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Assume that the Phillips curve is given as: t= t e+ 0.1 - 2ut a) What is the natural unemployment rate? Now the expectations for inflation are: te= (1 - ) (bar)+ t 1 (bar) can be any constant value, either 0 or another number above zero. First assume that = 0 and that t = n (ie the unemployment rate is equal to the natural unemployment rate). In year the government authorities decide to reduce unemployment to 3% and keep it there forever. b) Determine inflation for the years + 1, + 2, + 3, + 4, + 5. c) Do you think the answer you gave to b) makes sense? d) Now suppose that the year + 6 increases from = 0 to = 1 and that the authorities are still determined to keep at 3% forever. e) Why does it make sense to increase in this way? f) Determine inflation for the years + 6, + 7 and + 8. Also show your answer diagrammatically.

A carbon tax and Tobin's q We consider a small open economy (henceforth called SOE) with perfect mobility of financial capital but no mobility of labor. The SOE faces a constant real interest rate 0 given from the world market for financial capital. The technology of the representative firm is given by a neoclassical production function with constant returns to scale, = ( ) 0 0 for = Here is output gross of installation costs, imports, and physical capital depreciation is capital input, and is labor input, whereas is an imported fossil energy source, say oil. We also assume that the three inputs are direct complements in the sense that 0 6= The firm faces strictly convex capital installation costs and the installation cost function is homogeneous of degree one: = ( ) () (0) = 0 (0) = 0 00 0. In national accounting what is called Gross Domestic Output (GDP) is aggregate gross value added, i.e., = (1) where 0 is the exogenous real price of oil which we treat as a shift parameter. The labor force of the SOE is a constant There is perfect competition in all markets. There is a tax, 0 on use of fossil energy (a "carbon tax"). In equilibrium with full employment the following holds:

We write the marginal product of capital ("Marginal Product of ") as a function ((1 + )) ( ((1 + ))) It can be shown that 0 0 (3) a) Set up the value maximization problem of the representative firm and derive the first-order conditions. b) Given the general information put up, briefly explain by words why GDP takes the form in (1), why the partial derivatives in (2) must have the shown signs, why we must have 0 as indicated in (3), and why, at first glance, the sign of might seem ambiguous.1 The dynamics of the capital stock is given by = (() ) 0 0 given, (4) where (1) = 0 0 = 100 and is the capital depreciation rate whereas is the shadow price of installed capital along the optimal path, satisfying the differential equation = ( + ) ((1 + )) + (()) ()( 1) (5) Moreover, a necessary transversality condition is lim = 0 c) Briefly interpret (4) and (5): what is the economic "story" behind these equations? d) Assuming satisfies the Inada conditions, construct a phase diagram for the system (4) - (5). Hint: it can be shown that at least in a neighborhood of the steady state, the slope of the = 0 locus is negative.2 e) For an arbitrary 0 0 indicate in the diagram the evolution of the pair ( ) in general equilibrium. Does the convergent solution path satisfy the transversality condition? Is the solution to the model unique? Hint: it can be shown that the divergent solution paths violate the transversality condition.

f) Assume that until time 0 0 the economy has been in its steady state. Then, unexpectedly the government raises the carbon tax to 0 Illustrate graphically what happens on impact and gradually over time. Comment on the effect of the tax rise on investment. g) Illustrate in another figure the time profiles of and for 0 Briefly explain in words. h) We now go a little outside the present simple model. Suppose, that a non-fossil energy source is available whose application requires a lot of specially designed capital equipment. Although before the tax rise this alternative technology has not been privately cost-efficient, after the tax rise it is. Moreover, the representative firm expects internal positive learning-by-doing effects by adopting the alternative technology. Briefly do an informal analysis of how the rise in the carbon tax could in this broader setup affect capital investment. VI.2 A convenient specification of the capital installation cost function. In this exercise we consider an economy as described in Exercise VI.1, ignoring its last question. Assume () = 1 2 2 0 Assume further that until time 0 0 the economy has been in its steady state with carbon tax equal to Then, unexpectedly the government raises the carbon tax to 0 at the same time as it introduces an investment subsidy 0 1 so that to attain an investment level purchasing the investment goods involves a cost of (1) The subsidy is financed by some tax not affecting firms' behavior (for example a constant tax on households' consumption). a) Given 0 , find the required in terms of ((1 + 0 )) such that the economy stays with its "old" steady-state capital stock, Comment. b) How, if at all, is the steady-state value of affected by the change in fiscal policy? VI.3 (question a) - i) from exam Jan. 2016) When capital installation costs are independent of the stock of already installed capital. Consider a single firm with production function = ( )where and are output, capital input, and labor input per time unit at time , respectively, while is a neoclassical production function with CRS and satisfying the Inada conditions. Time is continuous. The increase per time unit in the firm's capital stock is given by = 0 0 0 given, where is gross investment per time unit at time and is the capital depreciation rate. There is perfect competition in all markets and no uncertainty. The real interest rate faced by the firm is a positive constant . Cash flow (in real terms) at time is = ( ) () where is the real wage and () represents capital installation costs and is a function satisfying (0) = 0 (0) = 0 00() 0 a) Set up the firm's intertemporal production and investment problem as a standard optimal control problem, given that the firm wants to maximize its market value. b) Let the adjoint variable be denoted Derive the first-order conditions and state the necessary transversality condition, TVC. Hint: the TVC has the standard form for an infinite horizon optimal control problem with discounting. c) What is the economic interpretation of ? d) On the basis of one of the first-order conditions, show that the firm's chosen labor input is such that the capital-labor ratio at time is a function of = (). Hint: it is convenient to express the marginal productivity of labor by the production function in intensive form. Suppose from now on that = for all 0 where is a positive constant. Let the corresponding optimal capital-labor ratio be denoted () e) Show that the optimal investment level, can be written as an implicit function of and that can be expressed in terms of and ().

f) Derive and construct a phase diagram for the ( ) dynamics, assuming that a steady state with 0 exists. Let the steady state value of be denoted . For an arbitrary 0 0 indicate in the diagram the movement of the pair ( ) along the optimal path. Hint: since the setup differs from the examples in syllabus, the phase diagram may look somewhat different from what you are used to; moreover, as you know from one of the policy regimes in Blanchard's dynamic IS-LM model, the saddle path in a phase diagram may coincide with the line representing no change in the asset price. g) In another diagram draw the time profiles of and Comment on why, in spite of the marginal productivity of capital in the steady state exceeding + there is no incentive to increase above h) It is common to call the "desired capital stock". Express the desired capital stock as an implicit function of and How does the desired capital stock depend on and respectively? Indicating the sign is enough. Hint: a simple approach can be based on curve shifting. i) Show that optimal net investment, equals ( ) This rule is called the capital adjustment principle. What is intuitively the contents of the rule? j) Let be Cobb-Douglas with CRS and let ()=(2)2 0 Find and along the optimal path. Hint: the differential equation ()+() = with 6= 0 has the solution ()=((0)) +, where = k) Evaluate the model.