# RDP 8907: Tax Policy and Housing Investment in Australia Appendix II

### THE VALUE OF A HOUSE

The value of a unit of housing stock V, is most easily calculated by starting with the arbitrage condition. As different people face different taxes they have different arbitrage conditions and different forms of the value function. We start with the general arbitrage condition:

Where a prime indicates the first derivative with respect to time. This states that the total after-tax return is equal to the cost of capital. This can be rearranged as:

We multiply both sides of the above equation by the integrating factor, exp-{(i−cπ)/(1−c)} and integrate:

We apply the transversality condition that as t approaches infinity the present value approaches zero:

Therefore by introducing the right hand side, multiplying by the integrating factor and integrating we have the result that,

In order to gain an intuitive feel for the results it is helpful to assume that the real price of renting pHS remains constant and that Ps the price index grows at the rate of inflation π. We assume that Ps = 1 when s=t.

Therefore,

Inserting this in the above equation gives us,

which has the solution:

The above equation is the general equation for valuing a house. We turn now to specifics. For an investor i = (1−u)r + (1−g)π. Therefore,

For the owner-occupant who is fully financed by debt i = r+π because he earns no tax deductions from his interest payments. However he pays no income tax on his imputed rent pHS (u = 0) nor does he pay any tax on capital gains so that c = 0.

The happy person who owns his house outright has a discount rate i = (1−u)r + (1−g)π. This is not because the person is negative gearing but rather because this represents the opportunity cost of investing his funds elsewhere. No capital gains tax is paid so c=0, and no tax is paid on the imputed rent (u = 0).

Substituting numerical values into these equations it becomes apparent that the present regime is extremely biased towards equity buyers and investors. This bias is manifest in the market by the withdrawal of the first home buyer from the property market. The first home buyer normally has to finance his purchase predominantly with debt.