5.
A planner makes resource allocation decisions for a closed economy, maximizing the utility
function of a representative household:
where
c
is consumption and
l
is the labor input. Labor is the only factor of production, and the
economy’s production function is
t
t
t
Y
=
A
l
,
t
t
where
A
is the exogenous level of productivity.
At time
t
, a fraction
g
of the economy’s output
must be allocated to government purchases.
So, the planner’s optimization is subject to
t
t
t
c
#
(1
-g )
Y
A.
(7 points) Find the solution for
Y
as a function of
A
and
g
.
Answer
:
The optimality conditions for consumption and labor imply
In equilibrium, this gives
and
B.
(7 points) Suppose
g
is constant most of the time but unusually high occasionally.
How
does a high value of
g
affect the optimal solutions for consumption and employment?
Explain
the effects using the relevant derivations and give some intuition for the results.

Answer
:
The optimal solution for output implies
and
When government purchases rise, the planner cuts consumption and raises employment.
Since
the government requires more goods, households should either work more to make more goods
or cut back on consumption.
The planner (optimal) solution involves adjusting on both
margins (work more and consume less).
C.
(5 points) What is the basic message of this exercise for fiscal multipliers in an economy
with flexible prices and perfectly competitive firms (operating the same production function as
above)?
Make sure you state clearly why this exercise is informative about fiscal multipliers in
such an economy.
Answer
:
Since the competitive equilibrium is efficient, it replicates the planner’s solution.
This means
the fiscal multiplier is less than one.
A fiscal expansion increases output, but by less than the
increase in government purchases (because consumption falls).
D.
(4 points) Comparing the setup of this exercise to standard Keynesian models (like the
short-run IS-LM model), what are the key differences in assumptions that matter for the
models’ implications about fiscal multipliers?
Answer
:
Short-run Keynesian models are different because they assume some rigidity in wages and / or
prices.
So, the resulting equilibria are not efficient.
Many simple models (like the IS-LM) also
incorporate an arbitrary consumption function that links consumption to current disposable
income.
In these models, a fiscal expansion increases consumption, and this makes the fiscal
multiplier larger than one.

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