question archive Consider the model of learning by doing with spillovers (Arrow & Romer) presented in class and assume that the production function is Cobb-Douglas, that is, Y m m α m 1 α t = (Kt ) (htLt ) − However, assume there are diminishing returns to technological progress, γ ht = ηkt , for some constants η > 0, 0 < γ < 1, where Km kt = t Lm
Consider the model of learning by doing with spillovers (Arrow & Romer) presented in class and assume that the production function is Cobb-Douglas, that is, Y m m α m 1 α t = (Kt ) (htLt ) − However, assume there are diminishing returns to technological progress, γ ht = ηkt , for some constants η > 0, 0 < γ < 1, where Km kt = t Lm
Subject:BusinessPrice: Bought3
Consider the model of learning by doing with spillovers (Arrow & Romer) presented in class and assume that the production function is Cobb-Douglas, that is, Y m m α m 1 α t = (Kt ) (htLt ) − However, assume there are diminishing returns to technological progress, γ ht = ηkt , for some constants η > 0, 0 < γ < 1, where Km kt = t Lm . t i. We want to write the equilibrium dynamics are functions of c and k alone: (a) Express the return R that firms are willing to pay in equilibrium as a function of kt alone. (b) Express the resource constraint in terms of c and k. ii. Imagine the continuous time version of the dynamics in part (a) and draw the phase diagram. iii. Repeat parts (a) and (b) for the social planner's problem (Hint: this is similar to the Ramsey model). iv. How does the phase diagram of part (c) compare to that of part (b)? which line changes, the c? = 0 locus or the ?k = 0 locus? What happens to the steady state levels of c and k? v. If the equilibrium allocations differ from the planner's allocations, describe a policy that would restore efficiency. 1 Problem 2: Tax smoothing Consider a two-period economy. Households preferences are given by U = u (c1, c2, n1, n2) = c1 − n 2 1 + β c 2 2 − n2 , where ct ≥ 0 is consumption in period t ∈ {1, 2} and nt ≥ 0 is lab or supply. Labor is used to produce output with the technology yt = Ant (there is no capital). The wage is thus given by wt = A, for t ∈ {1, 2}. The government taxes labor income at rates τt in period t, so households' intertemporal budget constraint is given by 1 1 c1 + c2 = (1 τ1)An1 + (1 τ2)An2 1 + r − 1 + r − The government has constant expenditues, gt = g for t ∈ {1, 2}. Its intertemporal budget constraint is thus given by 1 IBC ≡ (τ1An1 − g1) + (τ2An2 g2) = 0 1 + r − Finally, the resource constraints in the economy are y1 = An1 = c1 + g and y2 = An2 = c2 + g. 1) Consider the household's optimal consumption and labor-supply problem. Argue that the solution is interior only if the interest rate r is such that 1 = β. 1+r Assume that this is the case for the rest of the exercise. 2) Solve for the household's optimal n1 and n2 as functions of τ1 and τ2. 3) Use the two resource constraints to replace ct = Ant−g into U. Next, use the previous result to replace nt with a function of τt . You should now have expressed the household's utility U as a function of the two tax rates: U = U(τ1, τ2) 4) Do the same for the government's intertemporal buget: replace nt with the function of τt that you found in part 2 so as to express IBC in terms of τ1 and τ2 : IBC = IBC(τ1, τ2) 5) It follows that the optimal policy is given by the combination of τ1 and τ2 that solves the following problem: maxU(τ1, τ2) s.t. IBC(τ1, τ2) = 0 Prove that the optimal policy satisfies τ1 = τ2 (tax smoothing). 6) Suppose that we increase g1 but reduce g2 so that g1 + βg2 stays constant. What happens to the optimal taxes? Explain
1. Assume that corn is produced in a perfectly competitive market. Farmer Roy is a typical producer of corn.
(a) Assume that Farmer Roy is making zero economic profit in the short run. Draw a correctly labeled side-by side graph for the corn market and for Farmer Roy and show each of the following.
(i) The equilibrium price and quantity for the com market, labeled as PMI and QMI, respectively
(ii) The equilibrium quantity for Farmer Roy, labeled as QFI
(b) For Farmer Roy's com, is the demand perfectly elastic, perfectly inelastic, relatively elastic, relatively inelastic, or unit elastic? Explain.
(c) Corn can be used as an input in the production of ethanol. The demand for ethanol has significantly increased.
(i) Show on your graph in part (a) the effect of the increase in demand for ethanol on the market price and quantity of corn in the short run, labeling the new equilibrium price and quantity as PM2 and QM2' respectively.
(ii) Show on your graph in part (a) the effect of the increase in demand for ethanol on Farmer Roy's quantity of corn in the short run, labeling the quantity as QF2
(iii) How does the average total cost for Farmer Roy at QF2 compare with PM2?
(d) Corn is also used as an input in the production of cereal. What is the effect of the increased demand for ethanol on the equilibrium price and quantity in the cereal market in the short run? Explain
7) The rate at which one good can be converted technologically into another is called
A) the marginal rate of transformation.
B) the marginal rate of substitution.
C) the marginal product of labour.
D) rate of conversion.
E) the marginal product of capital.
8) The Solow residual is a measure of
A) average labour productivity. B) average capital productivity.
C) total factor productivity.
D) the rate of growth of real GDP.
E) the marginal product of labour.
9) Consider the choice of the consumer in the endowment model of Chapter 9. The consumer receives 50 units of goods in the current period, and pays a lump-sum tax of 10 units in the current period. If the consumer consumes 60 units in the current period, then her/his savings is
A) -20
B) 10
C) 0
D) 10
Exercise 1: One-Consumer, Two-Producer Economy
We consider an economy in which there are a representative consumer and two firms j = 1,2 producing two goods = 1,2. The consumer supplies an exogenous quantity of labour L. Firm 1 produces good I by using a production function F. (Z?, L?) where Z?, L1 are respectively the quantity of good 2 and labour. Similarly, firm 2 produces good 2 with a production function F?(Z?, L?) where Zi, L2 are respectively the quantity of good 1 and labour. Let denote by w, P1, P2 >>0 wages and prices of good 1 and good 2. The consumer is also the firms owner.
1. Let denote R, the consumer's total revenue. Her/his utility function is given by u(11, 1?) =
(i) Derive the consumer's Walrasian/Marshallian demand functions. Compute her/his indi rect utility function.
(ii) Derive the consumer's Hicksian demand functions. Compute her/his expenditure function.
2. Assume that F?(-) = 32/L³ and F?(-) = 32/L³. Derive the factor demand functions 1/3, and supply functions of good 1 by firm 1 and firm 2. Compute their profit.
3. Define and then compute the general equilibrium for this economy.
(Hint):Do we have a GE for this economy
[4:52 PM, 1/18/2022] Muthuri: Question:
5.12 A survey conducted by Certified Practising Accountants Australia to investigate the reaction to new audit standards issued by the Australian Auditing and Assurance Standards Board found that only 53% of auditors believed that the new standards had improved audit quality. Fifty-eight per cent of auditors felt that the standards had improved confidence in financial reporting. Assume that these two events (improved quality and improved confidence) are independent. Suppose 20 auditors from Australia are selected at random.
a. What is the probability that fewer than 15 of them believe that the standards have improved confidence in financial reporting?
b. What is the expected number of auditors who believe that the standards have improved confide...
[4:52 PM, 1/18/2022] Muthuri: Question:
Hassan and Dana had bought a property valued at $1,225,000 for 20% down and a mortgage amortized over 25 years on March 1, 2018. They made equal end-of-month payments towards their mortgage. Interest on the mortgage was 3.29% compounded semi-annually and the mortgage was renewable after five years.
What is the size of each monthly payment?
What is the cost of the mortgageforthe first 5years?
In November2020,theydecidedtorefinancetheirmortgagefortworeasons:rateswere down by quite a lot, and they also wanted to pay off some Line of Credit debt they had accumulated. Suppose the new rate they qualified for was 1.74% compounded semi-annually and they could borrow $1,060,000 from the bank to cover their remaining mortgage balance and LOC debt. The new mortgage is amortized over 25 years, but they also need to pay a penalty for breaking the old mortgage early.
If the penalty is the interest differential over the remaining term of the old mortgage (under the old and the new rates), and if the penalty is also added to the new mortgage, what is the size of their new monthly payment?
[4:52 PM, 1/18/2022] Muthuri: Question:
5.12 A survey conducted by Certified Practising Accountants Australia to investigate the reaction to new audit standards issued by the Australian Auditing and Assurance Standards Board found that only 53% of auditors believed that the new standards had improved audit quality. Fifty-eight per cent of auditors felt that the standards had improved confidence in financial reporting. Assume that these two events (improved quality and improved confidence) are independent. Suppose 20 auditors from Australia are selected at random.
a. What is the probability that fewer than 15 of them believe that the standards have improved confidence in financial reporting?
b. What is the expected number of auditors who believe that the standards have improved confide...
[4:52 PM, 1/18/2022] Muthuri: Question:
Hassan and Dana had bought a property valued at $1,225,000 for 20% down and a mortgage amortized over 25 years on March 1, 2018. They made equal end-of-month payments towards their mortgage. Interest on the mortgage was 3.29% compounded semi-annually and the mortgage was renewable after five years.
What is the size of each monthly payment?
What is the cost of the mortgageforthe first 5years?
In November2020,theydecidedtorefinancetheirmortgagefortworeasons:rateswere down by quite a lot, and they also wanted to pay off some Line of Credit debt they had accumulated. Suppose the new rate they qualified for was 1.74% compounded semi-annually and they could borrow $1,060,000 from the bank to cover their remaining mortgage balance and LOC debt. The new mortgage is amortized over 25 years, but they also need to pay a penalty for breaking the old mortgage early.
If the penalty is the interest differential over the remaining term of the old mortgage (under the old and the new rates), and if the penalty is also added to the new mortgage, what is the size of their new monthly payment?
