Solve all problems. 

Solve the following 2 period model for equilibrium. Preferences are given as u(c?,c?)=ln(c?)+?*ln(c?), where c? is consumption today and c? is consumption tomorrow, and ? is the time preference factor. The young household supplies one unit of labor inelastically and earns the wage rate w for it in the first period. In addition, the household pays a proportianl tax t on labor in the first period. The household can save in the first period and earn interest r when entering the second period. In the second period the household retires and has no further labor income. The only income the household receives is from savings. There are N? young households in the economy and N? old households. There is a representative firm that produces output using the following production function:

F(K,L)=A*(K^?+L^(1-?)),

where A is the total factor productivity, K is aggregate stock of capital, and L is aggregate labor. Assume that capital depreciates fully between the two periods so that ?=100%.

The household budget constraint in the first period is which of the following:

a) c? =(1-t)w - s

b)

c?+s=(1-t)w

c)

c? + s = t + w

d)

c? + s = -t + w

1.2. We now consider a consumer with utility function a : R# -> R : (x, .....xx) H u(x,,...,XN). which is in particular assumed to be locally nonsatiated, where (x,,...,*) = x denotes her consumption vector. We denote by p= (p,.....P,)ER., the price vector for the consumption goods, and by w > 0 the wealth of the consumer. We consider the following optimization problems. Problem UMAX[p, w]. Given (p, w), choose x = (x,,...,xx ) in order to maximize u(x) subject to p. xS w. Denote by x(p,w) and by v(p,w) the solution and the value, respectively, of this problem, and by 2(p,w) its Lagrange multiplier. Problem EMIN[p, u]. Given (p, u), choose x = (X,,...,Xx ) in order to minimize p . x subject to u(x) 24. Denote by h(p,u) and by e(p,u) the solution and the value, respectively, of this problem, and by u(p,u) its Lagrange multiplier. Now a new one: Problem FRISCHIP, ~]. Given pe R. and re R. , choose x = (x,,...,x,) in order to maximize ru(x)- p. x. Denote by Q(p,r) and by 2(p,r) the solution and the value, respectively, of this problem. 1.2(a). Interpret Problem FRISCH[p, ~] and the parameter r. 1.2(b). What can you say about "(p,r)? Justify your answer. ap, Page 2 of 7 1.2(c). Some of the optimization problems in this part (Part 1.2) are formally (mathematically) identical to some of the ones in Part 1.1 above. Which ones are those? Explain. 1.2(d). Interpret the Lagrange multipliers A(p,w) and u(p,u). 1.2(e). Show that if u* = v(p,w*), then Ap,w* )=- H(p,u*) Interpret. (Hint. Use duality.) 1.2(f). Let x* = q(p,r) and u* = u(x*). Show that r = u(p,u*) and interpret. 1.2(g). Let x* = Q(p,r), u* = u(x*) and w* = e(p,u*) . Show that r = "Mp,w* ) and interpret. 1.3. Comment on the results of parts 1.1-1.2 above.QUESTION 26 Springfield, Inc. started the month with no beginning inventory. During the month, the firm produced 11,000 units, sold 9.000 units, and incurred the following costs: Direct materials per unit $18 Direct labor per unit $15 Variable MOH per unit $7 Total fixed MOH $55,000 Total selling and admin. costs $80,000 Springfield's product cost per unit under variable costing is: O A $46 OB $40 O C. $52 OD $45QUESTION 27 In its first month of business Topeka, Inc. made and sold 560 units and reported the following information: Sales price $150 per unit Direct materials $10 per unit Direct labor $20 per unit Variable MOH $30 per unit Fixed MOH $22,000 per month Variable selling and admin. costs $5 per unit Fixed selling and admin. costs $ 10,000 per unit What is Topeka's product cost per unit for external reporting purposes? (Round any intermediate calculations and your final answer to the nearest cent.) O A $59.29 O B. $120.00 Q C. $69.29 O D. $99.29Chapter 8 Using the following information, answer the questions below: Direct Materials $15 Direct Labor $20 Variable MOH $10 Fixed MOH $50,000 Variable Selling & $1 Administrative Fixed Selling & Administrative $30,000 Units Produced and Sold 10,000 Selling Price $100 per unit A. What is the cost per unit under the absorption method? Prepare the income statement. B. What is the cost per unit under the variable method? Prepare the income statement