Discrete-Time Digression: Optimization
[For now: As motivation for continuous time. For later: Preview of discrete-time macro.]
Consider optimal consumption and capital accumulation problem over T periods:
T
U = t 1u(ct ) = u(c1 ) + u(c2 ) + ... + T 1u(cT )
- Preferences:
t =1
yt = f ( kt ) = ct + [ kt+1 (1 )kt ]
- Initial condition: Take k1 > 0 as given.
- Budget equations:
[Simplify the example: A=L=1]
- Finite horizon T => Capital is useless after period T => Terminal condition kT +1 = 0 .
- Choice variables: k2 ,..., kT and c1 ,..., cT . Finite number.
Optimization: Maximize utility subject to the budget equations.
- Define T Lagrange multipliers t, t=1,,T, for the T budget equations.
- Standard Lagrangian expression:
T
t 1
t =1
t =1
for t=T; and k1
L =
where kt+1
=0
T
u(ct ) + t { f ( kt ) ct + (1 ) kt kt+1 }
> 0 is given for t=1.
- Optimality conditions:
(i) L / ct
= 0 for t=1,,T; (ii) L /kt = 0 for t=2,,T; (iii) satisfy the budget equations.
- Claim: These conditions are analogous to the conditions of the Maximum principle.
(3d)-P.1
Discrete-time Optimality Conditions
T
Repeat: L =
t =1
t 1
(3d)-P.2
T
u(ct ) + t { f ( kt ) ct + (1 ) kt kt+1 }
t =1
i. Differentiate with respect to ct (for t=1,,T):
L
c t
= t 1u ' (ct ) t = 0
=>
t = t 1u ' (ct )
ii. Differentiate with respect to kt (for t=2,,T):
- Note that kt appears twice: in the period-t constraint and in the period-(t-1) constraint.
L = [ { f ( k ) c + (1 ) k k }] + [ { f (k ) c
t
t
t
t+1
t 1
t 1 + (1 ) k t 1 kt }]
k t k t t
k t t 1
= { t f ' ( kt ) + (1 )} t 1 = 0
t t 1 = t [ f ' ( kt ) ]
- Write as:
iii. Differentiate with respect to the multipliers t (for t=1,,T):
L
t
= f ( kt ) ct + (1 ) kt kt+1 = 0
=> kt+1 k t
= f ( k t ) k t ct
Three parts each analogous to the Maximum Principle:
(i)
Optimal consumption =>
Marginal utility = Shadow value.
(ii)
Optimal capital
=>
Change in multiplier proportional to return on capital.
=>
Recovers the budget equations
(iii) Optimal multiplier
Important insight: Discrete-time Euler Equations
[Result worth remembering for later.]
Derivation: combine
and
t = t 1u ' (ct ) => u ' (ct ) = t / t 1
t 1 = t { f ' ( kt ) + (1 )} = t {1 + f ' ( kt ) } = t (1 + rt )
- Apply to any periods t and t+1:
u ' (ct ) = t / t 1 = (1 + rt +1 ) t +1 / t 1 = (1 + rt +1 ) t +1 / t = (1 + rt +1 ) u ' (ct +1 )
Result:
or
=>
u' (ct ) = (1 + rt+1 ) u' (ct+1 )
u ' (ct ) = [1 + f ' (kt +1 ) ] u ' (ct +1 )
Marginal utility now = Discounted marginal utility next period * (1 + interest rate)
where (1 + interest rate) = gross return on capital = 1+ MPK depreciation.
Equivalent:
=>
u ' ( ct )
u ' ( ct +1 )
= 1 + rt +1 = 1 + f ' (kt +1 )
Marginal rate of substitution now vs. next period = Gross return on capital.
Intuition: Slope of indifference curve = Slope of budget line.
Equivalent:
u ' ( ct )
u ' ( ct +1 )
= (1 + rt +1 )
Marginal utilities declining over time iff (1 + rt +1 ) > 1 Consumption increasing.
=>
Consumption growth is positive if and only if (1 + rt+1 ) >
1
.
(3d)-P.3