Substitute 1st Order conditions into intertemporal budget c 2 3

# Substitute 1st order conditions into intertemporal

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Substitute 1st Order conditions into intertemporal budget constraint: ( 29 ( 29 ° ° + + + + + + = + + + + + + + + + 2 3 2 1 0 2 1 2 1 1 1 1 1 1 ) 1 ( ) 1 ( ) 1 ( ) 1 ( r Y r Y Y A r C r C C r r ρ ρ ° ° + + + + + + = ¸ ¸ ¹ º » » ¼ ½ + + + + + 2 3 2 1 0 2 1 ) 1 ( ) 1 ( ) 1 ( 1 ) 1 ( 1 1 r Y r Y Y A C ρ ρ Solve for C 1 : ¸ ¸ ¹ º » » ¼ ½ + + + + + + + = ° 2 3 2 1 0 1 ) 1 ( ) 1 ( 1 r Y r Y Y A C ρ ρ Problem - still can't measure future incomes ( Y 2 , Y 3 ,...) Expected Future Income E i ( Y j ) - expectation in year i of income in year j Solution - ¸ ¸ ¹ º » » ¼ ½ + + + + + + + = ° 2 3 1 2 1 1 0 1 ) 1 ( ) ( ) 1 ( ) ( 1 r Y E r Y E Y A C ρ ρ Year 2: ¸ ¸ ¹ º » » ¼ ½ + + + + + + + = ° 2 4 2 3 2 2 1 2 ) 1 ( ) ( ) 1 ( ) ( 1 r Y E r Y E Y A C ρ ρ Rational Expectation - will look at optimal forecast; E 1 ( Y 3 ) contains all pertinent information in year 1 regarding income in year 3; E 2 ( Y 3 ) updates this information so
7 of 10 expected value could be different, but the difference should only result from new information (i.e., no lagged variables should be significant) Know Now or Later - some use Y 1 , others use E 1 ( Y 1 ), depends on assumption of when information becomes available; not critical for results Assume r = 0 (for simplicity) and consume same amount each period (permanent income theory), now: [ ] T Y E Y E Y E A C / ) ( ) ( ) ( 3 1 2 1 1 1 0 1 ° + + + + = [ ] ) 1 /( ) ( ) ( ) ( 4 2 3 2 2 2 1 2 - + + + + = T Y E Y E Y E A C ° Substitute A 1 = A 0 + Y 1 - C 1 and add terms like E 1 ( Y i ) - E 1 ( Y i ) = 0 ( i = 2, 3,...) [ ] ) 1 /( )] ( ) ( [ )] ( ) ( [ ) ( ) ( 3 1 3 1 2 1 2 1 3 2 2 2 1 1 0 2 - + - + - + + + + - + = T Y E Y E Y E Y E Y E Y E C Y A C ° ° Swap the E 2 ( Y i ) and E 1 ( Y i ) terms [ ] ) 1 /( )] ( ) ( [ )] ( ) ( [ ) ( ) ( 3 1 3 2 2 1 2 2 3 1 2 1 1 1 0 2 - + - + - + + + + - + = T Y E Y E Y E Y E Y E Y E C Y A C ° ° Forecast Revision - E 2 ( Y i ) - E 1 ( Y i ) Add E 1 ( Y 1 ) - E 1 ( Y 1 ) [ ] ) 1 /( )] ( ) ( [ ) ( ) ( ) ( 2 1 2 2 1 1 2 1 1 1 1 1 0 2 - + - + + + + + - + = T Y E Y E Y E Y E Y E C Y A C ° ° Rearrange terms and substitute E 2 ( Y 1 ) = Y 1 (perfect info after the fact so year 2's expected value of income in year one is the actual income from year 1) [ ] ) 1 /( )] ( ) ( [ )] ( ) ( [ ) ( ) ( 2 1 2 2 1 1 1 2 2 1 1 1 0 1 2 - + - + - + + + + + - = T Y E Y E Y E Y E Y E Y E A C C ° ° T C 1 Revision on all forecasts Plug in T C 1 and collect terms [ ] ) 1 /( )] ( ) ( [ )] ( ) ( [ ) 1 ( 2 1 2 2 1 1 1 2 1 2 - + - + - + - = T Y E Y E Y E Y E T C C ° Break up sum to cancel (T - 1) in first term [ ] ) 1 /( )] ( ) ( [ )] ( ) ( [ 2 1 2 2 1 1 1 2 1 2 - + - + - + = T Y E Y E Y E Y E C C ° Finding - only change consumption if there's been some change in expected future income (i.e., C 2 = C 1 unless forecasts change) What Changes Forecasts - from rational expectations (& math shown above), only new information changes forecasts (hence consumption); any lagged terms should be insignificant: Ct = C t -1 + aY t -2 should yield a = 0 because C t -1 already incorporates Y t -2 Regression - used quarterly consumption data (1948:1 to 1977:1) for services and non- durables Consumption Categories - national income and product accounts include 3 types of consumption: durables, non-durables, and services No Durables - Hall left out durables because he argued they are more like savings or investment; purchase is done at one time, but consumption is taken over a period of time; Example: car provides transportation service; NIPA records sale in year 1 as consumption, but services consumed last longer (10 years in Bomberger's case); so consumption of service is smooth over time, but purchases aren't Confirmation - C t = 1.02 C t -1 - 0.01 Y t -1 (model used Y 1