical experiments andfoundthat the sequence given (17) did indeed converge to a simple-looking function.
The iteration is basedon a linear"nite element discretization. Let us now brie#y describe this procedure. The"rst step is, for a given integerm, to divide the interval [0,1] intom!1 with the following nodal points:
x
G"(i!1)h, i"1, 2,2,m withh" 1
m!1.
We then consider continuous piecewise (with respect to the above subintervals) linear function to approximate the functiong. Let us denote the approximation byg
F. Then the"xedpoint iteration can be realizedby the following relation:
gI>
F (xG)"F(gIF(xG)), k"1, 2,2 .
A natural initial guess of this iteration is the constant function g F"KM . Our numerical experiments have shown that this procedure converges rapidly. From the practical point of view, it is perhaps su$cient to use this numerically computedfunction, but we carriedout our investigation one step further. Judging from the shape of the computed function (see Fig. 1), we used the date-"tting technique to try to guess the exact formulation of this function. We foundan exact"tting ("rst for"
) by using the following function:
Fig. 1. The equilibrium functiong(x).
where ". As a matter of fact, it is rather easy to verify directly that the function given by (18) does indeed satisfy (15) and both conditions C1 and C2. With a more careful theoretical manipulation, furthermore, it is actually pos-sible toderivethis close-form solution analytically. We shall not get into the details of these speci"c theoretical techniques here andinsteadwe shall stay focusedon the numerical methods that can be appliedto this type of problems andcan also be easily extendedto more general problems.
4. Evaluation of non-steady-state equilibrium path
In this section we present some numerical results by using di!erent monetary injections.
To be speci"c, we present our results for some special values of and , namely"3/4 and".99. In this case,
KM "0.3039.
4.1. One injection
Now we consider the"rst case. We assume that in period0, the economy is at the steady state, namelyK
"KM , andassume that
M
"(1#2%)M, MR>"MR,t51. In view of (12) and(13), we havet
"2 andEq. (12) is reducedto an equation forK
With the explicit solutionggiven by (18), we can solve forK
from (19) by the
given above, we then use (14) to evaluateKRfort52. The resulting equilibrium path is illustratedin Fig. 2.
4.2. Two injections
Now we consider the second case. Again we assume thatK"KM , but for periods 1 and 2, we assume that
M"(1#1%)M, M"(1#1%)M, M
R>"MR,t51. In view of (12) and(13), we have t
Fig. 2. Open-market monetary injection 2% in period1.
A direct calculation shows that
K
"K , K"K , (23) where
"
(1! )#
, " ! #
.
With K
and K given above, we then use (14) to evaluate KR for t53. The resulting equilibrium path is illustratedin Fig. 3.
We observe that when the money is injectedto the market two times, then the equilibrium capital path has two periodoscillations before it approaches the steady state.
The e!ect of the open-market operation we obtain here is in agreement with the result in Rotemberg (1984), where a shooting methodwas usedto solve the underlying"nite di!erence equations.
Fig. 3. Open-market monetary injections 1% in both periods 1 and 2. namely to solve Eqs. (12) and(13) for anyt51 andany distributions ofM
Rfor 14t4t
.
5. Concluding remarks
One major mathematical problem involvedin the treatment of forward -looking models is the lack of determination and the absence of dynamic recursivity: forward-looking models typically yield "nite-di!erence systems whose dynamic order is greater than the dimension of the given boundary values.
Acknowledgements
I wouldlike to express my sincere thanks to the referee for several helpful comments andsuggestions.
References
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Grossman, S., Weiss, L., 1983. A transactions-basedmodel of the monetary transmission mecha-nism. The American Economy Review 73 (5), 880}971.
Grossman, S., 1982. A transactions basedmodel of the monetary transmission mechanism: Part II, Working paper No. 974, National Bureau of Economic Research, Cambridge.
Grossman, S., 1985. Monetary dynamics with proportional transaction costs and"xedpayment periods. Working paper No. 1663, National Bureau of Economic Research, Cambridge. Li, J.X., 1998. Numerical Analysis of a Nonlinear Operator Equation Arising from a Monetary
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Rotemberg, J., 1984. A monetary equilibrium model with transactions costs. Journal of Political Economy 92, 40}58.
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