chapter  10
13 Pages

How risk factors affect growth in Mexico: a free- market liberalism approach: Francisco Venegas- Martínez

ByFRANCISCO VENEGAS - MARTÍNEZ

Let us consider now a Brownian motion, dVt, that is, dVt ~ N(0, dt) E[dVt] = 0 and Var[dVt] = dt. We assume that the consumer perceives that the expected inflation rate, dpt/pt, and consequently the expected rate of depreciation, dEt/Et, follows a geometric Brownian motion with Poisson jumps in accordance with

(10.3)

where a is the mean expected rate of depreciation conditional on no jumps, mP is the instantaneous volatility of the expected price level, and * is the mean expected size of an exchange-rate jump. Process Vt is supposed to be independent of Qt. In what follows, a, mP, h, and * are all supposed to be positive constants. The agent holds real cash balances, mt = Mt/pt, where Mt is the nominal stock of money. The stochastic rate of return of holding real cash balances, dRm, is given by the percentage change in the price of money in terms of goods, dmt/mt. By applying Itô’s lemma for diffusion-jump processes to the inverse of the price level, with (10.3) as the underlying process, it can be shown that

(10.4)

The agent also holds capital, kt, that pays a risk-free real interest rate r, which is constant for all terms, satisfying dkt = rktdt, where k0 is given. The representative agent takes r as given. Let us consider now a Brownian motion dWt, that is,

(10.5)

We assume that the representative consumer perceives that his/her wealth is taxed at an uncertain rate, nt, in accordance with the following stochastic equation:

(10.6)

where n0 > 0 and

(10.7)

with _ κ(–1, 1). Here,

_ τ is the mean expected growth rate of the taxes on wealth,

mP is the volatility of the tax rate on wealth, and _ κ is the correlation between

changes in inflation and changes in wealth taxes. Notice that an increase in the rate of depreciation will produce a higher depreciation in real cash balances. This, in turn, will reduce real assets, which could lead to the fiscal authority to modify tax rates. Processes Qt, Vt, and Wt are supposed to be pairwise independent. Consider a cash-in-advance constraint of the form:

(10.8)

where ct is consumption, and ] > 0 is the average time that money must be held to finance consumption. Condition (10.9) is critical in linking exchange-rate dynamics with consumption. Finally, observe that

(10.9)

In the sequel, we will assume that the error o(]) is negligible.