Optimal Deficit Ratio

Abstract

In sections 1 and 2, we started from the premise that private agents optimize within an infinite horizon. In the current section, instead, we postulate that it is the government who does the optimization. In full analogy to section 1, the short-run equilibrium can be encapsulated in a system of seven equations:

$$y={{k}^{\alpha }}$$

(1)

$$r=\alpha y/k$$

(2)

$$\overset{\centerdot }{\mathop{k}}\,=y-c-nk$$

(3)

$$\overset{\centerdot }{\mathop{c}}\,=(r-n-\rho )c$$

(4)

$$c=(1-s)(1+b)y$$

(5)

$$\overset{\centerdot }{\mathop{d}}\,=by-nd$$

(6)

$$by+t(y+rd)=gy+rd$$

(7)

Here α,ρ,c, d, g, k, n and s are exogenous, whereas b, ċ, $\overset{\centerdot }{\mathop{d}}\,$, $\overset{\centerdot }{\mathop{k}}\,$, k, r, s, t and y are endogenous. As compared to section 1, the sole difference is that here the saving ratio is given, while the government optimizes the deficit ratio dynamically.