1. Consider the following growth model where the production function is given by the Constant Elasticity of Substitution (CES) specification, g o1 o1 o1 Y, =|aK,\" +(1a)L,' where a (0,1) and where the elasticity of substitution between capital and labot, o > 0. 1(a) Show that the CES production function exhibits Constant Returns to Scale (CRS). 1(b) Express this production function in per capita terms. That is, express output per worker, Y = _', as a function of the capital-labor ratio, k =_". I, ' L 1(c) Derive an expression for the marginal product of capital (MPK) as a function of the capital-labor ratio, k =_". [Hint The algebra may be less cumbersome if t you denote p =-#%and work in terms of p.] 1(d) Show that the MPK exhibits diminishing returns. 1(e) Examine the limits of the MPK as ki 0 and ki 0. In particular, consider the cases when0 1. 1(f) Write down Solow's Fundamental Equation (as defined in class) for this economy. 1(g) What does this model predict about growth for this economy? Does this economy have a steady state, k* ? If so, derive an expression for k* and describe its properties. [Hint: Consider the cases when 0 1] Page 1 of 2 2. Goods and services are produced in this economy using the following Cobb-Douglas production function, Y, = K*(HL)% a (0,1) H is a measure of the average level of human capital (e.g., average years of schooling) in the workforce. 2(a) Express this %,roduction function in per capita terms. That is, express output i % . = t 5 per worker, Y = as a function of the per worker capital stock, kr =__. Provide an t t economic interpretation for the parameter . 2(b) For the Cobb-Douglas case as given above: (1) write down Solow's Fundamental Eqnation (as defined in class), (i) using a diagram, show how we obtain the steady state per worker capital stock, and (i) detive an expression for the steady state per worker capital stock. 2(c) The South African Health Ministry has become increasingly worried about the growing AIDS epidemic in that country. You have been rushed in to provide the Ministry with advice on the likely impact of AIDS on South Africa's long-run economic performance. After careful study, you conclude that AIDS will impact the economy in two ways: @) South Africa will experience a dramatic reduction in the population growth rate due mainly to higher child mortality rates and shorter average life expectancy. (i) South Africa will experience over time a decrease in the average number of years of schooling in the population. There are at least two reasons for this: (a) there is likely to be a significant increase in the number of orphans (whose parents have died of AIDS). These orphans are unlikely to be given access to much schooling opportunities. (b) Because of the mode of transmission and long gestation period, AIDS tends to kill off the segment of the adult population that is relatively young and which in recent times has had the most access to educational opportunities. Using a diagram to support your analysis, present the Health Ministry with a coherent explanation of the likely impact of AIDS on South Africa's steady state per capita income. 2(d) A prominent Economist interjects in the middle of your presentation. He offers the following comments: \"I don't think we need to do anything about the AIDS epidemic at alll In fact, my own research finds that AIDS will 7aise the long-run level of per worker income in South Africa.\" Is it possible that the AIDS epidemic could actually raise South Africa's long-run level of per worker income? Explain your answer using a diagram. Suppose you concluded that his analysis of the situation could be correct (note: this should not be taken as a hint! It could well be a red herringl), do you agree nevertheless with his policy recommendation (i.e., the part about us having no need to do anything about the epidemic)? Explain. Page 2 of 2