4.
The government budget constraint is satisfied, therefore G = T. The taxes paid by the
consumer are equal to the exogenous quantity of government spending.

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Income – Expenditure Identity
In competitive equilibrium, the income expenditure identity is satisfied, so that
Y = C + G.
In
this economy, there is no investment expenditure, as the economy is closed. Therefore
I = 0
and
NX = 0
.
The Production Function and the Production Possibility Frontier
In this figure, a shows the equilibrium relationship
between the quantity of leisure consumed by the
representative consumer and aggregate output. The
relationship in b is the mirror image of the
production function in a. In c, we show the
production possibility frontier (PPF), which is the
technological relationship between C and I,
determined by shifting the relationship on b down
the amount of G. The shaded region in c represents
the consumption bundles that are technologically
feasible to produce in this economy.
Competitive Equilibrium
This figure brings together the representative consumer’s preferences and the representative
firm’s production technology to determine a competitive equilibrium. Point J represents the
equilibrium consumption bundle. ADB is the budget constraint faced by the consumer in
equilibrium, with the slope of AD equal to minus the real wage and the distance DB equal to
dividend minus taxes.
In this figure, the PPF is given by the curve HF. From the relationship between the production
function and the PPF we can determine the production point on the PPF chosen by the firm,
given the equilibrium real wage rate
w
. The representative firm chooses the labour input to
maximise profits in equilibrium by setting
MP
N
= w
, and so in equilibrium minus the slope of
the PPF must be equal to
w,
because
MRT
l,c
= MP
N
= w
in equilibrium. Therefore, if w is an
equilibrium real wage rate, we can draw line AD that has a slope
–w
and that is tangent to the
PPF at point J, where
MP
N
= w.
Then the firm chooses labour demand equal to
h – l*
and

produces
Y* = zF(K,h-l),
from the production function. Maximised profits for the firm are
*
= zF(K, h – l*) –w(h – l)
(total revenue minus the cost of hiring labour), or the
distance DH . Now DB is equal to
π
* - G =
π
* - T
, from the government constraint
G
=
T
.
ADB is the budget constraint that the consumer faces in equilibrium, because the slope of AD
is –w and the length of DB is the consumer’s dividend income minus taxes. Because J
represents the competitive equilibrium production point , where C* is the quantity of
consumption goods produced by the firm and h – l* is the quantity of labour hired by the
firm, it must be the case that that C* is also the quantity of consumption goods that the
representative consumer desires and that l* is the quantity of leisure the consumer desires.

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- Summer '18