WHERE SOME FORMULAS COME FROM IN THE TWO PERIOD ...



WHERE SOME FORMULAS COME FROM IN THE TWO PERIOD CONSUMPTION MODEL

▪ PVLR = present value of lifetime resources

▪ Defined as the present value of the income that a consumer expects to earn in current (y) and future periods (yf) , plus initial (a) and expected wealth (af).

▪ PVLR = y + a + (yf + af)/(1+r)

▪ PVLC = present value of lifetime consumption equals current consumption (c) plus the present value of future consumption (cf).

▪ PVLC = c + cf /(1+r)

▪ PVLC = PVLR…….this is the condition ensures that you are consuming (using) all your available resources across the two periods (i.e., you are on your budget constraint).

▪ c + cf/(1+r) = y + a + (yf + af)/(1+r)

▪ If we re-arrange the equality above, we can solve for a 'nice' expression of the budget constraint:

▪ cf = [(1+r) (y + a) + yf + af] - (1 + r) c

Where the GREEN is the all important intercept and the YELLOW 'tells' us the (relative) price of one unit of current consumption in terms of future consumption - the higher the r, the higher the price of current consumption!

Perfect Smoothing Preferences suggest c = cf = c*

▪ Once we have preferences, we can set up the PVLC = PVLR equation as:

▪ c* + c*/(1+r) = y + a + (yf + af)/(1+r)

▪ Solving the above equation for c* we have:

▪ c* = [(1+r) (y + a) + yf + af] / (2 + r)

THIS IS THE 'PERFECT SMOOTHING EQUATION (DAGWOOD TYPE PREFERENCES)

Homer type preferences - he prefers to consume twice as much today (current consumption) relative to next period (future consumption

In this case, we can define:

c = c* cf = (1/2)c* example if c = c* = 50 then cf= (1/2)50 = 25

Let's multiply both by sides by 2 so we have:

c = 2c* and cf = c*

now we set up equation and solve for c* remembering importantly that when we obtain c* we need to multiply it by 2 to get c (recall c = 2c*)

▪ PVLC = PVLR

▪ 2c* + c*/(1+r) = y + a + (yf + af)/(1+r)

▪ [2c*(1+r) + c*]/(1+r) = PVLR

▪ factor c* out of numerator

▪ [c*(3 + 2r)]/ (1+r) = PVLR

▪ c* (3 + 2r) = [(1+r) (y + a) + yf + af]

▪ c* = [(1+r) (y + a) + yf + af] / (3 + 2r)

This is the formula for Homer preferences

NOTE AGAIN THAT WHEN WE OBTAIN c*, that is equal to cf, to get c we need to multiply c* by two

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