# ECON Elements of Economic Analysis IV. Problem Set 1

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1 ECON Elements of Economic Analysis IV Problem Set 1 Due Thursday, October 11, 2012, in class 1 A Robinson Crusoe Economy Robinson Crusoe lives on an island by himself. He generates utility from leisure and the consumption of coconuts. Coconuts will not grow by themselves on this island so Crusoe has to work to grow coconuts. He has one unit of time to allocate between working and resting. Crusoe consumes all the coconuts he grows. If Crusoe decides to allocate l units of time to working, he will grow y f(l) =l units of coconuts. If he consumes c units of coconuts and rests for r units of time, he will obtain utility u(c, r) =c r 1. Obviously, r + l 1 and c = y. Crusoe makes his consumption and labor/leisure decisions optimally to maximize his utility. Exercise 1 Set up Crusoe s utility maximization problem and solve for his optimal consumption c and labor supply l. 8 [c] : >< c 1 (1 l) 1 = ; >: [l] :(1 )c (1 l) = l 1. ) l = L = c (1 l) 1 + (l c) 1+ 1 l ) 1 l = l1 1 ) 1=l 1+ 1 ) c = A Simple General Equilibrium A simple economy has 100 identical households and one firm. Focus on the households first. 1

2 Each household has one unit of time to allocate between working and resting. The wage rate for the labor the households supply is w and all households take w as given. Each household also has one unit of capital to rent out to the firm at rental rate r, which all households take as given. Each household also owns a 1% share of the firm, which means that each can claim 1% of the firm s profit. To sum up, each household has three sources of income, the labor income, the rental income, and the firm s profit. All households take the wage and rental rates (w and r) and the firm s profit ( ) as given. Each household generates utility from leisure and the consumption of apples. The price of apples is normalized to 1. Each household has to pay for the apples they purchase with their income as described above. If a household chooses to consume c units of apples and work for l units of time, it will obtain utility u(c, l) =c (1 l) 1. All households choose their consumption and labor/leisure to maximize their utilities. Exercise 2.1 Set up the households utility maximization problem and solve for their optimal consumption c and labor supply l as functions of w, r, and and the parameters. L = c (1 l) 1 + wl + r c ) 8 c >< = w + r + >: l =1 (1 ) ; r w w We now turn to the firm. The firm hires labor and rents capital from the households to produce apples. The firm s production technology is given by y F (L, K) =AL K 1, where y is the amount of apples produced if the firm hires L units of labor and K units of capital. A is the parameter representing the firm s productivity in transforming inputs into the output. The firm operates competitively in the sense that it takes the wage and rental rates and the price of apples as given. The firm s objective is to maximize its profit. Exercise 2.2 Set up the firm s profit maximization problem and calculate the first-order conditions as functions of w and r and the parameters. max F (K, L) wl rk ) {K,L} 8 < [L] :A L 1 K 1 = w; :[K] :A(1 )K L = r. Bringing the previous two parts together, we can solve for the general equilibrium in this simple economy. Exercise 2.3 Define the general equilibrium for this economy. 2

3 A general equilibrium of this economy is a combination of wage and rental rates w and r, consumption demand c, labor supply l, capital demand K, labor demand L, and the firm s profit such that: 1. (c,l ) maximizes households utility given wage and rental rates w and r and the firm s profit. 2. (K,L ) maximizes the firm s profit given wage and rental rates w and r. 3. L = 100l, K = 100, 100c = F (K,L ), and = F (K,L ) w L r K. Exercise 2.4 Solve for the general equilibrium. In particular, solve for the equilibrium wage (w ) and rental (r ) rates, the firm s profit ( ), and the aggregate amount of labor (L ) and capital (K ). First, calculate the firm s profit in equilibrium by imposing the first-order conditions of the firm s problem = AK (1 ) L w L r K = AK (1 ) L AK (1 ) L (1 )AK (1 ) L. Therefore, = 0. Now, divide the two first-order conditions and impose the market clearing conditions K = 100 and L = 100l. r w = 1 L 100 = 1 l = 1 r 1 (1 ) w +1 ) r (1 ) = w 1+. The goods market clearing condition implies that 100 (w + r )=A 100 (1 ) r w, which implies 1+ w = A 1 ) r = A (1 ) 1+ ) L = Exercise 2.5 Does the households labor/leisure and consumption decisions depend on the share of the firm s profit they are given? What is the key reason for your answer? No, they do not since the firm s maximized profit in this case is always 0. The reason for the zero profit is because the firm s production technology exhibits constant returns to scale. 3 G.E. Model with Lump Sum Taxes Exercise 3 Using exactly the same model as we developed in class, calculate the steady state level of capital assuming that the households choose consumption and investment instead of consumption and capital. 3

4 Start with period-t resource constraint in the economy: Y t = Y P t The law of motion for capital states + Y G t = K t + G = C t + I t + G ) K t =(C t + I t +(1 )G) 1/ K t+1 =(1 )K t + I t ) (C t+1 + I t+1 +(1 )G) 1/ =(1 )(C t + I t +(1 )G) 1/ + I t. This transformed law of motion for capital is our period-t constraint in the new Lagrangian. L = 8 [C t ]: >< t u(c t )+ >: [I t ]: µ t 1 (C t + I t +(1 )G) (1/ ) 1 = µ t µ t h (C t+1 + I t+1 +(1 )G) 1/ I t (1 )(C t + I t +(1 )G) 1/ i t u 0 (C t )+ µ t 1 (C t + I t +(1 )G) (1/ ) 1 =(1 ) µ t (C t + I t +(1 )G) (1/ ) 1 ; apple 1+ 1 (C t + I t +(1 )G) (1/ ) 1. To solve for the steady state, we have established in class that in the steady state, all variables must be constant over time. First, combine the two first-order conditions above and calculate the Euler equation, t u 0 (C t )= µ t ) t u 0 (C t ) t+1 u 0 (C t+1 ) = µ t µ t+1. Therefore, in steady state, µ t /µ t+1 =1/. Now, impose steady state to [I t ] and notice that (C t + I t +(1 )G) (1/ ) 1 =(K t ) (1/ ) 1 = K 1 t. ) µ t 1 Kss 1 = µ t K1 ss ) 1= just like what we found in class. 1 ss ) K ss = K 1 1+1/ 1/(1 ) 4 Taxation of Labor Income Consider the following one-period problem. There are two politicians and one household in this economy. The household can supply labor in a competitive labor market at wage rate w to receive income to purchase consumption goods. The price of consumption goods is 1. The household obtains utility from consumption and leisure with utility function 2 p c l, where c is the amount of consumption goods the household consumes and l the amount of labor it supplies. The household s labor income is subject to a proportional tax. However, the two politicians disagree over the tax rate. The first politician proposes a 1 = 50% tax rate while the second proposes an even more hefty 2 = 75%. Given any tax rate, the household will maximize its utility subject to its budget constraint that the expenditure over the consumption goods must be equal to its labor income after tax. Exercise 4.1 Let denote a generic labor income tax rate. Set up the household s utility maximization 4

5 problem and solve for its optimal consumption c and labor supply l as functions of and wage rate w. 8 >< 1 [c] : p = ; c >: [l] :1= w(1 ) L =2 p c l + (wl wl c) ) p c = 1 = w(1 ) ) c = w 2 (1 ) 2 ) l = w(1 ). Exercise 4.2 Calculate the household s maximized utility level v and the total amount of tax paid T as functions of and w, based on your results from the previous part. v =2 p c l =2w(1 ) w(1 ) =w(1 ) and T = wl = w 2 (1 ). Now, since this economy happens to exist in a fair democracy, the household has the right to decide which politician is to be elected. Suppose the proposed tax rate is the only di erence between the two politicians campaigns, the household really is choosing a tax rate when it is voting. The household knows that it will maximize its utility given the tax rate chosen by itself, regardless of which politician it elects. Exercise 4.3 If the household votes to obtain the higher level of utility, which politician will it vote for? Why? Obviously, the household s maximized utility level v is decreasing in so the first politician with lower tax rate at 50% will be elected. Exercise 4.4 If the household votes to pay the lower total amount of labor income tax, which politician will it vote for? Why? The amount of tax the household pays is a quadratic function in the tax rate with peak at = 50% which means that second politician will be elected with tax rate 75%. Specifically, T 1 = w >T 2 = w Suppose now the two politicians decide to merge their parties and tax the household s labor income with one single tax rate ˆ. In this new single-party political system, the politicians objective is to maximize the amount of tax revenue it can collect. On the other hand, the household s objective is to maximize its utility. Exercise 4.5 Consider the politicians and the household as two economic agents in one economy, define Pareto Optimum for this economy in terms of the tax rate ˆ. 5

6 A Pareto Optimum in this economy is represented by a tax rate ˆ such that no other tax rate can result in a higher level of utility for the household and a higher tax revenue for the government simultaneously. Exercise 4.6 What is the Pareto Optimal range of ˆ according to your definition of Pareto Optimum for this economy above? From our previous results, we know that the household s utility decreases with the tax rate while the government s tax revenue peaks at the tax rate 50%. Therefore, for any tax rate in the range of [50%, 100%], the household s utility can be increased and the government s tax revenue can be higher at the same time if the tax rate is lowered. Hence, no tax rate in this range can be a Pareto Optimum. Thus, the range ˆ 2 [0, 50%] is the Pareto Optimal range of the tax rate. Remember to confirm this statement directly according to definition of a Pareto Optimum we presented above. 5 Taxation and Transfer In this problem, we will examine the e ect and configurations of government taxation and transfer programs. Consider an economy populated by a measure 1 of consumers with identical preference. Each consumer has 1 unit of time to allocate between working and resting. Consumers care about consumption and leisure/labor according to the following utility function, u(c, l) =c (1 l) 1. Consumers have to work to obtain income so that they can purchase consumption goods. The price of consumption goods is normalized to 1. Consumers di er in the wage rate they can get when working. Among the measure 1 of consumers, an share of them are skilled workers with wage rate w a and the rest 1 share are unskilled ones with wage rate w b <w a. To achieve some distributional objective, which will be clarified shortly, the government decides to tax the labor income of the skilled workers with a proportional tax with rate and transfer the tax revenue directly to the unskilled workers in a lump sum. The government must balance its own budget, so the total amount of lump sum transfers to the unskilled workers must be equal to the total tax revenue collected from the skilled workers. Let v be the lump sum transfer received by a representative unskilled worker. Exercise 5.1 Set up the utility maximization problem of a representative skilled worker and solve for her optimal consumption c a and labor supply la as functions of w a and and the parameters. 8 L = c a (1 l a ) 1 + ((1 )w a l a < c a = (1 )w a c a ) ) : la =1 (1 )=. Exercise 5.2 How much tax is a representative skilled worker paying, given the tax rate being? What is her maximized utility? Let µ a denote this maximized level of utility (as a function of and w a and the 6

7 parameters). What is the value of µ a when = 0? Call this value µ a. t a = w a l a = w a. µ a =( (1 )w a ) (1 ) 1 =[(1 )w a ] (1 ) 1 ) µ a = (1 ) 1 w a. Exercise 5.3 Set up the utility maximization problem of a representative unskilled worker and solve for her optimal consumption c b and labor supply l b as functions of w b and v and the parameters. c b = (w b + v) and lb =1 (1 ) 1+ vwb. Exercise 5.4 What is the maximized utility of a representative unskilled worker given the lump sum transfer v? Let µ b denote this maximized level of utility (as a function of v and w b and the parameters). What is the value of µ b when v = 0? Call this value µ b. µ b = (1 ) 1 (w b + v)w 1 b ) µ b = (1 ) 1 w b. Exercise 5.5 Write down the constraint the government is facing to balance the budget in this tax-andtransfer program. w a l a (1 )v ) w a (1 )v Suppose now the government s purpose is to maximize the aggregate utility of the consumers. government treats each consumer equally, hence the aggregate utility of the consumers is simply The µ a +(1 )µ b. Exercise 5.6 Set up the government s problem and solve for the optimal tax rate that maximizes the aggregate utility of the consumers. 7

8 The government s problem is L = (1 ) 1 ((1 )w a ) +(1 ) (1 ) 1 (w b + v)w 1 b + ( w a (1 )v) Note that the constant (1 ) 1 does not a ect the solution so we ignore it in the transformed Lagrangian L = ((1 )w a ) +(1 )(w b + v)w 1 b + ( w a (1 )v) 8 < [ ] : w a (1 ) = w a ) wa 1 (1 ) :[v] :(1 )w 1 b = (1 ) ) = wb 1 1 = ; ) 1 = w b w a ) =1 w b w b = w a. w a w a Exercise 5.7 Calculate the maximized aggregate utility of the consumers and compare it to the aggregate utility of the consumers without the tax-and-transfer program ( = v = 0). What is the intuition behind the result of your comparison above? 8 µ a ( >< )= (1 ) 1 w b < µ a >: µ b (v )= (1 ) 1 w 1 b w b + (w a w b ) 1 > µ b. The intuition behind the comparison is that through the tax-and-transfer program, the government actually transfers utilities from the skilled workers to the unskilled ones via the format of transfer of income. In this particular example, the labor supply of the income tax is not distortionary toward the skilled workers labor supply, which implies that the income for the skilled workers must decrease after the taxation. On the other hand, any transfer to the unskilled workers will increase their full income, which necessarily leads to an increase in their maximized utility given the Cobb-Douglas structure of their utility function. The aggregate utility of all consumers is simply (1 ) 1 w b + (w a w b ) w 1. We do not even have to calculate the aggregate utility of the consumers without the program to know that this aggregate utility with the program will be higher with since is the tax rate that is maximizing aggregate utility and it is nonzero. Consider instead the situation where the government intends to maximize the aggregate consumption b c a +(1 )c b. Exercise 5.8 Set up the government s problem and solve for the optimal tax rate ˆ that maximizes the aggregate consumption. 8

9 max (1 )w a +(1 ) (w b + v) s.t. w a =(1 )v. {,v} Substitute the constraint into the objective function, max {v} w a (1 )v +(1 ) w b +(1 ) v. Note that now the objective function is strictly decreasing in v, meaning that the solution must be ˆ =ˆv = 0. 6 Social Security In this problem, we consider and compare two social security systems in an Overlapping Generation (OLG) framework. Consider an economy where all consumers live for two periods. All consumers only work and receive income in the first period of their lives. For simplicity, we assume the income of the workers are exogenously given. Each consumer s income depends on the generation she is in. For someone born in period t, wereferto her as being in generation t 0. Each consumer of generation t receives income y t in period t. Consumers of generation t generate utility from consumptions in periods t and t + 1. We assume a natural-logarithmic form for the period utility function for all consumers. All consumers, regardless of generation, discount the second period with respect to the first at rate. We use the symbol c t s to denote the consumption of a representative consumer of generation t in period s. Naturally, s can only be either t or t + 1. In short, for a representative consumer of generation t who consumes c t t and c t t+1, herutilityis given by ln c t t + ln c t t+1. To capture some of the fundamental features of the processes of output and population in reality, we assume consumers income grows at a constant rate g from one generation to another. That is y t+1 =(1+g)y t. Similarly, assume population grows at a constant rate n, i.e. N t+1 =(1+n)N t. n is mostly likely di erent from g. The income and population of generation zero (y 0 and N 0 ) are exogenously given. All consumers have access to a perfect credit market where they can save or loan at a constant interest rate r. Exericse 6.1 Set up the utility maximization problem of a representative consumer of generation t and solve for her optimal consumptions c t t and c t t+1. Calculate her maximized utility. Call this utility level v 1. 9

10 Exploiting the natural-logarithmic period utility function (which turns into Cobb-Douglas for lifetime utilities), the consumptions when young and old are simply c t t = y t 1+ and ct t+1 = y t(1 + r) ) v 1 =(1+ )ln 1+ yt 1+ + ln( (1 + r)). At first, the government proposes a fully-funded social security system which works as follows. The government taxes all consumers their income at a constant rate in the first period of their lives. It then invests the tax revenue in the credit market and return all proceed back to the same consumers when they are in the second period of their lives. Specifically, consider a representative consumer of generation t. Ifshe pays the tax of amount T t in period t, she will get back (1 + r)t t in period t + 1. All consumers still have access to the credit market where they can save or loan at the same interest rate r. Exericse 6.2 Set up the utility maximization problem of a representative consumer of generation t and solve for her optimal consumptions c t t and c t t+1. As shown in class, the fully funded system will simply result in the same solution for the consumers. ) c t t = y t 1+ and ct t+1 = y t(1 + r). 1+ Exercise 6.3 How much does a representative consumer save in a fully-funded social security system? This amount is what we call the private savings. For what rate is the private savings zero? Explain the intuitions. The private savings in a fully funded system is simply the di erence between the consumers after-tax income and the amount they consume when they are young, i.e. which will be equal to zero when s t =(1 )y t c t t = y t 1 = = 1+. The fully-funded pension system is essentially a form of enforceable private savings that requires consumers to save aside a certain portion of their income for retirement. Since the return on the pension is no di erent from that on the private savings at all, however much goes into the pension must be however much comes from the private savings. Hence, the tax rate at which the private savings will be zero is simply the portion of income consumers will set aside for retirement without a pension program and thanks to the Cobb-Douglas nature, this rate is /(1 + ). Suppose, instead, the government is considering an alternative pay-as-you-go social security system that works as follows. In any period t + 1, the government taxes the incomes of consumers of generation t

11 at rate and distribute the tax revenue equally among the consumers of generation t who are now in the second period of their lives. Notice that with this system, all consumers have some income in the second period of their lives even if they do not have any savings in the first period. This is a model that is supposed to mimic the actual social security system in reality where the young workers (consumers in the first period of their lives in our model) support the old retirees (consumers of an earlier generation now in their second period). Moreover, the government must balance its budget in every period. In period t + 1, the tax revenue to be collected will amount to y t+1 N t+1. Therefore, a representative consumer of generation t will receive y t+1 N t+1 N t in period t + 1 from the government. All consumers still have access to the credit market where they can save or loan as they will. Exericse 6.4 Set up the utility maximization problem of a representative consumer of generation t and solve for her optimal consumptions c t t and c t t+1. Calculate her maximized utility. Call this utility level v 2. With the pay-as-you-go system, the present discounted value of the income of generation t consumers is (1 )y t + y t+1n t+1 N t (1 + r). With this, the consumptions are easy to calculate with c t t = 1 1+ apple (1 )y t + y t+1n t+1 N t (1 + r) v 2 = ln (1 + )ln(1+ )+ln and c t t+1 = 1+ apple (1 )y t + y t+1n t+1 N t (1 + r) apple (1 )(1 + r)y t + y t+1n t+1 N t apple + ln (1 )(1 + r)y t + y t+1n t+1. N t Exericse 6.5 Compare the maximized utility level you obtain above in 6.4 (v 2 ) to the one in 6.1 (v 1 ). Under what condition, in terms of g, n, and r, isv 2 v 1? We will now compare v 1 and v 2. Focus on v 2 first and substitute in y t+1 =(1+g)y t and N t+1 =(1+n)N t, v 2 = ln (1 + )ln(1+ )+(1+ )lny t +(1+ )ln 1 + (1 + n)(1 + g) + ln(1 + r). 1+r 11

12 Therefore, the di erence between v 2 and v 1 is This di erence is positive if and only if v 2 v 1 =(1+ )ln 1 + (1 + n)(1 + g) 1+r (1 + n)(1 + g). 1+r > 1 ) gn + g + n>r. 12

13 ECON Elements of Economic Analysis IV Problem Set 2 Due Thursday, October 18, 2012, in class 1 A Simple Infinitely Lived Economy Suppose there exists an infinitely lived economy populated with identical households. A representative household in this economy derives utility from consumption of coconuts and leisure. If the representative household consumes c t and works for l t in time period t, it derives period utility of u (c t,l t )= c1 t 1 l t. Households discount the future at rate. Therefore, the discounted utility for a representative household with consumption and labor stream {c t,l t } 1 is given by! t c 1 t l t. 1 The wage rate, exogenously given, at time t is w t for all households and the price of coconut is normalized to 1 for all time periods. Households have access to a perfect credit market where they can save or borrow, along with the labor income they make, to fund their consumptions. The interest rate in the credit market is r for all periods. Let b t denote the savings a representative household makes in period t. Assume b 1 = 0. The households problem is to maximize the discounted utility subject to their budget constraint for all time periods. The government funds its expenditure solely with taxation. It taxes the households labor income at a constant rate. We now look at the e ect of this tax scheme on the households. Exercise 1.1 Write down the inter-temporal budget constraint for a representative household in period t. c t + b t =(1+r)b t 1 +(1 )w t l t. Exercise 1.2 What is the transversality condition for the household s lifetime budget constraint? What is the interpretation of the transversality condition? Transversality condition: lim t!1 b t (1 + r) t =0. The interpretation is that the household will not save in the infinite future an amount that has a positive present discounted value. Neither can the household fund its consumption by running a Ponzi scheme where the amount the household is borrowing increases exponentially with time at the gross interest rate. 1

14 Exercise 1.3 From the inter-temporal budget constraints, derive the household s lifetime budget constraint by imposing the transversality condition. t 1 c t apple (1 ) 1+r t 1 w t l t 1+r Exercise 1.4 Set up the household s utility maximization problem with the lifetime budget constraint you obtained above and calculate the first-order conditions. L = t c 1 t 1! ( t 1 l t + (1 ) w t l t 1+r 8 t 1 [c t ]: >< t c t = ; 1+r 1 X ) t 1 c t 1+r >: [l t ]: t 1 t = (1 ) w t. 1+r From now on, assume the wage rate is constant over time at w and the household s labor supply is also constant over time. Exercise 1.5 Using the first-order conditions obtained above, argue that the household s consumption of coconuts must also be constant over time. Calculate this constant level and call it c ss. Dividing the two first-order conditions above, we have c t =(1 )w t. So, if the wage rate is constant over time at w, consumption must also be constant over time at c ss =((1 )w) 1/. Exercise 1.6 Calculate the constant level of labor supply by the household, using the household s lifetime budget constraint and results you obtained in the previous parts. Call this level l ss. The present discounted value of the household s consumption is t 1 c ss = 1+r ((1 )w) 1/. 1+r r On the other hand, the present discounted value of the household s income is t 1 (1 ) wl ss = 1+r (1 )wl ss. 1+r r The household s lifetime budget constraint requires these two to be the same, which implies (1 )wl ss =((1 )w) 1/ ) l ss =((1 )w) 1/ 1. 2

16 ) c 1 +(1 )b 1 =(1+r)b 0 +(1 )w 1 l 1 ) c 1 +(1 )b 1 =(1+r)w 0 l 0 1+r 1 c 0 +(1 )w 1 l 1 which implies So the lifetime budget constraint is c r c 1 +(1 ) 1 1+r b 1 =(1 ) w 0 l r w 1l 1. t 1 t 1 c t =(1 ) w t l t 1+r 1+r with the transversality condition being Z 1 lim b Z =0. Z!1 1+r 2 Ricardian Equivalence Consider an economy where there is a single representative household and a government. The household earns an exogenously given income stream {y t } 1 and derives utility from consumption in a natural-logarithmic manner. It also discounts utility at a constant rate. The government has an exogenously given stream of expenditure {G t } 1 it has to make and it can only fund this expenditure stream lump-sum taxes on the household and debt. The government and the household both have access to a perfect credit market where they can borrow or lend at constant interest rate r. Let s start with the household s problem. Let b t denote the savings the household makes in period t and T t the lump-sum tax it pays to the government. The period-t inter-temporal budget constraint of the household is b t + c t + T t apple y t +(1+r)b t 1. The household s objective is to maximize its discounted utility given by t ln (c t ). Exercise 2.1 Write down the transversality condition for the household s lifetime budget constraint. Derive the household s lifetime budget constraint from its inter-temporal budget constraint, assuming b 1 = 0. Use Y to denote the present discounted value of the household s income stream and T the present discounted value of the lump-sum tax stream. The transversality condition for the household s lifetime budget constraint is lim t!1 b t (1 + r) t =0. With this condition, the household s lifetime budget constraint will simply be, by recursive substitution, c 1 t (1 + r) t + X T 1 t (1 + r) t = X y 1 t (1 + r) t ) X c t (1 + r) t + T = Y. 4

17 Let B t be the debt the government takes on in period t. Assume B 1 = 0. The government s intertemporal budget constraint in period t is B t + T t = G t +(1+r)B t 1. Exercise 2.2 Write down the transversality condition for the government s lifetime budget constraint and derive the government s lifetime budget constraint from its inter-temporal budget constraints. Use G to denote the present discounted value of the government s expenditure stream. What is the relationship between G and T? The transversality condition for the government s lifetime budget constraint is And the lifetime budget constraint is lim t!1 B t (1 + r) t =0. G 1 t (1 + r) t = X T t (1 + r) t ) G = T. Exercise 2.3 Set up the household s utility maximization problem and take the first-order conditions. Use the relationship you obtained above between G and T to substitute out the T in the household s lifetime budget constraint. L = t ln(c t )+ Y G [c t ]: t c t = (1 + r) t. c t (1 + r) t!. From now on, assume (1 + r) = 1. Exercise 2.4 Derive an expression for the value of the Lagrangian multiplier in the household s problem. (1 + r) = 1 implies that the first-order conditions can be converted into c t =1/. Substitute this condition into the household s lifetime budget constraint, Y G = 1+r r ) = 1+r r(y G). Exercise 2.5 Calculate the household s optimal consumption stream {c t } 1. Simply take the reciprocal of, c t = r(y G), 8t. 1+r Exercise 2.6 Now suppose that at time t 0, unexpectedly, the government decides to lower the lump-sum tax in period t 0 by while adjusting taxes and debts in periods after t 0 so that the government s lifetime 5

18 budget constraint still holds. Does c t 0 change and if so by how much? What about the savings the household makes in period t 0, b t0? Does Ricardian equivalence hold in this case? As we have solved above, the optimal consumption at t 0 will not change only the savings by the household in t 0 will change in the opposite direction to the tax adjustment by exactly the same amount and the Ricardian equivalence still holds since the transversality conditions for the household and the government are still assumed to hold. 3 Ramsey Taxation Consider instead the government proposes to fund its expenditure stream with a proportional consumption tax. In this problem, we will look at how this tax scheme works and how to decide the optimal tax rate. We look first look at the household s problem. Just as in the previous exercise, the representative household maximizes the discounted value of its utility stream that is solely determined by its consumption stream. In particular, the household s discounted utility is t ln(c t ). Unlike the previous problem, the household is now subject to a proportional consumption tax with timevarying tax rate t in period t chosen by the government. The household s income stream is exogenously given as in the previous problem. The household also has access to a prefect credit market where the interest rate is constant at r. Therefore, the household s period-t inter-temporal budget constraint is given by (1 + t ) c t + b t apple (1 + r) b t 1 + y t. Throughout the problem, we will assume consider convenient. (1 + r) = 1, so feel free to utilize this relationship wherever you Exercise 3.1 Derive the household s lifetime budget constraint from its inter-temporal budget constraints. Use Y for the present discounted value of the household s income stream. Just for clarity, we demonstrate several steps in the recursive substitution to obtain the household s lifetime budget constraint. Starting with periods 0 and 1. (1 + 0 )c 0 + b 0 = y 0 and (1 + 1 )c 1 + b 1 =(1+r)b 0 + y 1 =(1+r)(y 0 (1 + 0 )c 0 )+y 1. Notice that now the coe cient for c 0 is (1 + r)(1 + 0 ) instead of (1 + r). A couple more steps will yield the pattern for the household s lifetime budget constraint, c t (1 + t ) (1 + r) t = y t (1 + r) t = Y. Exercise 3.2 Set up the household s problem and calculate the first-order conditions. Solve for the optimal consumptions {c t } 1. L = t ln(c t )+ Y! c t (1 + t ) (1 + r) t 6

20 [ t ]: t 1+ t = ry(1 + t) t ry (1 + r) t+1 (1 + t ) 2 ) t = r 1+r Y 1 Exercise 3.7 Use the first-order conditions in 3.6 to conclude that the tax rate should be constant over time. Call this rate. This is the result we often refer to as tax smoothing. From the first-order condition above, we have seen that the optimal value of t only depends on, r, and Y, hence must be a constant over time. Exercise 3.8 Solve for the optimal with the results you obtained in r r G = r 1+r 1+ Y ) = G Y G. 4 Path of Government Debt According to the argument for tax smoothing, the ratio between the government s tax revenue to the output of the economy should be constant over time. In this problem, we will look at the evolution of government debt with di erent output and expenditure processes. In period t, the government s inter-temporal budget constraint is G t +(1+r)B t 1 = T t + B t, where B t is the amount of government debt in period t. From various previous exercises, we have concluded that, for infinitely lived governments, the present discounted value of the tax revenue stream must be equal to that of the government expenditure stream. We will utilize this piece of information along with the argument that the tax rate must be constant over time to calculate the process of government debt. 4.1 Constant output and time-varying expenditure Suppose the government s expenditure is given by G 0 =2,G 1 =1,G 2 =1,G 3 =1, That is, the government has to spend 2 at time 0 and 1 for each period afterwards. Moreover, assume that output {Y t } 1 is constant at Ȳ > 2. Recall the argument for tax smoothing: T t Y t must be constant over time, which naturally implies that the tax revenue must also be constant over time. Call this constant tax revenue T. Let interest rate be r>0. Also, assume B 1 = 0. Exercise 4.1 Derive the government s lifetime budget constraint. Your answer should be in terms of T and the interest rate r. 2r +1 r = T 1+r. r 8

21 Exercise 4.2 Calculate T. Calculate B 0 according to the following transformation of the government s period-0 inter-temporal budget constraint, B 0 = G 0 +(1+r)B 1 T. T = 2r +1 r +1 ) B 0 =2 2r +1 r +1 = 1 1+r. Exercise 4.3 Calculate B 1 through B 3 with the relationship that B t = G t +(1+r)B t 1 T. Do you see any pattern? Conjecture the process of government debt after period 3 and graphically represent your conjecture with time on the horizontal axis and government debt on the vertical axis. 8 >< B 1 = G 1 +(1+r)B 0 T =1+1 2r +1 r +1 = 1 1+r ; B 2 = G 2 +(1+r)B 1 T =1+1 2r +1 r +1 = 1 1+r ; >: 2r +1 B 3 = G 3 +(1+r)B 2 T =1+1 r +1 = 1 1+r. It is quite obvious that all the government debt for the periods after 3 will be constant at 1/(1 + r). Now, suppose the government s expenditure stream is G 0 =2,G 1 =1,G 2 =2,G 3 =1, That is, the government spends 2 in even-numbered periods and 1 in odd-numbered periods. The rest of the primitives stay the same as before. Exercise 4.4 Repeat 4.2 with this new expenditure stream. First of all, the present discounted value of the government s expenditure stream is 2 1 1/(1 + r) + 1 1/(1 + r) 2 1 1/(1 + r) 2 = 2r2 +5r +3 r 2 = +2r The government s lifetime budget constraint implies (2r + 3)(r + 1). r(r + 2) T 1+r r = (2r + 3)(r + 1) r(r + 2) ) T = 2r +3 r +2. B 0 =2 T = 1 r +2. Exercise 4.5 Repeat 4.3 with this new expenditure stream. 9

22 8 >< B 1 = G 1 +(1+r)B 0 T =1+ 1+r r +2 2r +3 r +2 = 0; B 2 = G 2 +(1+r)B 1 T =2+0 2r +3 r +2 = 1 r +2 ; >: 1+r 2r +3 B 3 = G 3 +(1+r)B 2 T =1+ r +2 r +2 =0. Thus, we can easily conjecture that the government debt will be zero in odd-number periods while 1/(r + 2) in even-numbered ones. 4.2 Constant expenditure and growing output Consider an alternative set of primitives of the economy: output is growing at a constant rate g<rwhile the government expenditure is constant at 1 for all time periods. From the tax smoothing result, we know that the tax revenue must also grow at the constant rate g to maintain the constant tax rate over time. In other words, T t = T 0 (1 + g) t. B 1 is still assumed to be 0. Exercise 4.6 Calculate the present discounted value of the government expenditure stream. Your answer should be only in terms of the interest rate r. G = 1+r. r Exercise 4.7 Calculate the present discounted value of the tax revenue stream. Your answer should be in terms of T 0, g, and r. 1 T = T 0 1 (1 + g)/(1 + r) = T 1+r 0 r g. Exercise 4.8 Recall that the government s lifetime budget constraint requires that the present discounted values of the government expenditure stream and the tax revenue stream are equal. Equate the answers you obtained in 4.6 and 4.7 and solve for T 0. T 0 = r g. r Exercise 4.9 Solve for B 0 with the T 0 you obtained in 4.8 and B 0 = G 0 +(1+r)B 1 T 0 =1 T 0. B 0 =1 r g r = g r. 10

24 In a world where economies evolve di erently, a flexible exchange rate may serve as an early information source about the disparities. With administratively fixed exchange rate, this signal is suppressed. The government may tend to uphold the fixed exchange rate for too long until imbalances accumulate, which then results in a crisis and the need for a rapid and large adjustment. Exercise 5.4 A problem with adjusting internal prices in response to external imbalances is their inflexibility. Describe how such an adjustment process operates when wages are inflexible. Why is such an adjustment undesirable? If deficits are to be counterbalanced by declining prices, a natural consequence is a decline in nominal wages (firms need to cut nominal costs since nominal revenues are decreasing). If wages are inflexible, the cost cutting proceeds through layo s, which results in unemployment. But unemployment is a pretty costly way of balancing the deficits. Exercise 5.5 Why is it not possible to use monetary reserves to finance large and longlasting imbalances in the foreign transactions? In case of a trade deficit, the use of monetary reserves is limited by the amount of foreign reserves. In case of a surplus, the government must be willing to accumulate foreign currency and provide domestic currency in a noninflationary way, if it wants to keep the price level steady. The government needs to sterilize the currency operations by issuing debt, thus withdrawing the domestic currency from circulation. These inflation prevention steps are however not likely to be desirable in the long run. References Friedman, Miltion, The Case for Flexible Exchange Rates, Essays in Positive Economics, University of Chicago Press, pp

25 ECON Elements of Economic Analysis IV Problem Set 3 Due Thursday, October 25, 2012, in class 1 A Simple Model of International Trade Suppose there are only two economies in the world, the U.S. and the E.U. Both economies have 100 units of labor to be divided between the production of two types of consumption goods A and B. Both countries gain utilities from the consumption of these two goods but maybe in di erent ways. Specifically, let CJ i be the consumption of good J 2{A, B} by economy i 2{U, E} (U for the U.S. and E the E.U.), 8 Û C >< A U,CB U = 1 2 ln CU A ln CU B, >: Ũ CA E,CB E = 1 3 ln CE A ln CE B. The productivities of the two economies in the two consumptions goods are summarized in the following table, where the number in each cell represents the number of units of labor required to produce one unit of the corresponding good in the corresponding economy. Good A Good B U.S. 1 1 E.U. 2 1/2 Exercise 1.1 Which economy has the comparative advantage in producing good A? The U.S. has the comparative advantage in producing good A. In this problem, we will go directly to the scenario where there exist some trade barriers between the two economies. Suppose both the U.S. and the E.U. impose a proportional tari of >0 on the goods they import. That is to say, for example, if the E.U. is importing good A from the U.S. and if the world price of good A is p A, the E.U. is paying (1 + )p A for each unit of A it imports. However, the U.S. is only getting p A for each unit it exports. Moreover, the tax revenue from the tari belongs to the E.U. in this case and it can freely spend this revenue on the purchase of the consumption goods. Let p A denote the world price of good A and p B that of good B. Assume from now on that both economies specialize in producing the goods they have comparative advantage in. Exercise 1.2 Which good does the U.S. specialize in producing? Set up and solve the utility maximization problem of the U.S. and obtain the optimal consumptions by the U.S. Your answer should include CA U and CB U as functions of p A, p B, and. Hint: Part of your answer should be CB U = 100p A /((2 + )p B ). The U.S. is producing good A, which means the income of the U.S. after selling all its product in the world market 1

26 is 100p A. Moreover, note that the U.S. also has the income from the tari revenue from taxing the import the E.U. made in good A. Let this revenue be T U. Now, notice that the U.S. can buy good A at price p A but good B from the E.U. only at price (1 + )p B. Hence, the U.S. problem is L =lnc U A +lnc U B + (100p A + T U p A C U A (1 + )p B C U B ). Utilizing the Cobb-Douglas structure of the utility function, we know that CA U = 1 100p A + T U and C U 1 B = 100p A + T U. 2p A 2(1 + )p B Now, what is T U? It is simply the tari rate multiplied by the price of A then by the amount of A that the E.U. imports. So how much A does the E.U. import? Since whatever gets produced must be consumed, CA E = 100 CU A.Thus,wehave CA U = 1 100p A + (100 CA U ) p A = 100(1 + ) CU A 2p A 2 ) C U A = Solving for C U B becomes simple with the fact that p A 100 C U A = p A 100/(2 + ). ) C U B = 100p A(2 + 2 ) 2(2 + )(1 + )p B = 100p A (2 + )p B. 100(1 + ). 2+ Exercise 1.3 Which good does the E.U. specialize in producing? Set up and solve the utility maximization problem of the E.U. and obtain the optimal consumptions by the E.U. Your answer should include CA E and CB E as functions of p A, p B, and. This follows exactly the same process as in the previous part except for we want to start with good B, CB E = 2 200p B + p B (200 CB E ) ) C E 400(1 + ) B = ) CA E = 200p B. 3p B 3+2 (3 + 2 )p A Exercise 1.4 Use your results from the previous parts and the market clearing conditions in the markets of the consumption goods to solve for the equilibrium relative price between A and B (p A /p B ) in terms of. C U A + C E A = 100 = 100(1 + ) p B (3 + 2 )p A ) p B p A = 3+2 2(2 + ). Exercise 1.5 For what range of will the current specializations be maintained? For the U.S. to specialize in A, p A p B > 1 ) 3+2 <4+2. Hence, any will work for the U.S. to remain producing A only. For the E.U. to specialize in B, p B p A > 1 4 ) 6+4 >2+ ) > 4 3. Since the tari rate only makes sense when positive, so any positive tari rate will work. Exercise 1.6 Within the range you calculated in 1.5, how do the maximized utilities of the U.S. and the E.U. and the sum of the two utilities change with respect to? 2

27 Start with the U.S. C U A = 100(1 + ) (2 + ) and C U B = Therefore, the direction in which the maximized utility of the U.S. is changing with respect to will be the same as the one in which (1 + )/((2 + )(3 + 2 )) is, which leads to d(1 + )/((2 + )(3 + 2 )) d For the E.U., applying the same approach, the key quantity now is = (1 + ) 2 (2 + )(3 + 2 ) 2. (1 + 2 ) 2 (2 + ) 2 (3 + 2 ) 2 < 0. It turns out this key quantity will increase with respect to when 0 apple apple ( p 17 3)/2 and decrease afterwards. Exercise 1.7 What is the that will maximize the sum of the maximized utilities of the two economies and how does your conclusion relate to the potential cost of trade barriers. When combining the two economies, the quantity of interest to determine the e ect of the tari becomes apple apple 1+ (1 + 2 ) 3 ln +2ln (2 + )(3 + 2 ) (2 + )(3 + 2 ) 2. The sign of derivative of this quantity with respect to is the sign of < 0. Thus, the combined utility of the two economies is always decreasing for positive tari rate. This demonstrates the argument that although tari could benefit some economies while hurting the others, the gain for the benefited economies can never o set the loss in utility incurred on the harmed economies. The cost of trade barrier exhibits itself in moving the world economy away from Pareto optimal allocations. 2 Stochastic Money Demand with Perfect Foresight Suppose in an economy, the real money demand follows the stochastic di erence equation m d t p t = E t [p t+1 p t ], where >0 is a constant parameter, p t the natural-logarithm of the price level in period t, and m d t the naturallogarithm of the nominal money demand also in period t. Let m s t denote the natural-logarithm of the nominal money supply, chosen by the central bank, in period t. Exercise 2.1 What does the parameter represent? Justify your answer. represents the semi-elasticity of real money demand with respect to expected inflation. To see this, notice that = d ln M d t /P t de t [ln(1 + t )]. We know that the derivatives of natural-logarithms represent elasticities while in this case the denominator is really approximately t instead of ln t, which makes the the semi-elasticity. Exercise 2.2 What is the equilibrium condition in the money market? 3

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