An Endowment Economy
•Discrete time t=0,1,2,...
•Single agent is endowed with a stream of goods
y0,y
1,y
2,...
representing the amount of resources they have at each date
•Write this endowment stream
y={y0,y
1,y
2,...}
9
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macroeconomic macroeconomic macroeconomic macroeconomic, Schemes and Mind Maps of Law
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Typology: Schemes and Mind Maps
2018/2019
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An Endowment Economy
- (^) Discrete time t = 0, 1 , 2 ,...
- (^) Single agent is endowed with a stream of goods
y 0 , y 1 , y 2 ,...
representing the amount of resources they have at each date
- (^) Write this endowment stream
y = {y 0 , y 1 , y 2 ,... }
Preferences
- (^) Single agent has preferences over streams of consumption
c 0 , c 1 , c 2 ,...
- (^) Write this consumption stream
c = {c 0 , c 1 , c 2 ,... }
- (^) Preferences are represented by a utility function
U (c)
which ranks consumption streams
- (^) Suppose for simplicity U (c) strictly increasing, concave
Borrowing and lending
- (^) Suppose agent can borrow or lend at real interest rate r > 0
- (^) Agent with assets at at the beginning of t has interest payments
rat
which may be positive or negative
- (^) If at > 0 , interest income rat > 0 adds to endowment yt
- (^) If at < 0 , debt servicing rat < 0 subtracts from endowment yt
Flow budget constraint
- (^) Budget constraint for date t
ct + at+1 (1 + r)at + yt
with the understanding that a 0 = 0
- (^) Since utility function is strictly increasing in each ct, budget
constraint holds with equality
ct + at+1 = (1 + r)at + yt
- (^) We will assume that it is not possible to rollover debt forever (‘no Ponzi schemes’)
Iterating forward
- (^) Collecting terms and rearranging
R^2 c 0 + Rc 1 + c 2 + a 3 = R^2 y 0 + Ry 1 + y 2
- (^) Iterating this out to some arbitrary date T > 2
RT^ c 0 +RT^ ^1 c 1 +RT^ ^2 c 2 +· · · +cT +aT +1 = RT^ y 0 +RT^ ^1 y 1 +...+yT
- (^) Dividing both sides by RT^ gives
c 0 +R ^1 c 1 +R ^2 c 2 +· · · +R T^ cT +R T^ aT +1 = y 0 +R ^1 y 1 +...+R TyT
- (^) Or more compactly
X^ T
t=
R tct + R T^ aT +1 =
X^ T
t=
R tyt
Intertemporal budget constraint
- (^) Taking the limit as T! 1 and supposing that
lim T!
R T^ aT +1 = 0
we get
X^1
t=
R tct =
X^1
t=
R tyt
- (^) This is known as the agent’s intertemporal budget constraint
(as opposed to their single-period budget constraint)
Standard consumer problem
- (^) Now looks like a standard consumer choice problem
- (^) Choose bundle c 0 to maximize
U (c)
subject to the budget constraint
p · c = p · y
- (^) Lagrangian with single multiplier 0
U (c) + p · (y c)
Standard consumer problem
- (^) System of first order necessary conditions
@ @ct
U (c) = pt, t = 0, 1 , 2 ,...
- (^) This system pins down optimal consumption choices given and p
c( , p)
- (^) Budget constraint then pins down multiplier given y and p
p · c( , p) = p · y ) (y, p)
- (^) Solution can then be written
c⇤^ = c( (y, p), p)
- (^) Prices p matter both directly (substitution effects) and indirectly
via (income/wealth effects)
Time-separable utility
- (^) We will typically use the time-separable utility function
U (c) =
X^1
t=
tu(ct), 0 < < 1
with strictly concave period utility, u^0 (c) > 0 , u^00 (c) < 0
- (^) Future utility is discounted by constant factor
1 , , ^2 , ^3 ,...
- (^) Marginal utility of date-t consumption
Uc,t :=
@ct
U (c) = tu^0 (ct)
depends only on t and ct, not consumption on any other date
Marginal rates of substitution
- (^) System of first order conditions is then
tu^0 (ct) = pt
- (^) Marginal rate of substitution between t + 1 and t is then
t+1u^0 (ct+1) tu^0 (ct)
u^0 (ct+1) u^0 (ct)
- (^) So our tangency condition can be written
u^0 (ct+1) u^0 (ct)
pt+ pt
= R ^1
Qualitative dynamics
- (^) Recall that u^00 (c) < 0. This implies
ct+1 > ct , u^0 (ct+1) < u^0 (ct) , R > 1
- (^) Recall R = 1 + r and likewise define the pure rate of time
preference ⇢ > 0 by = (1 + ⇢) ^1
- (^) Then we can write
ct+1 > ct , r > ⇢
- (^) Consumption is growing when real interest rate is high relative to
the rate of time preference
Example
- (^) Suppose period utility function is
u(c) =
c^1 ^ 1 1
- (^) Consumption Euler equation
c t = Rc t+1
or ct+ ct
= ( R)^1 /
- (^) Hence consumption growth is just
log
ct+ ct
log( R) ⇡
r ⇢
Elasticity of substitution
- (^) Three important special cases
(i) perfect substitutes, = 0
log
⇣ (^) c t+ ct
⌘ very sensitive to even small changes in r
(ii) log utility, = 1 [use l’Hôpital’s rule]
log
⇣ (^) c t+ ct
⌘ responds 1-for-1 to changes in r
(iii) perfect complements, = 1
log
⇣ (^) c t+ ct
⌘ insensitive to even large changes in r
Example
- (^) Euler equation gives us consumption growth
ct+ ct
= ( R)^1 /
- (^) Hence iterating forward from date t = 0 we have
ct = ( R)t/ c 0
- (^) Pin down consumption level using intertemporal budget constraint
X^1
t=
R t( R)t/ c 0 =
X^1
t=
R tyt