# Lecture 1 - NOTES ON CONSUMPTION AND SAVING I Two-period...

• Notes
• 13

This preview shows pages 1–3. Sign up to view the full content.

NOTES ON CONSUMPTION AND SAVING I. Two-period model. We assume that are only two periods—the current period and the future period. Consumers must decide how to allocate their current and future income across current and future consumption. Utility Function: U = C 1 a C 2 1-a . C 1 = real consumption in period 1 (current period) C 2 = real consumption in period 2 (future consumption). r = real rate of interest Y 1 = real income in period one (current period) Y 2 = real income in period two (future period) S 1 = real private saving = Y 1 – C 1 . If S 1 > 0 this represents lending and if S 1 < 0 means borrowing. The “a” in the utility function is a taste parameter. To find the utility-maximizing combination of C 1 and C 2 we would: 1. Find the MRS (the slope of the IC). MRS= (MU C1 /MU C2 ) = ( U/ C 1 )/ ( U/ C 2 ) = = [aC 1 a-1 C 2 1-a /1-a C 1 a C 2 -a ]= [a/(1-a)] / [C 2 /C 1 ] 2. Set the MRS equal to the slope of the budget line (which is 1+r) and solve for C 2 : [a/(1-a)] / [C 2 /C 1 ] = (1+r) barb2right C 2 = [(1-a)/a] C 1 (1+r) 3. Show the budget constraint: C 1 + C 2 /(1+r) = Y 1 + Y 2 /(1+r). This equation tells us that the present value of lifetime consumption equals the present value of lifetime income. 4. Plug (2) into (3) and solve for C 1 : C 1 + [(1-a)/a] C 1 = Y 1 + Y 2 /(1+r) barb2right [a/a] C 1 + [(1-a)/a] C 1 = Y 1 + Y 2 /(1+r) barb2right [1/a]C 1 = Y 1 + Y 2 /(1+r) barb2right C 1 = a [Y 1 + Y 2 /(1+r)] This equation tells us the consumer’s optimal C 1 for given values of a, Y 1 , Y 2 and r. [Notice that a = C 1 /[Y 1 + Y 2 /(1+r)] In other words, the exponent “a” represents current consumption’s share in total expenditures. This is useful to know because it provides a convenient short-cut for finding (4)] 5. Plugging the result from (4) into (2): C 2 = [(1-a)/a] C 1 (1+r) barb2right C 2 = (1-a) [Y 1 (1+r) + Y 2 ] This equation tells us the consumer’s optimal C 2 for given values of a, Y 1 , Y 2 and r. [C 2 /(1+r) = (1-a) [Y 1 + Y 2 /(1+r)] barb2right (1-a) = C 2 /(1+r) divided by [Y 1 + Y 2 /(1+r)] . So it can be seen that (1-a) represents future consumption’s share in total expenditures. Again, this is useful because it provides a short-cut for finding (5)] Comparative statics: 1.dC 1 /dY 1 = a > 0 . An increase in current income results in an increase in current consumption, although by some fraction “a” of the change in income. Intuition: Because the consumer desires to smooth consumption over time, only part of the increase in current income results in an increase in current consumption. The remaining fraction will be saved, thereby allowing the consumer to increase consumption in the future.

This preview has intentionally blurred sections. Sign up to view the full version.

2. dC 1 /dY 2 = a/(1+r) > 0. Increase in future income results in an increase in current consumption (by some fraction). Again, because of the desire to smooth consumption over time, the consumer will “bring forward” some of the increase in future income by saving less today (which could be interpreted as borrowing more).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern