Budget Today or Tomorrow Here we study the properties of the budget line in the context of consumption over time.

Budget Line In the context of consumption today or consumption tomorrow, the budget line is a bit different than in the typical consumer behavior model. One difference is we have consumption today (C1) on the horizontal axis and consumption tomorrow (C2) on the vertical axis. Another difference is the presence of an “endowment” point. It is assumed the consumer has an initial set amount of possible consumption today and a set amount of consumption next period and these come from income each period, called M1 and M2. The budget line must go through the endowment point, but borrowing or lending (saving) can move the consumer away from the endowment.

Budget Line C tomorrow C2 paid back + interest Pay back + interest Endowment point (M1, M2) C today C1 Borrow today Lend today

Budget Line On the previous screen the endowment point shows what the person can have in each period. If the person borrows - takes more today than the endowment - then next period 1 + r must be paid back for every 1 taken today. If the person lends - gives some of today’s consumption up - then next period 1 + r is received for every 1 given up today. The slope of the budget is –(1 + r), or 1+ r in absolute value.

Change in the interest rate The original line has a lower interest rate because when borrowing occurs less is given back next period, or if lending occurs less is paid back. So, the higher the interest rate the steeper the curve through the endowment point. C tom. New original C today

Example The endowment point here means the income is 50000 today and 60000 tomorrow (Next year). Say the interest rate is 20% C tomorrow Vertical intercept. paid back + interest Pay back + interest Endowment point (50000, 60000) C today Borrow today Lend today Horizontal intercept

Example continued The vertical intercept means you consume nothing today and everything next period. The 50000 this period will earn interest and thus the total will be 50000(1.2) + 60000=120000. The horizontal intercept means you spent all you current income and as much as you can borrow and use next years income to pay it off. This would be 50000 +(60000/1.2) = 100000. This horizontal intercept is the present value of lifetime income.

Example continued When the interest rate is 20% and the endowment point is (50000, 60000) the budget line can be thought of in math form in the following ways: 1) In terms of the future C2 + 1.2C1 = 120000, 2) In terms of the present C1 + (C2/1.2) = 100000, or 3) As we do in the graph C2 = 120000 – 1.2C1. The slope is -1.2 and means if you spend $1 today you give up the ability to consume $1.20 next year.

Equation for budget line When we have endowments M1 and M2 and interest rate r the equation for C2 and C1 is C2 = M2 + M1(1 + r) – (1 + r)C1. Thus, 1) If C1 = 0, C2 = M2 + M1(1 + r). This is the vertical intercept. 2) If C2 = 0, 0 = M2 + M1(1 + r) – (1 + r)C1, or C1 = M1 + M2[1/(1 + r)]. This is the horizontal intercept and is called the present value of lifetime income. Note the slope of the budget line is –(1+r).

Indifference Curves Here we study indifference curves in the context of consumption over time.

Indifference Curves Indifference curves in this context are basically the same as we saw in the past. The curves slope downward, do not cross, fill the graph (although we do not always draw many in a graph), and are convex (meaning they get flatter as you move down the curve.)

Present or Future Oriented C tom. C today

Present or Future Oriented On the previous screen we have a curve for two separate people. Each one gives up a unit of C today. The flat curve person does not need much C tomorrow back in return for the C today given up. This type of person is tomorrow oriented or patient. The steep curve person needs more C tomorrow (relative to flat curve) in return for the C today given up. This type of person is today oriented or impatient. The more a person has to get tomorrow to give up a unit today means the person is more present oriented.

Marginal rate of time preference (MRTP) The absolute value of the slope of an indifference curve at a point is called the MRTP. The slope is change in C2 divided by the change in C1. If the MRTP > 1 the consumer has a positive time preference meaning when giving up 1 unit of C1 more than 1 unit of C2 must be given back to have the same utility. If the MRTP < 1 the consumer has a negative time preference meaning when giving up 1 unit of C1 less than 1 unit of C2 must be given back to have the same utility. If the MRTP =1 1 the consumer has a neutral time preference meaning when giving up 1 unit of C1 1 unit of C2 must be given back to have the same utility.

Equilibrium Given an interest rate, we see here the point consumers end up at in order to maximize their utility.

A borrower Notice at the endowment the consumer’s indifference curve goes through the budget steeper then the budget- they are willing to pay back more than they have to, so they borrow today and become happier than at the endowment. C tom. Endowment point C today

A lender Notice at the endowment the consumer’s indifference curve goes through the budget flatter than the budget - They get more in the future than they require to have the same utility so they lend today and are happier doing so. C tom. Endowment point C today

Note Both the lender and the borrower at the point of equilibrium have the MRTP = 1+the interest rate and this is greater than 1. This means that both borrowers and lenders have positive time preferences in equilibrium. Again, this means when giving up 1 unit of C1 more than 1 unit of C2 must be given back to have the same utility. Also note that with a given interest rate some people are lenders and some borrowers based on their preferences. Later on we show how folks might change from being a lender to a borrower, and vice versa, depending on changes in the interest rate.

Changes in Equilibrium Here we study how the consumer position changes given changes in the interest rate.

Change in the interest rate C tom. l n Endowment point m C today

Change in the interest rate We have seen in the past that as the interest rate falls the budget line becomes flatter. At the highest interest rate in the example on the previous screen, we see the individual go to point l (and is actually a lender.) This point has a certain amount of C today involved (as well as a certain of C tom.) As the interest rate falls the consumer moves to point m and then point n. So the amount of current consumption rises as the interest rate falls. The point here is that the demand for current consumption is a function of the interest rate. In fact, we say as the interest rate falls the quantity demanded for current consumption rises.

Change in the interest rate This is the demand for current consumption curve and is derived from the graph two slides before this one. r l m n C today

Permanent income hypothesis Let’s consider an example where your income will 1000 in each of two years and the interest rate will be 25%. C2 = M2 + M1(1+r) – (1+r)C1. a. Suppose that you save all of your money to spend next year. How much will you be able to spend next year? This is the same as asking on the budget what is C2 when C1 = 0? C2 would be 1000 + 1000(1.25) = 2250. How much will you be able to spend today is like what is C1 if C2 = 0. C1 would be 1000 + (1000/1.25) = 1800.

b. Suppose you borrow $800 and spend $1800 today b. Suppose you borrow $800 and spend $1800 today. How much will you be able to spend next year? If C1 = 1800, C2 = 1000 + 1000(1.25) – 1800(1.25) = 0. c. The graph is on the next slide with C1 on the horizontal and C2 on the vertical axis. Note the vertical intercept is (0, 2250), the horizontal intercept is (1800, 0) and the endowment point is (1000, 1000) The slope = (2250-0)/(0-1800) = -1.25, so the slope shows that the price of spending $1 today means you can not spend $1.25 next year. Note if C1= M1, then C2 = M2, and vice versa. This means the person can have their endowment point and neither borrow or lend.

c2 (0, 2250) (1000, 1000) (1800, 0) c1

Say you find $400 in your desk drawer Say you find $400 in your desk drawer. Your endowment today becomes 1400. How does the budget shift? Note the new intercepts and endowment point. c2 (0, 2750) (0, 2250) (1400, 1000) (1000, 1000) (2200, 0) (1800, 0) c1

Say you will get $500 more in pay next year but this year you only have 1000. Your endowment next year becomes 1500. How does the budget shift? Note the new intercepts and endowment point. The budget shifts just like in the previous example. c2 (0, 2750) (0, 2250) (1000, 1500) (1400, 1000) (1000, 1000) (2200, 0) (1800, 0) c1

Now say the person has endowment 1000 and 1000 initially Now say the person has endowment 1000 and 1000 initially. They consume at point A. If C1 and C2 are normal goods, then if the endowment in period 1 rises to 400, or the endowment in period 2 rises to 500, the individual will end up here. c2 (0, 2750) (0, 2250) A (1400, 1000) (1000, 1000) (2200, 0) (1800, 0) c1

In this example we see if income in period 1 goes up 40% consumption in period 1 is not likely to go up 40%. Some of the increase is spread out into the next year. Similarly, if income next period goes up 50% (Say you expect to graduate and make more money) your consumption in period 2 is not likely to go up 50%. Permanent income is the present value of our lifetime income and we saw this has the horizontal intercept. Given our preferences, permanent income is what determines our consumption pattern. Another way to say this is that our consumption pattern over time is influenced not only by the income in he period in which we consume, but by the income in every period.