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

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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.
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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).
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