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Consider the extended IS-LM model analyzed in the course and assume that there is an increase in the risk premium: a) How can the central bank react to offset the effect of the increase in the risk premium on the level of output? Analyze graphically and explain briefly. b) Briefly explain what the zero-bottom constraint is, and why it is possible that this constraint limits the ability of the central bank to constraint may limit the ability of the central bank to fully offset the impact the impact of the rising risk premium on the level of output.

Consider an orange juice producer's production process. Suppose a firm can produce juice from oranges that are available in two sizes, large and small, where for the purposes of juice manufacturing a large orange provides exactly twice the amount of juice as a small orange. Assume the firm perceives no other differences between large and small varieties of oranges (such as quality, cost of handling or processing...). Suppose, in addition, the juice of 4 large oranges is required to produce one full carton of orange juice. a) Describe in words the fundamental economic nature of this production process? b) Write down the production function (the mathematical relationship between inputs and output). Denote the number of large oranges as "L", the number of small oranges as "S", and the number of full cartons of orange juice as "q". c) Carefully draw two (appropriately labeled) isoquants that correspond to 5 and 10 full cartons of orange juice, respectively. Label axes intercepts, if any. Put large oranges on the horizontal axis and small oranges on the vertical intercept. d) What is the marginal product of small oranges? Of large oranges? e) What is the marginal rate of technical substitution (large oranges for small)? Explain how this is related to the isoquants drawn in part c).

Let {} be a stochastic process in discrete time. Suppose = + , where = 1 + and and are white noise. a) Is {} a random walk? Why or why not? b) Is {} a random walk? Why or why not? c) Calculate the rational expectation of conditional on all relevant information up to and including period 1. d) What is the rational expectation of conditional on all relevant information up to and including period 1? e) Compare with the subjective expectation of based om the adaptive expectations formula with adjustment speed equal to one. VII.2 Consider an old-Keynesian model of a closed economy with constant wages and prices (behind the scene), abundant capacity, and output determined by demand: = = + + (1) = + 1 0 0 1 (2) = (1 ) + 1 + 0 0 1 (3) where the endogenous variables are = output (= income), = aggregate demand, = consumption, and 1 = expected output (income) in period as seen from period 1 while , which stands for government spending on goods and services, is considered exogenous as is , which is white noise. Finally, investment, , and the parameters and are given positive constants. Suppose expectations are "static" in the sense that expected income in period equals actual income in period 1. a) Solve for . b) Find the income multiplier (partial derivative of ) with respect to a change in 1 and respectively Suppose instead that expectations are rational. c) Explain what this means. d) Solve for e) Find the income multiplier with respect to a change in 1 and respectively. f) Compare the result under e) with that under b). Comment. g) Briefly, evaluate the model from a "modern" point of view. VII.3 Consider arbitrage between equity shares and a riskless asset paying the constant rate of return 0. Let denote the price at the beginning of period of a share that at the end of period yields the dividend . As seen from period there is uncertainty about + and + for = 1 2.... Suppose agents have rational expectations and care only about expected return (risk neutrality). a) Write down the no-arbitrage condition. Suppose dividends follow the process = + where is a positive constant and is white noise, observable in period but not known in advance. b) Find the fundamental solution for and let it be denoted . Hint: given = +1 + the fundamental solution is = + P =1 +Suppose someone claims that the share price follows the process = + with a given 0 0 and, for = 0 1 2..., +1 = 1+ with probability 0 with probability 1 where = () 0 0 c) What is an asset price bubble and what is a rational asset price bubble? d) Can the described process be a rational asset price bubble? Hint: a bubble component associated with the inhomogenous equation = +1 + is a solution, different from zero, to the homogeneous equation, = +1. VII.4 The housing market in an old city quarter (partial equilibrium analysis, discrete time) Consider the housing market in an old city quarter with unique amenity value (for convenience we will speak of "houses" although perhaps "apartments" would fit real world situations better). Let be the aggregate stock of houses (apartments), measured in terms of some basic unit (a house of "normal size", somehow adjusted for quality) existing at a given point in time. No new construction is allowed, but repair and maintenance is required by law and so is constant through time. Notation: = the real price of a house (stock) at the start of period = real maintenance costs of a house (assumed constant over time) = the real rental rate, i.e., the price of housing services (flow), = = the net rental rate = net revenue to the owner per unit of housing services. Let the housing services in period be called Note that is a flow: so and so many square meter-months are at the disposal for utilization (accommodation) for the owner or tenant during period We assume the rate of utilization of the house stock is constant over time. By choosing appropriate measurement units the rate of utilization is normalized to 1, and so = 1. The prices and are measured in real terms, that is, deflated by the consumer price index. We assume perfect competition in both the market for houses and the market for housing services.

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? Comment. VII.5 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 .