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8

The First Best

In period zero, the planner collects each saver’s unit endowment and invests so per saver

in short-term assets and 1 o - s per saver in long-term assets. In period one, when the short-term

rate ri is realized, the planner can reinvest some fraction si in short-term assets until period two.

Reinvestment will be state-contingent; hence the subscript. The planner also chooses

consumption in each period, and consumption will be state-dependent as well. The period-one

consumption constraint in state i is

1c ( )/ . i oi = - s s r

Early consumers get the output from short-term investments less the amount that is reinvested.

(Consumption is scaled up by 1/ ρ because the planner invests o s , but only r consume

early.) Late consumers get the returns from long-term assets and from any short-term assets

rolled over from period one; hence, period-two consumption in state i is

2 (1 )/(1 ). i o i i c s =-+ - rs r

The only other constraint facing the planner is that reinvestment in both states must be

nonnegative:

si ≥ 0.

Subject to these three constraints, the planner chooses {s sc c oi i i ,, , } 1 2 to maximize the

representative saver’s expected utility in period zero:

12 1 2 ( ) [ ( ) (1 ) ( )] (1 )[ ( ) (1 ) ( )]. EU ti l l h h c = +- +- +- pr r U c U c p r r U c U c

Except for the state-contingent aspect, this is a straightforward programming problem. The firstorder conditions can be combined as follows:

s 0, i i l = (2)

1 2 ’( ) ’( ) / , U c ll l l - = rU c l p (3)
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The price of the long-term asset in the low-return state is increasing in the amount of period-zero

short-term investment and the fraction of late consumers. More short-term assets means late

consumers have more to ‘‘spend’’ on long-term assets and fewer to bid on. More late consumers

means more buyers and fewer sellers. Since 1, l p > (17) implies that . o s > r

The equilibrium price, portfolio mix, and other variables are determined as follows:

Given that 0, l s = we know that 1 / l o c s = r and 2 (1 ) /(1 ). l o c s =- - r Given 0, h s > we know

that 0 lh = and 1/ . p r h h = Substituting these values, along with (14), (15), and (9), into (11)

produces an equation in o s and pl . This equation and (17) determine period-zero short-term

investments and the period-one price of long-term assets when short-term returns are low. These

values determine the consumption allocations in each state. Finally, given that 1/ , p r h h = o s

and (12) determine lh and reinvestment in the high-return state: . h o hh s s pl = -

Risk-Neutral Savers

Recall that when U(c) = c, the planner invests only in the short-term asset even though it

has a lower expected return. Clearly when investors are risk neutral, this first-best solution is not

attainable through decentralized contracting (technically there is nothing to swap if 1 o s = ).

Even though individuals know that, ex ante, they would be better off promising to redistribute on

the basis of whether it is optimal to consume early or to reinvest, they know that after the fact the

ex post ‘‘losers’’ have no incentive to honestly reveal their type. If no transfers occur, then so =

1 yields (1 )((1 ) ) 1. EU h i =+- - + < r r pp r r


	
	
			
	
	
	
						

	
		
		
		  		
		
			
		

	
	
	
			
			
				



		          
		         		
	 



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