Managerial economics guide 7

Consumption over time: sequential model

Initial endowment and saving

T0: Consumer is endowed with but only consumes, the rest is saving. 0 0 = + 0 0

Investment and return

T1: He decided to use his saving to invest with return rate r, he also endowed with 1 = + (1 + )1 1.

Objective: maximizing consumption utility

Our goal is to maximize the consumption discounted utility in the two periods:

max ( , = ( + (0 1 0 1 = + 0 0. . { = + (1 + )1 1)↔ ( , = ( − ) + ( + (1 + ))0 1 0 1) ) ) ) )( ; ( ( ( (0 1 0 1 0 1[( )]; ↔ =0↔( + )=0↔− + (1 + ) =0max 0 1

Saving function

• This is the saving function
• Example 1 () = ln())( , = ( − ) + ( + (1 + ))0 1 0 11
• (1 + )′ ( )→ , = − +0 1
• − + (1 + )0 1(1 + ) 1 (1 + ) − 0 1′)
• ( )( , ↔ , = 0 ↔ = ↔=0 1 0 1 + (1 + ) − ( + 1)(1 + )1 0

If is equal to 1 (we get the same utility in both period) (1 + ) − 0 1→= ( + 1)(1 + )

Intertemporal model

Constraint without saving

If saving was not an option, we will have a special type of sequential model which only has one constraint:

0 0 ( )(1 (1 (1↔ = + − + ) ↔ + + ) = + + ){ 1 1 0 0 1 0 1 0 = + (1 + )1 1

Period constraints

Now we only have one constraint which is expressed in terms of values at period 1. Values at period 0 are multiplied by the interest rate so that they are homogeneous with those at period 1 and can be summed together. ( = ; = )

Endowment and forward market

If we decide to consume all our endowment and there exists a forward market:

0 0 1 1 consumption at time 0 may be exchanged directly with consumption at time 1, and that corresponding (nominal) prices is and1 2 0 0 + = + + = +

New constraint comparison

Our new constraint will be or 0 0 1 1 0 0 1 1 0 1 0 1 1 1. Comparing this with the original constraint we will find that a sequential model will be the same as an 01 + = intertemporal model only wh