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a) Consider an economy where all firms' technology is described by the same neoclassical production function, = ( ) = 1 2 with decreasing returns to scale everywhere (standard notation). Suppose there is "free entry and exit" and perfect competition in all markets. Then a paradoxical situation arises in that no equilibrium with a finite number of firms (plants) would exist. Explain. b) As an alternative to decreasing returns to scale at all output levels, introductory economics textbooks typically assume that the long-run average cost curve of the firm is decreasing at small levels of production and constant or increasing at larger levels of production. Express what this assumption means in terms of local "returns to scale". c) Give some arguments for the presumption that the average cost curve is downward-sloping at small output levels. d) In many macro models the technology is assumed to have constant returns to scale (CRS) with respect to capital and labor taken together.

What does this mean in formal terms? e) Often the replication argument is put forward as a reason to expect that CRS should hold in the real world. What is the replication argument? Do you find the replication argument to be a convincing argument for the assumption of CRS with respect to capital and labor? Why or why not?

f) Does the logic of the replication argument, considered as an argument about a property of technology, depend on the availability of the different inputs. g) Robert Solow (1956) came up with a subtle replication argument for CRS w.r.t. the rival inputs at the aggregate level. What is this argument? h) Suppose that for a certain historical period there has been something close to constant returns to scale and perfect competition, but then, after a shift to new technologies in the different industries, increasing returns to scale arise. What is likely to happen to the market form? Why?

Consider the production function = + ( + ) where 0 and 0 a) Does the function imply constant returns to scale? b) Is the production function neoclassical? Hint: after checking criterion (a) of the definition of a neoclassical production function in Lecture Notes, Section 2.1.1, you may apply claim (iii) of Section 2.1.3 together with your answer to a). c) Given this production function, is capital an essential production factor? Is labor?

d) If we want to extend the domain of definition of the production function to include ( ) = (0 0) how can this be done while maintaining continuity of the function? I.4 Write down a CRS two-factor production function with Harrodneutral technological progress look. Why is the assumption of Harrodneutrality so popular in macroeconomics? I.5 Refresher on stocks versus flows. Two basic elements in long-run models are often presented in the following way. The aggregate production function is described by = ( ) (*) where is output (aggregate value added), capital input, labor input, and the "level of technology". The time index may refer to period , that is, the time interval [ + 1) or to a point in time (the beginning of period ), depending on the context. And accumulation of the stock of capital in the economy is described by +1 = (**) where is an (exogenous and constant) rate of ( In continuous time models the corresponding equations are: (*) combined with () () = () () 0 ( ) () = () "free". a) At the theoretical level, what denominations (dimensions) should be attached to output, capital input, and labor input in a production function? b) What is the denomination (dimension) attached to in the accumulation equation (**)? c) Might there be a consistency problem in the notation used in (*) vis- vis (**) and in (*) vis--vis (***)? Explain.

Productivity speed up The basic model for this problem is the same as in Problem IV.4. Assume the economy has been in steady state until time 0 Then an unanticipated shift in to a higher positive level occurs. Hereafter everybody rightly expects to remain at this new level forever. a) What happens to the real interest rate on impact? Comment. b) Illustrate by a phase diagram the evolution of the economy for 0. There might be different possibilities to consider. Comment. c) What happens to the real interest rate in the long run? Comment. d) Compare two closed economies, and that can be described by this model and have the same production function, the same and the same initial conditions, 0 0, and 0, and the same The only difference is that country for some reason has a higher health level and therefore lower than country (and lower since is the same) "Country will in the long run experience a higher level of labor productivity, than country ". True or false? Why?

a) Make a list of motives for individual saving. Are some of these motives more in focus in an OLG framework than in a Ramsey framework? b) In standard long-run models with perfect competition (like Blanchard's OLG model with exponential retirement or the Ramsey model), the real rate of interest, , and the real rental rate, , for physical capital (i.e., a price on the market for capital services) may or may not coincide for all . Give a necessary and sufficient condition that they coincide. IV.7 Short questions. a) What is the golden rule capital intensity? b) A steady-state capital intensity can be in the "dynamically efficient" region or in the "dynamically inefficient" region. What is meant by "dynamically efficient" and "dynamically inefficient"? Give a simple characterization of the two regions. c) Compare some long-run properties of the Blanchard OLG model with the corresponding long-run properties of the Ramsey model. Hint: For example, think of the long-run interest rate and/or the possibility of dynamic inefficiency. d) The First Welfare Theorem states that, given certain conditions, any competitive equilibrium (Walrasian equilibrium) is Pareto optimal. Give a list of circumstances that each tend to obstruct the needed conditions and thus make the conclusion untrue. IV.8 Short questions. Consider the Blanchard OLG model with Harrodneutral technical progress at rate a) Can a path below the saddle path in ( ) space be precluded as an equilibrium path with perfect foresight in the Blanchard OLG model? Why or why not? b) Can a path above the saddle path in ( ) space be precluded as an equilibrium path with perfect foresight in the Blanchard OLG model? Why or why not?

a) "If and only if the production function is Cobb-Douglas, does the Blanchard OLG model predict that the share of labor income in national income is constant in the long run." True or false? Give a reason for your answer. b) Are predictions based on the Blanchard OLG model (with exogenous Harrod-neutral technical progress) consistent with Kaldor's stylized facts? Why or why not? c) Suppose we want a concise economic theory giving the long-run level of the average rate of return in the economy as an explicit or implicit function of only a few parameters and/or exogenous variables. Does the Blanchard OLG model give us such a theory? Why or why not? d) Briefly, assess the theory of the long-run rate of return implied by the Blanchard OLG model compared with that of the Ramsey model. That is, mention what you regard as strengths and weaknesses of the Blanchard theory. IV.10 Short questions a) What does Barro's dynasty model conclude about the hypothesis of Ricardian equivalence? b) What does Blanchard's OLG model conclude about the hypothesis of Ricardian equivalence? c) What is the basic reason that the two models lead to different conclusions in this regard? IV.11 Short questions a) "Considering the different Slutsky effects, the consumption function of the individual in the Blanchard OLG model (with logarithmic instantaneous utility) is such that a higher tax on interest income lowers current consumption." True or false? Why? b) "When the real interest rate remains above the GDP growth rate of the economy, then the NPG condition for the government is a necessary and sufficient condition for fiscal sustainability." True or false?