Question

Let us consider a production economy endowed with a single
perfectly competitive firm renting at every time t both labour and
physical capital from households at the real rental rates
?_{t}, ?_{t}, respectively. In equilibrium at time
t: ?_{t} = ???_{t} and ?_{t}=
???_{t} where ???_{t} and ???_{t} denote
the marginal product of labour and the marginal product of physical
capital, respectively. The aggregate output/income ?_{t} at
every time t is produced according to the following production
function:

?_{t}= ??_{t}^{1-?}
L_{t}^{?}

where *Z*>0 stands for the total factor productivity
parameter, ?_{t} represents the physical capital and
?_{t} denotes the number of workers with ? ∈ *(0,1)*
stands for the labour share of output parameter. Let us assume that
a constant fraction y∈ (0,1) of the total household population of
size ?_{t} works at every time t:

?_{t} = ??_{t}

The aggregate population of
households grows at a constant rate *n*∈ (*0,*
+∞*)*:

?_{t+1}= (1 +
?)?_{t}

The law of motion for the physical capital from time t to time t+1 can be written as:

?_{t+1}
=?_{t} + (1 −δ)?_{t}

where δ ∈ (0,1) represents the physical capital depreciation
rate parameter and ?_{t} denotes the aggregate investment
in physical capital at time t which is equal to the aggregate
saving in equilibrium:

?_{t} = ?_{t}

Let ?_{t} ≡ Y_{t}/N_{t} denote the
output/income per capita at time t and let ?_{t}
≡k_{t}/N_{t} stand for the physical capital per
capita at time t.

- Derive the marginal product of labour and the marginal product of physical capital schedules at time t.(Please specify what information you need for this question if needed)

- Show that in equilibrium, the aggregate output/income at time t
is the sum of the aggregate labour income:
?
_{t}?_{t}and the aggregate physical capital income: ?_{t}?_{t}.(please specify what information is needed for this if needed)

Answer #1

Consider a version of the Solow model where population grows at
the constant rate ? > 0 and labour efficiency grows at rate ?.
Capital depreciates at rate ? each period and a fraction ? of
income is invested in physical capital every period. Assume that
the production function is given by:
?t =
?ta(?t?t
)1-a
Where ??(0,1), ?t is output, ?t is
capital, ?t is labour and ?t is labour
efficiency.
a. Show that the production function exhibits constant...

1. (Static Cournot Model) In Long Island there are two suppliers
of distilled water, labeled firm 1 and firm 2. Distilled water is
considered to be a homogenous good. Let p denote the price per
gallon, q1 quantity sold by firm 1, and q2 the quantity sold by
firm 2. Firm 1 bears the production cost of c1 = 4, and firm 2
bears c2 = 2 per one gallon of water. Long Island’s inverse demand
function for distilled water...

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