RDP 8907: Tax Policy and Housing Investment in Australia 3. Numerical Solution of the Model
November 1989
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In this section we specify the complete model and discuss how we simulate the model using numerical techniques for solving models containing rational expectations. We also discuss the calibration of the model at a steady state. The complete model is specified in Table 1. It is an annual model.
Table 1 Model Equations
Housing Services: Demand = Supply
F(K,L) = β(p^{HS})^{η}Y
Capital Accumulation
dK/dt = J − δK
Land Accumulation
dL/dt = L^{B} – δL
Housing Investment Equation
J/K = (q^{*} – 1)/Ø
Adjustment Cost Equation
I = J(1 + 0.5ØJ/K)
Modified q
q^{*} =
m^{K}{(1−c(1+π−i)}/p^{H}(1−d)
Marginal Pricing: Land
p^{L} = m^{L}{1−c(1+π−i)}
Arbitrage Pricing Condition: Land
(1−c)dm^{L}/dt = m^{L}(i
+(1−c)δ −π) − (1−u)p^{HS}F_{L}
Arbitrage Pricing Condition: Capital
(1−c)dm^{K}/dt = m^{K}(i + (1−c)δ−π) –
(1−u)p^{HS}F_{K}
Home Production Function
F = AK^{α}L^{1−α}
Home Pricing Equation
p^{home} _{=} _{[m}L_{L}
_{+} _{m}K_{K}/(1−a)](1−c(1+π−i))/F(K,L)
Cost of Capital (after tax)
i = (1−u)r + (1−g)π
SYMBOL INDEX
A | = | constant |
d_{s} | = | present value of tax depreciation deductions from a unit of capital installed at time s; exogenous |
D | = | demand for housing services; endogenous |
F | = | supply of housing services; endogenous |
g | = | rate at which inflation component of interest income taxed |
H | = | current value Hamiltonian |
i | = | after-tax cost of capital; exogenous |
I | = | real investment expenditures; endogenous |
J | = | real increase in capital stock; endogenous |
K | = | quantity of housing capital; endogenous |
L | = | stock of land; exogenous |
L^{B} | = | purchases of new land; exogenous |
m^{L} | = | shadow price of land; endogenous |
m^{K} | = | shadow price of capital; endogenous |
p_{H} | = | real price of a unit of uninstalled housing capital; exogenous |
P^{HS} | = | real price of housing services; endogenous |
P^{Home} | = | average real price of a home; endogenous |
P^{L} | = | real price of a unit of land; endogenous |
q = | = | Tobin's q; endogenous |
q^{*} | = | modified q; endogenous |
r | = | real interest rate; exogenous |
Y | = | real per capita income; exogenous |
u | = | income tax rate; exogenous |
α | = | share of rental income going to capital; exogenous |
β | = | constant in the demand equation |
δ | = | rate of depreciation |
η | = | price elasticity of demand |
π | = | rate of price inflation; exogenous |
One problem which arises from the specification of this model is that the arbitrage conditions for land and housing prices involve expected variables. For convenience, we assume agents form these expectations rationally which implies that all available information is used efficiently in forecasting future variables. Because we also assume no uncertainty, we have to find a solution where the actual evolution of variables is equal to the expected values of these variables. To solve this we use the technique developed by Fair and Taylor (1983). This algorithm involves iterating on choices for future paths of expected variables until the expected path corresponds to the actual path.^{[3]}
To calibrate the model, we can find an analytical solution for the steady state of the model. Given the parameters such as tax rates, depreciation rates, price elasticity of housing demand, and profit share to house and land from a home package, we can solve for data that is consistent with a steady state of the model.^{[4]} Given the analytical solution for the steady state, we can calculate the values of all other variables. The values of each variable in the initial steady-state and the initial parameter assumptions are contained in Table 2.
F | = | 11,027,668 |
i | = | 0.078 |
I | = | 201,600 |
J | = | 180,000 |
K | = | 6,000,000 |
L | = | 5,580,000 |
m^{L} | = | 1.6469 |
m^{K} | = | 1.5316 |
P^{H} | = | 1.0 |
P^{HS} | = | 0.1 |
P^{L} | = | 1.0 |
p^{Home} | = | 1.01 |
q^{*} | = | 1.24 |
r | = | 0.07 |
A | = | 1.91 |
d_{s} | = | 0.1 |
g | = | 0.4 |
u | = | 0.4 |
α | = | 0.5 |
β | = | 0.08 |
δ | = | 0.03 |
Ø | = | 8.0 |
η | = | −0.5 |
π | = | 0.06 |
To verify that we have indeed found a steady state for the model we use the data to simulate the model and find that there is no tendency for real variables to change over time.
Once the model is calibrated to a steady state we can examine two effects of each change in tax policy and each exogenous shock. The first is the effect on the steady state of the model. The second is the dynamics of adjustment between steady states. The analytical solution for the steady state of the model enables us to calculate the new steady state following a shock. We then use the Fair Taylor algorithm to solve for the transition path between steady states.
Footnotes
The algorithm used is FAIRTAYLOR which is written in GAUSS and available from Aptech Systems. [3]
Details are provided in Appendix IV. [4]