11. Consider the following specification of the Cobb-Douglas production function, where K, denotes government capital (infrastructure, etc.): o I-ya y= AKIYK ' where both a and y are constants between 0 and 1, and 1-y-a is positive. (a) Using calculus, derive dy/dK,. Is it positive or negative? Does this result make economic sense? Explain. (b) Does the level of K affect the marginal product of labour? Do the calculus, and explain. 12. (a) Using the Cobb-Douglas production function, derive the equation for the rate of growth of real GDP. That is, take logs and then take the total differential. (b) Using growth accounting, explain what factors caused the average 3% rate of growth of real GDP in Canada over the last 50 yerars. (c) What is the Solow residual? 13. Explain, step by step, how one derived L in the calibration exercise. Go through and do the derivation. What factors cause L to increase and what factors cause it to decrease? Explain. How did one calculate total factor productivity in the calibration exercise? How did one derive y! ? Explain. 14. Let real labour income, the real wage rate times hours worked (denoted L), equal Z. Since the real wage equals the marginal product of labour, Z = fi L. Take the total differential of this equation (hold K and A fixed; just let L change), and divide through by dL. Does Z rise, fall, or remain the same as L rises? Explain. If K and A are fixed, does Z rise or fall in a recession