Question

Suppose there are only two time periods, today (period 1) and tomorrow (period 2), and only one consumption good, let’s call it food. Assume that food is a perfectly divisible good. Let c1 and c2 denote the amount of food consumed today and tomorrow, respectively. Note that here we use subscripts to denote time periods. But you should think of food today and food tomorrow as two different commodities. The price of food today is equal to p1 = P. As the rate of inflation is π > 0, the price of food tomorrow is p2 = (1 + π)P. Therefore, if Bill buys c2 units tomorrow, it will cost him p2 c2 = (1 + π)Pc2.

Bill has income m1 today and m2 tomorrow. Furthermore, as food is perishable, Bill cannot buy it today and save for tomorrow, but income can be saved or borrowed. If Bill saves some of his income in period 1, he earns interest on his savings at a rate of r > 0 per period. If Bill decides to borrow, then he has to pay the same rate r > 0.

1. (a) Draw Bill’s budget set in a (c1, c2)-plane, i.e., with c1 in the horizontal axis and c2 in the vertical axis, and mark the bundle that Bill consumes if he neither borrows nor saves.

2. (b) Indicate carefully the intercepts of the budget line with both axes.

3. (c) Determine the slope of the budget line.

Answer 1

Ans:-

Given, c = c1+c2

We are also given, bill neither borrows nor saves, and therefore, bill’s income in each period goes to food.

m1 = P*c1 or c1 = m1/P

And m2 = (1+)*P*c2 or c2 = m2/(1+)*P

When c2=0, c=c1

And when c1=0, c=c2

Therefore, points (m1/P, 0) and (0, m2/(1+)*P) lie on bill’s budget line

The equation of this budget line is

(y-y1)/(x-x1) = (y2-y1)/(x2-x1)

Therefore, we have the bill’s budget line as

(y-0)/(x-(m1/P)) = ((m2/(1+)*P)-0)/(0-(m1/P))

Solving, we get,

y = -(m2/(1+)m1)x + m2/(1+)P

The slope of the line is

-m2/(1+)m1