microecon 413 it is okay

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, = ( ) = 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? I.2 Consider a firm with the production function = where 0 0 1 0 1. a) Is the production function neoclassical? b) Find the marginal rate of substitution at a given ( ) c) Draw in the same diagram three isoquants and draw the expansion path for the firm, assuming it is cost-minimizing and faces a given factor price ratio. d) Check whether the four Inada conditions hold for this function? e) Suppose that instead of 0 1 we have 1 Check whether the function is still neoclassical? I.3 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 (physical) depreciation of capital, 0 1. Evolution in employment (assuming full employment) is described by +1 = 1 (***) 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.

d) Suggest an interpretation that ensures 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 does not use shift work, but country 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 is a neoclassical production function with CRS w.r.t. and . Compare the output levels in the two countries. Comment. g) In continuous time we write aggregate (real) gross saving as () () () What is the denomination of ()? h) In continuous time, does the expression () + () make sense? Why or why not? i) In discrete time, how can the expression + be meaningfully interpreted? I.6 The Solow growth model can be set up in the following way (discrete time version). A closed economy is considered. There is an aggregate production function, = ( ) (1) where is a neoclassical production function with CRS, is output, is capital input, is the technology level, and is the labor input. So is effective labor input. It is assumed that = 0(1 + ) where 0, (2) = 0(1 + ) where 0. (3) Aggregate gross saving is assumed proportional to gross aggregate income which, in a closed economy, equals real GDP, : = 0 1 (4) Capital accumulation is described by +1 = + where 0 1 (5) The symbols and represent parameters and the initial values 0 0 and 0 are given (exogenous) positive numbers.

a) What kind of technical progress is assumed in the model? b) To get a grasp of the evolution of the economy over time, derive a firstorder difference equation in the (effective) capital intensity () that is, an equation of the form +1 = ( ) From now on suppose is Cobb-Douglas. c) Construct a "transition diagram" in the ( +1) plane. d) Examine whether there exists a unique and asymptotically stable (nontrivial) steady state. e) There is another kind of diagram that is sometimes (especially in continuous time versions of the model) used to illustrate the dynamics of the economy, namely the "Solow diagram". It is based on writing the difference equation of the model on the form +1 = ( ) [(1 + )(1 + )] For the case of the general production function (1), find the function ( ) and the constant By drawing the graphs of the functions ( ) and in the same diagram, one gets a Solow diagram Indicate by arrows the resulting evolution of the economy. I.7 We consider the same economy as that described by (1) - (5) in Problem I.6. a) Find the long-run growth rate of output per unit of labor, . b) Suppose the economy is in steady state up to and including period 1 such that 1 = 0 (standard notation). Then, at time (the beginning of period ) an upward shift in the saving rate occurs. Illustrate by a transition diagram the evolution of the economy from period onward c) Draw the time profile of ln in the ( ln ) plane. d) How, if at all, is the level of affected by the shift in ? e) How, if at all, is the growth rate of affected by the shift in ? Here you may have to distinguish between temporary and permanent effects. f) Explain by words the economic mechanisms behind your results in d) and e).