_{1}

This paper provides a reformulation of Phillips’s multiplier-accelerator model with stabilization policy in terms of the Laplace transform. Applying the Laplace transform, the differential equations of the economy are transformed into the algebraic ones on a complex variable. The transfer functions of economic variables are defined by these algebraic equations. With this representation, we show the effects of Phillips-type policy on equilibrium level and derive the necessary and sufficient condition for asymptotic stability.

A. W. Phillips analyzed the stabilization policy for a multiplier-accelerator model [^{1}. The Laplace transform has been widely used in physics and engineering, especially in classical linear control theory. The Laplace transform is a linear operator, which transforms the linear ordinary differential equation into an algebraic equation. Based on this approach, we show the effects of Phillips-type policy on equilibrium level and derive its asymptotic stability condition.

We briefly explain the Laplace transform defined as a Riemann integral below^{2}. Let s be a complex variable.

where

If

where

We consider the Laplace transform of derivatives. Suppose that

We obtain

Then, it follows that

with initial condition

Let us consider the Laplace transform of integrals. Let

We obtain the Laplace transform of constants as follows:

We define a transfer function of economic variables. Let economic variables

Suppose that

with initial conditions

The Phillips’s multiplier-accelerator model is described by

where aggregate economic variables

The desired production level is taken as a reference,

The exogenous constant deviation in autonomous demand

where A is a bounded constant. Clearly the initial values are 0, i.e.,

If such an exogenous change in autonomous demand occurs, the equilibrium level of aggregate production will shift to another level, involving cyclical fluctuations. Thus, Phillips proposed a policy function

where

The policy lag until demand is affected is supposed as

where

Taking the Laplace transform of (10), (13), (14), we obtain^{3}

Thus, the transfer function of

We assumed that ^{4}. Here, all the initial values are 0.

We should qualitatively verify the capability of Phillips-type policy to achieve the asymptotic stability of original equilibrium level. First, we see the effect of exogenous change in

Theorem 1. An exogenous change in

Proof. Put

Thus, it follows from (5), (7), (19) that

W

Next, we analyze the effects of both the proportional and derivative policies

Theorem 2. For

Proof. Put

Thus, it follows from (5), (7), (21) that

The terms

Finally, we confirm the asymptotic stability by using the integral policy

Theorem 3. For

Proof. It follows from (5), (7), (17) that

W

Notice that since the final value theorem can be used when ^{5}.

Theorem 4. The necessary and sufficient condition for asymptotic stability of the multiplier-accelerator economy with policy (16) is as follows:

and all the leading principal minors of a matrix

are positive.

Proof. The Routh-Hurwitz theorem^{6} gives the necessary and sufficient condition for all the roots of a nth- degree polynomial

with real coefficients to have negative real parts. The condition is as follows: the coefficients

are positive. We can apply the Routh-Hurwitz theorem to the denominator polynomial of (17). From definition, we obtain

We have established a novel analytic framework on Phillips’s stabilization problem by using the Laplace transform method. On the basis of this formulation, the effects of Phillips-type policy on equilibrium level have been analyzed rigorously and qualitatively. Furthermore, we have derived the stability condition of the model by using the Hurwitz theorem. The present study has shown that the Laplace transform approach is powerful to analyze the stabilization problem. This method will give a fresh insight into the problem on stabilization policy design.

SatoruKageyama, (2015) On the Application of the Laplace Transform in the Study of Phillips-Type Stabilization Policy. Theoretical Economics Letters,05,691-696. doi: 10.4236/tel.2015.56080