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STOCHASTIC REFINEMENT OF THE Pecorino’s Optimal inflation-rate model

 

Dr. Costas Kyritsis                              Prof. Petros Kiochos

Software Laboratory                                     Department of Statistics

National Technical                                        and Insurance Science

University of Athens                                      University of Pireas

 

 

                                               ABSTRACT

In this short paper with title “Stochastic refinement of the  Pecorino’s Optimal Inflation-Rate model ”, we extend  the model  of Pecorino to include stochastic effects. The basic result is that under natural conditions the optimality of the model’s inflation rate, remains, even if we refine it to a stochastic model. The stochastic  refinement has in addition a  three-fold effect: a) It makes the model  mathematically more sophisticated and  advanced b) It gives to it statistical and econometric foundation; it is therefore related directly to empirical measurements. c) Many different cases can be considered to follow the model (up to a probability of an error ) that in the previous deterministic formulation should be excluded. (JEL:E3,E4,C3)

 

 

 

Introduction

In a recent paper (see [Pecorino,P. ,1997] ) it is described a model of the optimal inflation rate, when capital is taxed . Such models investigate the validity of the Friedman’s rule (see [Friedman,M.,1969]), which requires a rate of price deflation equal to the real rate of interest . In his model Pecorino’s optimal rate of inflation exceeds that implied by Milton Friedman’s money growth rate. Nevertheless his results indicate, as he points out himself , that it may not be far above it ,and clearly this rate must be below the revenue maximizing rate of inflation. In this paper, by refining Pecorino’s model, in a minimum way, to a stochastic model, we prove that his solution remains optimal even when including stochastic disturbances. In particular,  the optimal stochastic solution has a skeleton equation which coincides  with the corresponding in Pecorino’s deterministic  optimal solution.

The refinement has  a  three-fold effect: a) It makes the model  mathematically more sophisticated and  advanced b) It gives to it statistical and econometric foundation, thus, it is linked directly to empirical measurements. c) Many different cases can be considered to follow the model (up to a probability of error ), that in the previous deterministic formulation should be ruled out .

            The paper is kept  as short as possible and as easy-reading as possible .The  main point is not the particular details but the whole procedure that gives statistical and empirical econometric foundation to optimal control models.

1.The Pecorino’s  deterministic model of optimal inflation rate when capital is taxed.

We give , at first, a concise review of the Pecorino’s  model as it is described in [Pecorino,P.,1997].

The next symbols and magnitudes are used :                                       

Y=output of consumption/physical capital good.

A=technology shift parameter in the production function

K=capital stock

ñ=  consumer’s rate of time preference.

h(.)=function of the real per unit transaction costs.

S=total transaction costs.

P=price level.

M=money stock.

C=consumption.

I=investment.

V=velocity of money.

sI =real marginal transaction cost on capital purchases.

g=growth rate of output

R=nominal interest rate.

1/ó=intertemporal elasticity of substitution.

ì=money growth rate.

ô=income tax rate.

z=government transfers as percentage of gross output.

W=welfare.

Ø=satiation level of velocity.

VF=value of velocity such that…

n=shift parameter in the transaction cost function.

á=exponent from the transaction cost function.

 

There is a single consumption good and a single investment good, each produced by an identical technology,  as described by


                       

                                                                                                                                 (1)

Where A>ñ and ñ is the consumer’s rate of time preference.

The real per unit transaction cost and the total transaction costs satisfy the equation:

                                                                                                                                 (2)


with

h’£0 ,h’’³0,h’(0)=-¥


and it is defined a velocity Vt                                                                                                                                                                                                                       (3)

with a value of the velocity       VF such that:

h(1/VF)=h’(1/VF)=0.

The derivatives of the transaction cost function are:


                                                                                                                                 (4)

The real marginal transaction cost on capital sI   is defined as

                                                                                                                                
   (5)

The market clearing condition is:

                                                                                                                                 (6)


Consumer preferences ,budget constraint and capital accumulation constraint are formulated in the next equations respectively

                                                                                                                                 (7a)


                                                                                                                                 (7b)


                                                                                                                                 (7c)

The equation (7c) reflects an assumption of a zero rate of depreciation on physical capital. The time subscripts are dropped sometimes for simplicity.

The consumer’s intention is to maximize (7a) subject to (7b) and (7c) .

The Hamiltonian associated with this problem is given by

                                                                                                                                 (8)


 The Pontryagin’s maximum principle solves this control system and the solutions are

                                                                                                                     

 


(9a)

           


                                                                                                                                 (9b)


                                                                                                                                 (9c)

Some additional conditions are

 

ó>1 ,

which ensures that the interest rate exceeds the growth rate. The government budget constraint links the rate of money growth to the rate of income taxation

                                                                                                                                 (10a)


It is assumed that all government revenues are transferred lump sum back to consumers and that these transfers are constant as a percentage of gross output PAK (this percentage is denoted by z) .Thus


                                                                                                                                 (10b)

 

By solving (7) we get an expression for the welfare

 

                                                                                                                                 (11)


 

It s worth stating the Pontryagin’s maximum principle, in order to compare it with the stochastic maximum principle; (see e.g. [Chiang A.C. 1992 ]pp168-193 ) or [Kamien M.-Schwartz N.L. 1991] p 219  :

Deterministic maximum principle

«Let the optimal control problem :

Find piecewise continuous control u(t) and an associated continuous and piecewise differentiable state vector x(t) ,defined on a fixed time interval [t,T] (Finite Horizon) that will maximize the :

 

subject to he differential equation of evolution

x’(t)=g(t,x(t),u(t))  with initial condition x(0)=x0

 

In order that x*(t) ,u*(t) be optimal for the above optimal control problem ,it is necessary that there exist a constant p0   and continuos function p(t) ,where for all   t£s£T we have (p0  ,p(s))¹(0,0) and such that for every t£s£T ,

H(s,x*(s),u ,p(s)) £H(s,x*(s),u*(s),p(s))

where the Hamiltonian function H is defined by

H(s,x,u,p(s))= p0f(s,x,u) +pg(s,x,u)

Except at points of discontinuity of u*(s)

p’(s)=-

Furthermore p0=1 or p0=0

and, finally ,the following transversality condition is satisfied :

p(T)=0.

 

2.The stochastic maximum principle and the stochastic refinement of the model.

            We assume an Ito diffusion (a normal process ) that the skeleton equation  is  the equation of motion of the deterministic model and the variance is constant through out .We may write for it the Chapman-Kolmogorov equations and the stochastic differential equation for it. The Stochastic current value Hamiltonian  has an additional term depending on the variance and the derivative of the adjoint variable to the capital .

            The stochastic optimal solution is based on the Stochastic maximum principle (see [Mallaris A.et al 1982] proposition 10.1 pp112)   that goes as follows:

Suppose that x*(t) and u*(t) solve in [t,T] the system:

Maximize E (   ) ( and set J(x(s),s,T)=maxu E (   )  )

(By E(x) we denote the mean value of the random variable x)

subject to the  stochastic differential equation of evolution of an Ito diffusion (which is a normal process ,and z is a Brownian motion or «white noise»)

dx=g(x(s),u(s))dt+s(x(s),u(s))dz

 with initial condition x(0)=x0

Then there exists an adjoin variable p(s) such that for all s in [t,T]

u* maximizes H(x,u,p,)  for every random path x(s)  ,where

H(x,u,p,)=u(x,u)+pg(x,u)+1/2s2

the adjoin random  function p(t) satisfies the stochastic differential equation of an Ito diffusion

dp=-dt+s (x,u*)dz

and the transversality condition holds

p(x(T),T)=

p(T)x(T)=0

We notice immediately the similarity of the deterministic and the stochastic maximum principle . In particular if the  diffusion coefficient ó is constant in x  and time ,and also the  is constant ,then the maximization of the   value of the stochastic Hamiltonian, coincides with the maximization of the deterministic Hamiltonian.

This gives that the solution of the stochastic optimal control system is directly reduced to the solution of the deterministic optimal control system.

A main  difference of the stochastic solution from the deterministic is that it is  a random function (stochastic process ) .As the process is normal (at each time the random variable follows the normal distribution ) it is determined by the skeleton equation ( the mean value  curve) and the variance curve .

Thus we may deduce that:

Theorem

The optimality of the Pecorino’s model solution is preserved under the above conditions in a model including stochastic disturbances .

           

3. Statistical and Econometric empirical measurements and Optimal Control models.

Although deterministic optimal control systems are easier to solve, in Economics are much too theoretical, when we try to test them empirically.

This difficulty is resolved when we refine them to stochastic optimal control systems, through stochastic differential equations. All the techniques of Econometrics become in this way available to us. Statistical tests and estimates ,or regression analysis, permits decisions up-to-probability. Stochastic differential equations admit continuous time estimation,  simulation and forecasting (see [Kloeden et al 1997].

The advantage is that many distinct real cases may follow one only numerically defined stochastic model .

               

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