Inter-temporal budget set: an application

For now in our analysis we have focused on choices that concern only one period. But actually a great extent of choices regarding consumptions have to face with inter-temporal decisions. For example all the decisions on saving or investing .

As normal when more than one period is adopted there is an inter-temporal budget set, the simplest we can imagine is one with the possibility for the consumer to save money from one period to the other (if we hypothesize an economy with only two periods) without any transaction cost or any other frictions (for example it’s possible to assume that if she doesn’t consume a portion of a good it will go wasted). In this specific case she will be able to consume a maximum equal to her income in period 1 and a maximum of the sum of the incomes of the two periods in period 2 if she has saved all the income from period one. The budget set will be  a line connecting this two extremes points and with slope equal to one.

Now let’s move to a more complicated discussion using  a simplified budget set from Poverty traps: Ghatak and Jiang (1999) [For a brief discussion http://econ.lse.ac.uk/staff/mghatak/jde4.pdf]. Suppose there are two possibilities of subsistence: one that use only the wage and the other that use a portion of capital (I), that can be subtracted from initial wealth (a), and a unit of work but produces an income (Q). There is the possibility to save a portion (s, the other is used for consumption)and the wealth passed to next period grows at a  rate (r).  The budget set should be:

a(t+1)=s[r*a(t) +w]     if   a(t)<I and she can’t use the portion of capital

a(t+1)=s[r*(a(t)-I)+Q-w]  if a(t)>I

It’s possible to show, using appropriate hypothesis on utility function that we can have either a unique equilibrium or two different equilibrium, depending on the proportion of people who can use the portion of capital to invest. And as a consequence we can have two different state of economy, one in which there is only subsistence work and the other one where people can become entrepreneurs and surpass the poverty line. In fact a subject must have her wealth above a specified threshold to obtain a loan, if we simplify to a situation without possibility to access to credit markets, then the condition is . This means that we can have the possibility that the number of people able to become entrepreneur is higher than the number of people constrained or the opposite situation. Solving the model with the important condition of setting s*r<1 (not far from reality as the coefficients are respectively an interest rate and a saving coefficient) we can find stable points:

(SEE FIG.1 IN PAPER’S TEXT)

As shows the figure the first case is obtained when the number of constrained people exceed the number of entrepreneurs. The economy has two equilibria, one for “poor” and one for wealthy people. In the second case when the number of people allowed to become entrepreneur is higher we have only one equilibrium point where the wage collapse to one point and it is bigger than the wage in case of “poor people economy” (first type). That shows one important findind: without the credit markets (as we have supposed it’s not present in our hypotesis) is not possible for a first economy type to develope in a second economy type, where the wages are more sustainable for everyone.

ANGELO SAPONARA 1612