Answer all questions correctly. 

The representative firm has production function Y=zN, where z is labor productivity. The representative household has utility function U(c, l) = sqrt(c)+18sqrt(l). Let w denote the real wage rate.

a. Suppose that z=200. What is the representative firm's labor demand function? What must the wage rate be at the equilibrium? At this wage rate, how much profit can this firm send to household?

b. Given this wage rate and non wage income, what is the optimal time allocation and consumption for the representative household?

c. At the equilibrium, how big is labor input and how much output is produced?

d. Now the productivity drops. z=190. At the new equilibrium how big is labor input and how much output is produced?

e. Calculate the percentage decrease of productivity, labor input and output. Is the percentage decrease in output bigger or smaller than the percentage decrease in productivity? Explain.

f. Anticipating the productivity drop, government implements a stimulus package by increasing the government spending from 0 to 8, which is financed by borrowing. With this stimulus package, how big is labor input and consumption? How much output is produced?

g. Compared with the high productivity case, calculate the percentage change of labor input, output, and consumption under lower productivity with stimulus package.

h. Compare the changes of labor input, consumption, and output with or without stimulus package. What do you find?

b. Show that the actuarial present value can be expressed as 2 iii. k=1 If in Exercise 5.23 the yearly income does not cease at age I + u but continues at the level a while (I) survives therafter, the actuarial present value is de- noted by URI mgr}. a. Display the presentvvaiue random variable, Y, for this annuity as a function of the K and I random variables. 13. Show that the actuarial present value can be expressed as 11] 2 1453\"]- k=0 Verify the formula scam + TUT = as, where T represents the future lifetime of (I). Use it to prove that Ma), + (M), = ax. where (is)Jr is the actuarial present value of a life annuity to (at) under which payments are being made continuously at the rate of t per annum at time t. r i . . . . From a; = at\") + a \"'3 at\"), show that the assum hon of uniform distri- xl x a Jan P bution of deaths in each year of age leads to w 1' n 1 1 \"IE = f("ll [am + U npx J+n + (E _ W) U" up: Ax+n:|+ m 5.5 Establish and interpret the following formulas: a. 1 = it\" Egg\"! + xix b_ 1 = dun} aye} + Ax c. 3:? = (51in) as a. a5: = (were) Ema . I - m o m e. at; = (1 + a)\" ai'. Let H(m) = iii\" - 3E"). Prove that H{m) 2 0 and lim H{m) = 0. M aliaueous For 0 E t E 1 and the assumption of a uniform distribution of deaths in each year of age, show that __ (1 + it}.:'r' t(1 + i) b. \"a: = u*[(1 + an, to +13] Consider the following representative agent model. The representative consumer has preferences given by u(c,) : 0+ 66 where c is consumption, 8 is leisure, and ,3 > 0. The consumer has an endowment of one unit of time and 1% units of capital. The representative rm has a technology for producing consumption goods, given by Y = zKO'L1_' where Y is output, 2: is total factor productivity, K is the capital input, L is the labor input, and 0