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Working Paper Series
Notes on Collateral Constraints in a
Simple Model of Housing
WP 09-03
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/economic_
research/working_papers/index.cfm
Andreas Hornstein
Federal Reserve Bank of Richmond
Notes on Collateral Constraints
in a Simple Model of Housing
Andreas Hornstein
April 7, 2009
Federal Reserve Bank of Richmond Working Paper 2009-03
Abstract
These notes provide the derivations of results stated without proof in Hornstein
(2009). For a simple model of the demand for housing, it is shown that on a balanced
growth path, the rate at which the relative price of housing changes over time is
determined by the relative productivity growth rates of the housing sector and the
rest of the economy. The model is then modified to include a collateral constrained
consumer. We show that collateral constraints may affect the level of the housing price
path, but they do not affect the growth rate of housing prices.
JEL classification: E2, R2
Keywords: house prices, owner occupied housing, collateral constraints
1
Section 1 describes a simple model of the housing market where land is an essential input
to the production of houses. This model is based on work by Davis and Heathcote (2005) and
it attributes endogenous changes in the price of housing to changes in the relative scarcity of
land. In order to understand long run trends in house prices, the model’s balanced growth
path is characterized. In section 2 the environment is modified to model the demand for
mortgage financed housing. This section is based on the Campbell and Hercowitz (2006)
representation of collateral constraints. It is shown that collateral constraints do not affect
the growth rates of the balanced growth path, but they may affect the level of the balanced
growth path.
1. A simple model of housing
We describe a simple general equilibrium model of the demand for housing where the price
of housing is endogenous. A representative consumer has preferences over the consumption
of nondurable goods and housing services. Housing services are proportional to the stock of
housing. New housing is produced by combining new residential structures, structures for
short, with land. New structures, together with nondurable consumption goods, are produced from aggregate output. The rate of transformation between nondurable consumption
goods and structures is exogenous and determines the relative prices of structures. If aggregate output is increasing faster than the supply of land or if structures become relatively
more expensive, then the relative price of housing will increase over time. The rate at which
the relative price of housing increases depends on the share of land in the production of new
housing. The bigger (smaller) is the production share of land, the closer will be the growth
rate of relative house prices to the growth rate of aggregate output (the price appreciation
rate of structures).
1.1. The environment
Time is continuous and the horizon is infinite. A representative agent derives utility from
the consumption of a nondurable good, c, and the consumption of housing services, h. The
agent’s preferences are
Z
0
∞
e−ρ0 t {θ ln h0 (t) + (1 − θ) ln c0 (t)} dt
(1.1)
with time preference rate ρ0 > 0 and 0 < θ < 1. The consumption of housing services is
proportional to the stock of housing units owned by the agent. In the following we will use
the terms housing services and housing stock interchangeably.
The agent receives an exogenous endowment stream of a homogeneous good. The value
1
of the endowment in terms of the nondurable consumption good is y0 . In the following
we will express all prices in terms of the nondurable consumption good. The agent also
receives l units of new land and the price of new land is pl . The agent can use his income
for consumption, the purchase of new housing units, xh , at the relative price ph , or he can
save it at an interest rate r. The flow budget constraint of the household is
ȧ0 (t) + c0 (t) + ph (t) xh0 (t) = y0 (t) + pl (t) l (t) + r (t) a0 (t)
(1.2)
where a is the agent’s net-financial wealth.1 Housing depreciates at rate δ > 0 and the stock
of housing accumulates according to
ḣ0 (t) = xh0 (t) − δh0 (t) .
(1.3)
Note that old land embodied in the housing stock disappears with the depreciated structures.
The homogenous good can be used to produce the nondurable consumption good or it
can be used to produce structures, xs . The rate of transformation between nondurable consumption goods and structures is exogenous and determines the relative price of structures,
ps . The relative price of structures is the inverse of the relative productivity of the structures
sector. The aggregate resource constraint for nondurable consumption and structures is
c (t) + ps (t) xs (t) = y (t) .
(1.4)
Structures are combined with new land to produce new housing units using a Cobb-Douglas
technology
xh (t) = xs (t)β l (t)1−β ,
(1.5)
with 0 ≤ β ≤ 1. The production of all goods is competitive.
The representative agent owns all of the endowment of the homogeneous output good.
Market clearing for the output good, the nondurable consumption good, new housing, structures, and the credit market implies (1.4), (1.5), and
y (t) = y0 (t) , c (t) = c0 (t) , xh (t) = xh0 (t) , and 0 = a0 (t) .
(1.6)
We assume that the economy is growing over time. In particular, the endowments y and
l and the relative price ps all grow at constant rates, γ y , γ l , and γ s ,
y (t) = ȳeγ y t , l (t) = ¯leγ l t , and ps (t) = p̄s eγ s t .
1
(1.7)
The notation ż (t) = ∂z (t) /∂t denotes the time derivative of the variable z as a function of time t, and
the growth rate of the variable z is denoted ẑ (t) = ż (t) /z (t).
2
In the following we will drop the time index when not needed.
1.2. Optimal consumption and production
A convenient way to state the dynamic optimization problem in continuous time involves
the Hamiltonian function, see e.g. Kamien and Schwartz (1981). The Hamiltonian function serves a similar purpose as the Lagrange function for static constrained optimization
problems. The Hamiltonian for the representative agent is defined as
H0 = θ ln h0 + (1 − θ) ln c0
(1.8)
+λ0 (ra0 + pl l + y0 − c0 − ph xh0 ) + μ0 (xh0 − δh0 ) ,
where the constraint multipliers λ and μ denote the marginal utility value of wealth and
housing.
The first order conditions for the optimal allocation are
c0 : λ0 = (1 − θ) /c0
(1.9)
(1.10)
xh0 : ph λ0 = μ0
∂H0
= (ρ0 − r) λ0
∂a0
∂H0
: μ̇0 = ρ0 μ0 −
= (ρ0 + δ) μ0 − θ/h0
∂h0
a0 : λ̇0 = ρ0 λ0 −
(1.11)
h0
(1.12)
We can rewrite the first order conditions for wealth (1.11) and housing stock (1.12) as
capital value equations for the marginal utility of wealth and housing stock in utility terms
ρ0 λ0 = rλ0 + λ̇0
(1.13)
ρ0 μ0 = θ/h0 − δμ0 + μ̇0
(1.14)
Equation (1.13) states that the required return on a unit of wealth is equal to the interest paid
plus the capital gain. Equation (1.14) states that the required return on a unit of the housing
stock is equal to the marginal utility from housing less depreciation and the capital gain.
Equations (1.9) and (1.10) then state that for the marginal unit of nondurable consumption
or new housing stock the marginal costs and benefits are equalized. The marginal cost is the
good’s price times the marginal utility of wealth, and the marginal benefit is the marginal
utility for nondurable consumption goods or the capital value of a marginal unit of housing
stock.
Another way to look at the optimality conditions for optimal consumption of housing
3
and nondurable consumption goods starts with equation (1.12), rewritten as
θ/h0 = (ρ0 + δ) μ0 − μ̇0 .
(1.15)
This expression equates the marginal utility from housing with the user cost of housing in
utility terms. The user cost of housing is the implicit rental rate paid for the use of housing
stock. Given the utility price of housing μ0 , the user cost is the required return on housing
plus depreciation minus capital gains due to changes in the capital value of the housing stock.
Combining equation (1.15) with equations (1.9) and (1.10), we get that
θ/h0
(ρ + δ − μ̂0 ) μ0
= 0
= (ρ0 + δ − μ̂0 ) ph .
(1 − θ) /c0
λ0
(1.16)
Equation (1.16) states that for the optimal consumption allocation the marginal rate of
substitution between the consumption of housing services and nondurable consumption goods
is equal to their relative price where the relative price of housing services is equal to the user
cost of housing.
Competitive production of new housing implies that for the two inputs, structures and
new land, the value of an input’s marginal product is equal to its price
βph xh /xs = ps ,
(1.17)
(1 − β) ph xh /l = pl .
(1.18)
1.3. A balanced growth path
On a balanced growth (BGP) all variables grow at constant, but potentially different, rates.
We first characterize the growth rates of the BGP and in the next subsection the level of the
BGP.
The resource constraint for the output good (1.4) implies that on the BGP nondurable
consumption and the value of structures grow at the same rate as does output
ĉ = γ s + x̂s = γ y .
(1.19)
The production function for new housing, equation (1.5), implies that investment in new
housing grows at a rate that is a weighted average of the growth rates of new structures and
land,
x̂h = β x̂s + (1 − β) γ l .
(1.20)
The market clearing conditions (1.6) imply that the representative household’s choice vari-
4
ables grow at the same rates as the corresponding aggregate variables
ĉ0 = ŷ0 = γ y , and x̂h0 = x̂h .
(1.21)
The accumulation equation for the housing stock, (1.3), implies that the stock of housing
grows at the same rate as does investment in new housing,
ĥ = x̂h and
xh
= ĥ + δ.
h
(1.22)
The growth rates for the price of new housing and land are determined by the first order
conditions for optimal input use in the production of new housing, equations (1.17) and
(1.18),
p̂h + x̂h = p̂l + γ l = γ s + x̂s = γ y .
(1.23)
We can now express the growth rates for the housing stock and the relative price of
housing on the BGP as functions of the exogenous growth rates of output, the relative price
of structures, and the supply of new land. The impact of a higher output growth rate on
the rate at which relative house prices increase is immediate. Combining expressions (1.20)
and (1.22) yields the housing accumulation rate,
¡
¢
ĥ = β γ y − γ s + (1 − β) γ l ,
(1.24)
and combining expressions (1.20) and (1.23) yields the rate at which the relative price of
housing changes,
¡
¢
p̂h = βγ s + (1 − β) γ y − γ l .
(1.25)
On the one hand, a higher growth rate of aggregate output increases both the rate of home
price appreciation and the rate of housing stock accumulation. On the other hand, if the
relative price of structures increases at a faster rate or the rate at which new land becomes
available declines, the relative price of housing increases at a faster rate, but the housing
stock is accumulated at a slower rate.
The impact of changes in the exogenous growth rates on the house price appreciation
rate depends on the share of land in the production of homes. If land is not an input to
the production of homes, that is, β = 1, then home production is proportional to the use
of structures. Thus home price appreciation is determined by the rate at which the relative
price of structures changes and is independent of output growth and the availability of new
land. On the other hand, if new homes are in fixed supply, that is, β = 0, then home price
appreciation depends on the difference between the output growth rate and the land supply
growth rate.
5
Finally, we obtain the growth rates of the marginal value of wealth and housing from
equations (1.9) and (1.10), together with equations (1.21) and (1.25),
λ̂0 = −γ y ,
¡
¢
μ̂0 = β γ s − γ y − (1 − β) γ l .
(1.26)
(1.27)
1.4. The BGP level
We normalize all variables such that they remain constant on the BGP. If the variable z
grows at the rate ẑ on the BGP, we define its normalized value as
z̃ (t) = z (t) e−ẑt .
(1.28)
Essentially, the normalized value of a variable represents the level of its growth path. By
·
construction the normalized variables do not change on the BGP, that is, z̃ = 0. For the
exogenous shift terms this means that
ỹ = ȳ, p̃s = p̄s , and ˜l = ¯l.
(1.29)
We now characterize the competitive equilibrium allocation on the BGP in terms of the
normalized variables.
From equations (1.13) and (1.26) the equilibrium interest rate on the BGP is determined
by the household’s time discount factor and output growth
r = ρ0 + γ y .
(1.30)
Rearranging equation (1.16) and substituting for the housing stock from equation (1.22) we
get that the value of housing investment is proportional to nondurable consumption
p̃h x̃h0 =
δ + ĥ
θ
c̃0 = A0 c̃0 .
1 − θ ρ0 + δ + ĥ
(1.31)
We can use this expression, together with the resource constraint (1.4), in the condition for
the optimal use of structures as an input (1.17) and obtain that nondurable consumption
and the value of housing investment are proportional to output
1
ỹ = C0 ȳ,
1 + βA0
= A0 C0 ȳ.
c̃0 =
p̃h x̃h0
6
(1.32)
(1.33)
Since the agent is representative this is also the aggregate value of housing investment. To
separate out real housing investment and the price of housing we use the expression for
nondurable consumption, (1.32), together with the resource constraint for the output good,
(1.4), and the production function for new housing, (1.5), and we get
x̃h
p̃h
¸β
(1 − C0 ) ȳ ¯1−β
=
l
p̄s
¸1−β
∙
β ȳ (1 − C0 )
= p̄s
/β
¯l
∙
(1.34)
(1.35)
On the BGP the normalized home price and home investment depend on the normalized
output, price of structures, and new land supply. On the one hand, a permanently higher
level of output implies a higher level of home prices and home investment. On the other
hand, a permanently higher price of structures or a permanently lower supply of new land
implies a higher home price level and a lower home investment level. Through the equilibrium
share of nondurable consumption the normalized home price and investment also depend on
the rate of housing stock accumulation. If the BGP is such that the household accumulates
housing stock at a faster rate, e.g., output grows at a faster rate or the price of structures
grows at a slower rate, then the output share of nondurable consumption declines and the
normalized home price (investment) level decreases (increases).
A special case obtains when the supply of new homes is exogenous, that is, β = 0. For
this case the resource constraints simplify to c̃ = ȳ and x̃h = ¯l, and we get the home price as
ȳ
p̃h = A0 C0 ¯.
l
(1.36)
2. Collateral constraints for housing
We now introduce another representative consumer into the economy. This new consumer
is more impatient than the agent studied above so that at the equilibrium interest rate the
impatient agent will borrow from the patient agent. Henceforth we will therefore distinguish
between the lender, type 0 agent, and the borrower, type 1 agent. The impatient agent is
constrained in his borrowings by the collateral value of his housing stock.
2.1. The borrower
The representative borrower has preferences over the consumption of nondurable goods and
housing
Z
∞
0
e−ρ1 t {θ ln h1 (t) + (1 − θ) ln c1 (t)} dt
7
(2.1)
with time preference rate ρ1 > ρ0 . The borrower earns income y1 and he can borrow or
save at the interest rate r. The agent borrows (saves) if a < 0 (a > 0). The flow budget
constraint of the borrower is
(2.2)
ȧ1 (t) + c1 (t) + ph (t) xh1 (t) = y1 (t) + r (t) a1 (t) .
Housing depreciates at rate δ > 0 and the stock of durable goods accumulates according to
ḣ1 (t) = xh1 (t) − δh1 (t) .
(2.3)
The amount of credit that a borrower can obtain is limited by the collateral value of the
housing stock he owns. We assume that the required equity share of a borrower for a vintage
τ home is
ω (τ ) = 1 − (1 − π) e−(φ−δ)τ ,
(2.4)
with φ ≥ δ. The down payment requirement for the purchase of new housing is ω (0) = π ∈
[0, 1]. The required equity share remains constant if φ = δ, and increases with the age of the
vintage to one if φ > δ. Given the stock of undepreciated durable goods, the total equity
requirement for a borrower is
ω̄ (t) = ph (t)
= ph (t)
Z
∞
0
∙Z
£
¤
ω (τ ) e−δτ xh1 (t − τ ) dτ
∞
0
−δτ
e
xh1 (t − τ ) dτ − (1 − π)
Z
∞
−φτ
e
0
¸
xh1 (t − τ ) dτ .
The first term in the square brackets represents the undepreciated past purchases of housing,
that is, the stock of housing, h1 . The second term in the square brackets represents the part
of the housing stock against which the agent can borrow, q1 . The implied accumulation
equation for the collateralized housing stock is
q̇1 (t) = xh1 (t) − φq1 (t) .
(2.5)
In the following we will refer to the collateralized housing stock as the ‘collateral stock.’ We
can rewrite the expression for the total equity requirement as
ω̄ (t) = ph (t) [h1 (t) − (1 − π) q1 (t)] .
(2.6)
The collateral constraint then requires that the total equity value is at least as high as the
required equity value
ph (t) h1 (t) + a1 (t) ≥ ω̄ (t)
8
which simplifies to
(1 − π) ph (t) q1 (t) ≥ −a1 (t) .
(2.7)
We assume that the representative borrower interacts with the representative lender
from section 1 in a competitive equilibrium. Production of nondurable consumption goods,
structures, and new homes continues to be determined by equations (1.4) and (1.5). We
assume that the lender receives a fraction α of the endowment of the output good and the
remainder goes to the borrower,
y0 (t) = αy (t) and y1 (t) = (1 − α) y (t) .
(2.8)
We also continue to assume that the lender receives all of the endowment of new land. Market
clearing for the nondurable consumption goods and new housing now implies that
c (t) = c0 (t) + c1 (t) , xh (t) = xh0 (t) + xh1 (t) , 0 = a0 (t) + a1 (t) .
(2.9)
2.2. Optimal Consumption of the borrower
The borrower is assumed to maximize utility (2.1) subject to the budget constraint (2.2),
the accumulation equations for the housing and collateral stock, (2.3) and (2.5), and the
collateral constraint (2.7).
The Hamiltonian for the borrower’s optimization problem is defined as
H1 = θ ln h1 + (1 − θ) ln c1
(2.10)
+λ1 (ra1 + y1 − c1 − ph xh1 ) + μ1 (xh1 − δh1 )
+ϕ1 (xh1 − φq1 ) + κ1 [a1 + (1 − π) ph q1 ] ,
where the multiplier ϕ denotes the marginal utility value of the collateral stock, and κ denotes
the marginal value of loosening the collateral constraint.
The first order conditions for optimal consumption of the borrower are analogous to
the first order conditions of the lender with appropriate modifications for the collateral
constraint,
c1 : λ1 = (1 − θ) /c1
(2.11)
xh1 : ph λ1 = μ1 + ϕ1
(2.12)
a1 : λ̇1 = (ρ1 − r) λ1 − κ1
(2.13)
q1 : ϕ̇1 = (ρ1 + φ) ϕ1 − κ1 (1 − π) ph
(2.14)
h1 : μ̇1 = (ρ1 + δ) μ1 − θ/h1 .
(2.15)
9
For now we simply assume that the collateral constraint is binding, that is, equation (2.7)
holds as an equality. Below we will show that this is the case on the BGP.
We can rewrite the first order conditions for wealth, collateral stock, and housing stock
as capital value equations for the marginal utility of wealth, collateral, and housing,
ρ1 λ1 = rλ1 + κ1 + λ̇1
(2.16)
ρ1 ϕ1 = κ1 (1 − π) ph − φϕ1 + ϕ̇1
(2.17)
ρ1 μ1 = θ/h1 − δμ1 + μ̇1 .
(2.18)
Equation (2.16) states that the flow benefit of one more unit of wealth includes the value
of a marginal relaxation of the collateral constraint, in addition to the usual capital gains
and interest paid on wealth, see equation (1.13). Equation (2.17) defines the flow return on
the capital value of collateral as the marginal relaxation of the collateral constraint from an
additional unit of collateral, net of the depreciation and capital gains on collateral. Finally,
the borrower’s capital value equation for housing, (2.18), is the same as the corresponding
equation (1.14) for the lender.
Equation (2.12) equates the marginal cost and marginal benefit (in utility terms) of an
additional unit of housing stock. The marginal benefit now has two components. Besides the
capital value of an additional unit of housing it also includes the capital value of an additional
unit of collateral since having more housing allows the agent to borrow more. The borrower
also equates the marginal rate of substitution between the consumption of housing services
and nondurable consumption goods with the relative price of housing services,
¶
µ
θ/h1
ϕ1
.
= (ρ1 + δ − μ̂1 ) ph −
(1 − θ) /c1
λ1
(2.19)
Relative to the analogous equation (1.16) for the lender, the user cost of housing is reduced
because housing provides a collateral value for the borrower.
2.3. A BGP
The growth rates of aggregate variables on the BGP are determined as before by equations
(1.19) and (1.20) since the aggregate resource constraints have not changed. From the
definition of market clearing (2.9) it follows that on the BGP consumption of nondurable
goods and housing, wealth, etc., for borrowers and lenders grow at the same rates
ĉi = âi = γ y and x̂hi = ĥi = q̂1 = ĥ, for i = 0, 1,
and we normalize all variables as described by equation (1.28).
10
(2.20)
The interest rate on the BGP continues to be determined by the lender’s time discount
rate and the output growth rate, equation (1.30). One can show that on the BGP the
collateral constraint is binding for the borrower since the borrower’s marginal utility of
wealth is positive and he is more impatient than the lender. From equation (2.11) it follows
that on any BGP with positive consumption for the borrower, λ1 is positive. Equation (2.13),
together with the BGP value of the interest rate, equation (1.30), and the assumption that
the borrower is more impatient than the lender then implies that the collateral constraint
multiplier, κ1 , is positive, that is, the collateral constraint is binding.
On the BGP the borrower’s value of housing investment is proportional to his nondurable
consumption,
p̃h1 x̃h1
"
#−1
ρ1 − ρ0
δ + ĥ
θ
1 − (1 − π)
=
= A1 c̃1 .
1 − θ δ + ĥ + ρ1
ρ1 + φ + ĥ
(2.21)
We obtain equation (2.21) by substituting for the BGP values of the capital values of wealth
and collateral in (2.19). Combining the BGP expressions for the housing stock and the
collateral stock yields the collateral stock proportional to the housing stock,
q̃1 =
δ + ĥ
φ + ĥ
h̃1 .
(2.22)
Substituting this expression in the collateral constraint, (2.7), then yields total borrowing
proportional to the borrower’s home investment value
ã1 = −
1−π
φ + ĥ
p̃h x̃h1 = −B1 p̃h x̃h1 .
(2.23)
Finally, substituting expressions (2.21) and (2.23) into the borrower’s budget constraint,
(2.2) we obtain the borrower’s consumption of nondurable goods as a fraction of his flow
income,
c̃1 = [1 + A1 (1 + ρ0 B1 )]−1 ỹ1 = C1 ȳ1 .
(2.24)
So far we have been able to characterize the optimal consumption allocation for the
borrower, without any recourse to the decisions of the lender, except for the use of the
equilibrium interest rate (1.30). In order to derive the BGP allocation for the lender and the
equilibrium prices, we now use the lender’s optimal decision rule for investment expenditures,
(1.31), together with the lender’s budget constraint and the market clearing conditions. After
some more algebra we obtain the lender’s consumption of nondurable goods as a share of
11
aggregate output
c̃0 =
1 − α + αA1 C1 [1 − β + ρ0 B1 ]
ȳ = C0 ȳ.
1 + βA0
(2.25)
Aggregate expenditures on nondurable goods and new housing investment then are
c̃ = (C0 + αC1 ) ȳ = C ȳ
p̃h x̃h = [A0 C0 + αA1 C1 ] ȳ = Dȳ.
(2.26)
(2.27)
We obtain investment and the price of new housing from the production function for new
housing together with the resource constraint for the output good and the demand for
nondurable consumption (2.26)
x̃h = [(1 − C) ȳ/p̃s ]β ¯l1−β ,
³ ȳ ´1−β µ p̃ ¶β
s
p̃h =
D.
¯l
1−C
(2.28)
(2.29)
References
[1] Campbell, Jeffrey R., and Zvi Hercowitz (2006). The role of collateralized household debt
in macroeconomic stabilization. Federal Reserve Bank of Chicago WP 2004-24.
[2] Davis, Morris A. and Jonathan Heathcote. 2005. Housing and the business cycle. International Economic Review 46, 751-784.
[3] Hornstein, Andreas. 2009. Problems for a Fundamental Theory of House Prices. Federal
Reserve Bank of Richmond Economic Quarterly.
[4] Kamien, Morton I., and Nancy L. Schwartz. 1981. Dynamic Optimization: the calculus
of variations and optimal control. Elsevier: New York-Amsterdam.
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