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Exercises 15.1.1 Consider the function corresponding to the upper semicircle of the set of points in the ry-plane satisfying 12 + yz = 4. Find dy/dx (a) by finding and differentiating an explicit formula for y as a function of x; (b) by using the rule for implicit differentiation. 15.1.2 By analogy with the production function discussed in the text, a utility function of the form U(x, y) = xyl, where a and B are positive constants, is called a Cobb-Douglas utility function. For such a function, show that each indifference curve is negatively sloped, con- vex and has the axes as asymptotes. Hence sketch an indifference curve diagram.3. [Lower case letters denote per capita terms.) Say that a country faoe; an aggregate production mnction of y = 39%. The savings rate is 10% {Ill}. the rate of depreciation is 13% (. 1}, and the population growth rate is 5% {ELIE}. There is no technological progress. a.) n one graph {it can be drawn by hand}. show the output. break-even investment {depre- ciation plus population growth}, and savings functions for this economy (as a function of capital per worker}. Denote the steady state level of capita per worker in\". b.) What is the steady state level of capital per worker? Output per worker?I {You should be able to nd numerical answers for these.) e.) Say that the economy has an initial level of capital per wmker of 4 units. Explain in words how the economy works its way toward the steady state. d.) Demographic trends result in an increase in the rate of population growth to 10%. What do you expect the new steady-state levels of Iapital and output per worker to be? e.) Does the model just described generate long-run (Le... steady state} growth in output per worker? How is long-run growth generated in the .3on model? Consider how unemployment would affect the Solow growth model. Suppose that output is produced according to the production function Y = K\" [(1 ulLll'\2. Suppose that you are given the aggregate production function for Canada: Y = AK"NI-a The assumptions of the Solow Growth Model hold. The parameter values are: a = 0.5 s = 0.45 6 =0.1 n = 0.05 A = 2 a) What is the per capita production function? Hint: Divide the aggregate production function by / and express it in terms of k = 4. b) Recall that S = sY in the model. What is the per capita law of motion for capital? Hint: I am asking for k' as a function of k. Also, S = / in equilibrium. c) What is the steady state capital stock, output and consumption? d) Use the key growth model diagram to depict the steady state. Suppose that k starts below . Show using arrows how it will grow over time towards the steady state. e) Show the effects on your diagram the effects of a decrease in productivity. What effect will this have on consumption, capital and output per capita? You do not have to re-solve the model. Just discuss the effects. f) Suppose that Canada and South Korea each have the same productivity: ACanada = ASouth Korea
