Page 56 - IJPS-11-1

International Journal of
Population Studies Macroeconomic factors and housing dynamics

where χ stands for the weight of non-durable permanently and the government will not collect any
consumption. The relative risk aversion coefficient is property tax from it.
denoted as σ. The housing production function is defined as:

2.3. Production sector ν − 1 ν − φ
( ) ( ) ( )
Y = ZL 1 φ  K h N h  (VII)
h
h
There are two production sectors in our model: the t t t   t t  
non-durable goods sector and the housing sector. The
h
former produces non-housing goods and is referred to where Z refers to the technology in the construction
t
h
as the consumption sector. The latter one builds houses sector. The new residential investment at time t is Y . The
t
like construction companies and is referred to as the land share in housing production is denoted as 1−φ, and
construction sector. Both sectors maximize their profits the capital share is referred to ν. The land, capital, and labor
h
h
by choosing input factors. Both the capital market and input are denoted by L , K and N , respectively.
t
t
t
labor market are frictionless, and there are no restrictions The construction firms’ problem can be written as:
on labor flow and capital flow. Thus, in equilibrium, the
l
r +
returns on both input factors are paid the same among max D , ht k = Y p − h t h w N − t t h ( t δ )K − t h pL t (VI)
+
t
h
h
sectors. Land, only provided by the government, is taken { t K ,NL }
, t
t
as a production factor in housing construction. Every construction company tries to maximize the
h
2.3.1. Consumption sector profit D h,t+k in each period, where p is the relative house
t
price in units of the non-durable consumption good. p is
l
c
The output of non-durable goods denoted as, Y , is the relative land price, which is fixed and set by the
t
produced using physical capital and labor input according government. The government spending G at time t is fully
to the Cobb-Douglas production function: financed by the sale of the land p L . t t
l
( ) ( )
c
c
Y = ZK t c α N t c 1 α − (III) The evolution path in the construction sector is:
t
t
h
K h I 1 K h (VII)

where Z refers to the productivity in the consumption t 1 t t
c
t
h
c
sector. Capital and labor input are denoted by K and, where I stands for the investment in the housing
t
t
respectively. α is the capital share. sector.
Then the firm’s problem can be written as follows: The law of motion for the aggregate residential housing
stock Ht is:
max D = Y − c wN − c ( t δ )K c (IV)
r +
c
{ t KN c } , ct k+ t t t t H t + 1 = (1 δ− h )H + t Y t h (VIII)
, t
The houses depreciate at a constant rate δ Based on
where D (c,t+k) stands for the profit of the company. the demographic information, we can easily calculate the
h.
Physical capital depreciates at a constant rate δ, and r refers aggregate house stock at time t:
t
c
to the interest rate. I is the investment in the consumption jJ ij
t
sector. H ( it j t j (IX)
t
P h )
The evolution path of aggregate capital in the j1 i1
consumption sector is: where the h is the individual’s optimal housing service
j
t
c
c
K t 1 I 1 K t c (VI) consumption at age j and time t.

t
The price of non-durable goods is normalized to 1. 2.4. Government and social security
Government in our model has three functions: the first one
2.3.2. Construction sector is to collect payroll tax and run the PAYG social security
The construction firm’s problem is similar to the system; the second one is to distribute government
consumption firm’s problem, except the construction firm transfers; and the third one is to lease the lands to finance
uses an additional input: land. Land is only owned and government spending to maintain function. To simplify,
provided by the government at a predetermined fixed we assume that each spending is financed separately. The
price. The land stock is unlimited, and then the land aggregate social security pension is equal to the aggregate
demand of construction firms is determined by the land tax collected, the accumulated government transfers
price. When a house is built and sold to the household, are equal to the market value of all the wealth including
the land ownership is transferred to the house owner housing and financial assets left by the households who are
Volume 11 Issue 1 (2025) 50 https://doi.org/10.36922/ijps.3645