Chapter 1 Refresher on technology and firms I.1 Short questions (answering requires only a few well chosen sentences and possibly a simple illustration) a) Consider an economy where all firms' technology is described by the same neoclassical production function, Y = F(K;, L(), i = 1, 2, ..., N, with decreasing returns to scale everywhere (standard notation). Sup- pose there is "free entry and exit" and perfect competition in all mar- kets. 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? CHAPTER 1. REFRESHER ON TECHNOLOGY AND FIRMS f) Does the logic of the replication argument, considered as an argument about a property of technology, depend on the availability of the dif- ferent 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 argu- ment? h) Suppose that for a certain historical period there has been something dose 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? I.2 Consider a firm with the production function Y = AK"LP, where A>0, 0 0 and [5 > 0. a) Does the function imply constant returns to scale? b) Is the production function neoclassical? Hint: after checking criterion (a) of the denition of a neoclassical production function in Lectme Notes, Section 2.1.], 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 fac- tor? ls labor? d) If we want to extend the domain of definition of the production function to include (K, L) = (0,0), how can this be done while maintaining continuity of the function? 1.4 \"(rite down a CRS twofactor production function with Harrod- neutral technological progress look. Why is the assumption of Harrod- neutrality so popular in macroeconomics? 1.5 Reesher cm stools usrsus ours. TWO basic elements in long-run models are often presented in the following way. The aggregate production function is described by Y: = FUR L1, Ar): (it) where Y, is output (aggregate value added), It} capital input, L, labor input, and A; the \"level of technology\". The time index I! may refer to period t, that is, the time interval [E, E + l) , or to a point in time (the beginning of period t), depending on the context. And accumulation of the stock of capital in the economy is described by Km I1} = II an}, (M) where d is an (exogenous and constant) rate of (physical) depreciation of capital, 0 E 6 g 1. Evolution in employment (assuming full employment) is described by L1+1 L1 = \"Li: n 5 1- (\"\"3 In continuous time models the comesponding equations ale: (*) combined with rm) E \"{2}\" = m) arm), a 2 0, - dL E L(!) E % = -nL(!), n \"free\". a) At the theoretical level, what denominations (dimensions) should be attached to output, capital input, and labor input in a production lilnction? b) \"Fhat is the denomination (dimension) attached to K in the accumu- lation equation (**)'? c) Might there be a consistency problem in the notation used in (*) vis- a-v'is (**)andin(*)vis-a-1.ris(***)? Explain. d} Suggest an interpretation that ensn'es that there is no consistency problem. e} Suppose there are two countries. They have the same technology, the same capital stock, the same number of employed workers, and the same number of man-hours per worker per year. Country :1 does not use shift work, but country I) uses shift work, that is, two work teams of the same size and the same number of hours per day. Elaborate the formula (*} so that it can be applied to both countries. f) Suppose F is a neoclassical production function with CR5 w.r'.t. K and L. Compare the output levels in the two countries. Comment. g) In continuous time we write aggregate (real) gross saving as 30!) Y(!) CUE). What is the denomination of SUE)? h) In continuous time, does the expression KT!) + .51!) make sense'? \"'hy or why not'? i) In discrete time, how can the expression K; + S: be meaningfully inter- preted'? 1.6 The Solow growth model can be set up in the following way (dis- crete time version). A closed economy is considered. There is an aggregate production function, V; = meme), (1) where F is a neoclassical production function with CRS, Y is output, If is capital input, T is the technology level, and L is the labor input. So TL is effective labor input. It is assumed that T; = To(l g)', where g 2 0, (2) L1 = Lo[l n)', where n 2 0. (3) Aggregate gross saving is assumed proportional to gross aggregate income which, in a closed economy, equals real GDP, Y: S; = SK, 0 0, f" To- After this shock everybody rightly expects T to grow forever at the same rate, 9, as before. d) Illustrate by the phase diagram ( or a new one) what happens to & and con impact, i.e., immediately after the shock, and in the long run. e) What happens to the rate of return on impact and in the long run? f) Why is the sign of the impact effect on the real wage ambiguous (at the theoretical level) as long as f is not specified further?! g) What happens to the real wage in the long run? V.3 Fiscal sustainability. Consider the government budget in a small open economy (SOE) with perfect mobility of financial capital, but no mo- bility of labor. The real rate of interest at the world financial market is a positive constant r. Time is continuous. Let Y = GDP at time t, = government spending on goods and services at time t, T = net tax revenue (gross tax revenue - transfer payments) at time t, Be = public debt at time t. All variables are in real terms (i.e., measured with the output good as nu- meraire). Taxes and transfers are lump-sum. Assume there is no uncertainty and that the budget deficit is exclusively financed by debt issue (no money financing). a) Write down an equation describing how the budget deficit and the increase per time unit in public debt are linked. Suppose Y grows at a constant rate equal to g + n, where g is the rate of Harrod-neutral) technical progress and n is the growth rate of the labor force = employment). Suppose r > g +n > 0. Assume T, = TY, and G; = Y, where + and y are constant over time, 0