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Suppose the aggregate demand for housing services in period is ( ) 1 0 2 0 (*) where the stochastic variable reflects a factor that in our partial equilibrium framework is exogenous, say the present value of expected future labor income in the region. a) Set up an equation expressing equilibrium in the market for housing services. In the ( ) plane, for given illustrate how is determined. b) Show that the equilibrium net rental rate at time can be expressed as an implicit function of and written = R( ) () Sign the partial derivatives of R w.r.t. , and . Comment. Suppose a constant tax rate [0 1) is applied to rental income, after allowance for maintenance costs. In case of an owner-occupied house the owner still has to pay the tax out of the imputed income, per house per year. Assume further there is a constant property tax rate 0 applied to the market value of houses. Finally, suppose a constant tax rate [0 1) applies to interest income, whether positive or negative. We assume capital gains are not taxed and we ignore all complications arising from the fact that most countries have tax systems based on nominal income rather than real income. In a low-inflation world this limitation may not be serious. We assume housing services are valued independently of whether the occupant owns or rents. We further assume that the market participants are risk-neutral and that transaction costs can be ignored. Then in equilibrium, (1 ) + +1 = (1 ) (**) where +1 denotes the expected house price next period as seen from period , and is the real interest rate in the loan market. We assume that is constant over time, 0. c) Interpret (**). Assume from now that the market participants have rational expectations (and that they know the stochastic process which follows as a consequence of the process of ). d) Derive the expectational difference equation in implied by (**).

e) Find the fundamental value of a house, assuming does not grow "too fast". Hint: write (**) on the standard form for an expectational difference equation and use that the fundamental solution of the standard equation = +1 + is = + P =1 + Denote the fundamental value Assume follows the process = + (*) where is a positive constant and is white noise with variance 2. f) Find under these conditions. g) How does 1 (the conditional expectation one period beforehand of ) depend on each of the three tax rates? Comment. h) How does 1( ) (the conditional variance one period beforehand of ) depend on each of the three tax rates?

A housing market with bubbles (partial equilibrium analysis) We consider the same setup as in Exercise VII.4, including the equations (*), (*), and (**). Suppose that until period 0 the houses were owned by the municipality. But in period 0 the houses are sold to the public at market prices. Suppose that by coincidence a large positive realization of 0 occurs and that this triggers a stochastic bubble of the form +1 = [1 + + (1 )] + +1 = 0 1 2 (^) where +1 = 0 and 0 = 0 0 Until further notice we assume 0 is large enough relative to the stochastic process {} to make the probability that +1 becomes non-positive negligible. a) Can (^) be a rational bubble? You should answer this in two ways: 1) by using a short argument based on theoretical knowledge, and 2) by directly testing whether the price path = + is arbitrage free. Comment. b) Determine the value of the bubble in period , assuming known for = 0 1 .

c) Determine the market price, and the conditional expectation +1. Both results will reflect a kind of "overreaction" of the market price to the shock . In what sense? d) It may be argued that a stochastic bubble of the described ever-lasting kind does not seem plausible. What kind of arguments could be used to support this view? e) Still assuming 0 0 construct a rational bubble which has a constant probability of bursting in each period = 1 2 f) What is the expected further duration of the bubble as seen from any period = 0 1 2 given 0? Hint: P =0 (1 ) = (1 ) 1 g) If the bubble is alive in period what is the probability that the bubble is still alive in period + where = 1 2 ? What is the limit of this probability for ? h) Assess this last bubble model. i) Housing prices are generally considered to be a good indicator of the turning points in business cycles in the sense that house prices tend to move in advance of aggregate economic activity, in the same direction. In the language of business cycle analysts housing prices are a procyclical leading indicator. Do you think this last bubble model fits this observation? Hint: consider how a rise in affects residential investment and how this is likely to affect the economy as a whole.

"Under the hypothesis of rational expectations, speculative bubbles cannot arise in general equilibrium." True or false? Why?

The housing sector. We consider the housing sector in a small open economy facing a constant real risk-free interest rate, 0 given from the world financial market. Time is continuous. Let: = aggregate stock of houses, = aggregate housing services, = (real) price of houses, = (real) price of housing services, = expected increase per time unit in = aggregate gross housing investment (residential construction), = rate of physical depreciation on houses, = tax on (imputed) rental income after allowance for depreciation, = tax on housing property, tax on interest income. We assume that is proportional to , and we normalize the factor of proportionality to be one: = (1) = () 0 0a) Briefly explain (1) and (2), including why 0 0 is plausible. Suppose housing services are valued independently of whether you own or rent. Assume further there is no uncertainty and that there are no transaction costs. Then an equilibrium condition is (1 )(() + ) = (1 ) (3) b) Explain why (3) is an equilibrium condition. Figure 7.1 shows the year-to-year percentage change in and in Denmark since 1970. Data are from ADAM (Annual Danish Aggregate Model). c) Does the housing sector model of Chapter 15.2 in Lecture Notes fit with Figure 7.1? In Denmark, as in a lot of other countries, the housing market has changed somewhat in the last 15-20 years. In Denmark new loan types in terms of both adjustable-rate mortgage loans (from 1996?) and deferred amortization loans (from last quarter 2003) were introduced for the first time and from 2002 on the so-called 'tax freeze' implied that the property tax was frozen measured in nominal kroner (so it was falling in real terms). Figure 7.2 is from a simulation study by the Danish central bank (Nationalbanken) and indicates the estimated effects. d) For a given , what should the effects of the tax freeze be according to our model? Do the predictions fit qualitatively with the empirical study? e) For a given , what are the effects of the new loan types likely to be in a more realistic model? Do the predictions from the model fit qualitatively with the empirical study?