WHERE SOME FORMULAS COME FROM IN THE TWO PERIOD CONSUMPTION
FUNCTION
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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 (c f).
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 = c f = 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*)
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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 c f, to get c we need to multiply c* by
two