Corporate Income Tax (3): Dynamic Analysis

I would have to say that one of the main purposes of this blog is to let you have an interest in modern economics. Today I try to accomplish it.

Last time I talked about the static analysis of the corporate income tax by using the simple Keynesian cross.

This time I would like to explain the the dynamic analysis of the corporate income tax by using the simple neoclassical growth model(or the Ramsey model).

The model is simple. There's only one consumer with a perfect foresight in a perfect competitive market. To make this model simple, it is assumed to be an infinite horizon problem . Moreover, no externality, no uncertainty and no heterogeneity.

The consumer just aims to maximize his or her lifetime instantaneous utility that is assumed to be a strong increasing and strong concave function subject to the resource constraint.

The objective is,

max U(C(0))+βU(C(1))+β^2U(C(2))+....+β^tU(C(t))
+β^(t+1)U(C(t+1))+......,

where C(t) is his or her consumption level and β is a discount factor with 0<β<1.

The resource constraint is,

s.t. K(t+1)-K(t)=(1-y)(F+rK(t))-(1+c)C(t), K(0)=given

where K(t) is the capital stock, F, the income level, y, the (corporate) income tax rate, r, the real interest rate and c, the consumption tax rate.


To solve this problem, there are mainly three ways to do it, the Lagrangian method, the maximum principle (MP) and the dynamic programming (DP). Which way to choose relies on your preference, but these three algorithms definitely lead us to get the same solution.

The solution is called the Euler equation. By looking over it, we can see what policy affects what variable and the optimal path of the variable and some important implications, which are somewhat different from what we got from the static analysis of the corporate income tax.


(1)The Lagrangian function is:

L(C(t), K(t), λ(t))

= U(C(0))+βU(C(1))+β^2U(C(2))+....+β^(t-1)U(C(t-1)) +....

-λ(0)(K(1)-K(0)-(1-y)(F+rK(0))+(1+c)C(0))

-λ(1)(K(2)-K(1)-(1-y)(F+rK(1))+(1+c)C(1))+....

-λ(t-1)(K(t)-K(t-1)-(1-y)(F+rK(t-1))+(1+c)C(t-1))+....

This model is a discrete time version. So if we want to get the Hamiltonian dynamics for this problem, we should take the limitation to change the discrete to the continuous time version of this model.


(2)The current valued Hamiltonian function is:

H(C(t), K(t), λ(t))=U(C(t))+λ(t)(K(t)+(1-y)(F+rK(t))-(1+c)C(t))

(3)The Bellman equation is:

V(K(t))=max{ U(C(t))+βV(K(t+1))

s.t. K(t+1)-K(t)=(1-y)(F+rK(t))-(1+c)C(t), K(0)=given }

We solve for the function, and C(t)=P(K(t)) is called the "policy function".

Regarding (3), it is assumed that there's the value function that satisfies the above Bellman equation for this problem(That is, we assume that the Blackwell's theorem holds in this problem).

The Euler equation is,

U'(C(t))/U'(C(t+1)) = β{(1-y)r+1}

In addition to the transversality condition (TVC), we can get the optimal path of the consumption growth. The Euler equation is the ratio of the marginal utilities of the consumption between the period t and period t+1.

Looking at this equation,

(The marginal utility of t-period consumption) = U'(C(t)),

(The present value of the marginal utility of t+1-period consumption)

= U'(C(t+1))β{(1-y)r+1}

in our optimal solution we can see the equality between the marginal utility of t-period consumption and the present value of that of t+1-period consumption. This is the optimality condition that we've wanted to get for this problem.

We can see that only the rate of the corporate income tax affects the consumption growth(if we assume that the utility function is a logarithmic form, U(C(t))=lnC(t) ). The tax rate influences the efficiency of the economy! That is, the higher the income tax rate, the lower the consumption growth in the case of the logarithmic utility.

From the perspective of the dynamic analysis, the corporate income tax definitely affects the economy(more accurately, economic efficiency), whereas it doesn't in our static analysis. That's what I want to put emphasis on here.


06/17/08, Postscript,

Some people say that cutting the tax rate of consumption would increase the consumption or its growth rate. However, according to our analysis, cutting the consumption tax would neither increase nor decrease the consumption growth.

It seems peculiar, but it's a well-known result for such a problem. Why not? It's easy. The consumer in this problem can expect it accurately if the government increases the consumption tax. That's why he or she will change the optimal path of his or her consumption beforehand and thus there's no difference.