Instructions for this Section: Answer question 1 OR question 2

Y = K1/3(AL)2/3                                                                                (1)

where Y is output, K the capital stock, L the labour force and A (A > 0) is an index of productive efficiency. A increases at 2% per year, the labour force L increases at 1% per year and K depreciates at 5% per year. Capital accumulation is given by

K(.) = 0.24Y − 0.05K                                                    (2)

where K(.) =  .

(a)  Use eq. (1) to show that the average product of capital APK (APK = Y/K) is positive, diminishing in K and increasing in A and L.

9 marks

(b)  Now show that APK can only remain constant over time when K is growing at exactly 3% per year.

Is this result intuitive? Explain your answer.

9 marks

(c) Starting with eq. (2) and using your answer to part (b), show that APK must converge to a steady state value APK* = 1/3.

10 marks

(d)  Does this value look realistic to you? Explain your answer.

5 marks

Now suppose at t = 0 the growth rate of A falls to 1% and remains at that value from then onward.

(e)  Represent the impact of this shock on a “standard growth diagram”, with  = K/(AL)

9 marks

(f) Write output per worker, y(t) =  (t), as a function of A(t) and APK(t) only, and then explain what happens to APK(t) and to output per worker y(t) from t = 0 onward.

8 marks Total: 50 marks

2.  [50 marks] Consider an economy composed of two overlapping generations of households that live for two periods. Formally the decision of a working household born at time t is to

max     Ut = lnc1t +  lnc2t+1 − lt(2)

subject to the following lifetime budget constraint

c2t+1

with Ut the lifetime utility of the household born at t; r the real interest rate; c1t (c2t+1) con­ sumption at t (t +1) by the household born at t; 山lt labour income, the product of the wage per unit of labour 山 and lt, the amount of labour supplied by the household at t; and a the labour income tax rate (0 想 a 想 1).

(a)  Use a Lagrangian function to show that the solution to the individual’s consumption choices (c1t and c2t+1), work (lt) and savings (denoted at+1) is given by

c1(*)t =  (1 − a)山

l*t = 

at(*)+1 =  (1 − a)山

c2(*)t+1 =  (1 − a)山(1 + r)

13 marks

(b) Simple inspection of the household solution shows that in this economy a affects savings negatively but it has no impact on work. Explain why this is the case.

11 marks

From now onward assume that             and           .

(c)  Use the solution to the working household’s maximisation problem given in part (a) to show that their optimal lifetime utility Ut*  is negatively affected by a as follows:

Ut* = A ln(1 − a)+ B,              with A and B two constants.

8 marks

Now suppose that each generation is composed of two types of households, working house­

holds, described and behaving as above and making up proportion 1 − T of all households,

(T −1 − 1)a . The benefits are effectively just a transfer payment received from the govern­

are now assuming that 山 = 1). Each on­benefits household born at time t has to

max    UtB = ln c +  ln c+1

c1(B)t;c2(B)t+1

subject to

c +  = (−1 − 1)a

and this results in the following optimal consumption decisions and lifetime utility:

ct∗  =    (−1 − 1)a   ct =  (−1 − 1)a

U∗  =    lna+ln [ (−1 − 1)] +  ln [ (−1 − 1)]

Note: you are not being asked to show this.

(d) The government sets the labour income tax rate ato maximise the lifetime utility of the average household in the country, taking as given factor prices  and r.  Formally, the government’s decision is to

me(a)x wt = (1 − )Ut∗ + U∗

(3)

Show that the optimal choice of adepends on , the proportion of on­benefits house­ holds. Briefly explain the intuition of your result.

11 marks

(e)  Earlier on we showed that a, which is in this case being used purely for redistribution of income across households, would reduce the savings of working households. Shouldn’t

the government take that effect into account when maximising wt in eq. (3)? Explain your answer.

7 marks Total: 50 marks

Instructions for this Section: Answer question 3 OR question 4

3.  [50 marks] An individual chooses a consumption plan for c1 , c2 , ..., cT  to maximise her ex­ pected lifetime utility

T

U1 = E1 [X ps −1u(cs )] = E1  [u(c1 )+ pu(c2 )+ p2 u(c3 )+ ...¤

s=1

where

u(cs ) = (cs  − cs(2))

(4)

(5)

(s = 1, ...,T), p is a parameter (0 想 p ≤ 1) and E1  is the expectations operator conditioned on time t = 1 information, subject to the following lifetime budget constraint

T

 E1 cs  = c1 + E1 c2 + E1 c3 + ... = E1 w1

(6)

where E1 w1 denotes their expected lifetime wealth at t = 1 and T is the real interest rate and

is such that                        .

(a)  Find if the individual’s planned consumption profile is upward sloping, downward sloping, or horizontal. Briefly explain the intuition for your result.

8 marks

(b) Solve for the optimal level of consumption in period 1.

8 marks

(c) The (per­period) utility function in equation (5) does not allow for precautionary savings behaviour. However, had the utility function been u(cs ) = lncs , this would have resulted in precautionary savings behaviour. Explain why this is the case.

10 marks

(d) The government is considering a redistributive policy which involves temporarily increas­ ing taxes on the rich to finance cash transfers to the poor. Drawing from your answers to parts (b) and (c), discuss how this policy will affect aggregate consumption in both the CE­LC­PI model and the Modern lifecycle (MLC) model.

14 marks

(e) The US government has recently announced a fiscal package equivalent to 10% of GDP. A substantial part of this package takes the form of a one­time cash transfer to all households. Explain, using CE­LC­PI and MLC models, how these measures are likely to affect households’ consumption and savings decisions.

10 marks Total: 50 marks

4.  [50 marks] The central bank minimises the following loss function Lt  = a北t(2) +(Tt − T)2

subject to a Phillips curve

Tt  = Et Tt+1 + 入北t + ut

where 北t  is the output gap at time t; Tt  and  represent actual inflation and the positive inflation target, respectively; ut  is an exogenous cost­push shock; and Et is the expectations operator conditioned on time­t information (this includes the values of all variables and shocks up to time t). Assume that a and 入 are positive parameters.

(a)  The central bank will conduct monetary policy such that the following equation is satis­

fied:                                                                      a

入

Interpret this result in the context of a cost push shock.

7 marks

(b)  Explain using a graph on (北t ,Tt )­space how an increase in society’s aversion to inflation will affect the central bank’s response to a cost­push shock?

7 marks

Now suppose that the economy is well­described by the following 3­equation model where the central bank conducts monetary policy according to an interest rate rule and under perfect information:

北t     =   Et 北t+1 −  (Tt − 0.02)

Tt     =   Et Tt+1 +  北t + ut

Tt     =   0.02 + Et (Tt+1 − T)

where Tt  is the real interest rate while all other variables and parameters are as defined above. Moreover, ut follows an AR(1) process of the form:

ut  = ut −1 + t

with {t } ∼IID(0,auˆ(2)).

(c)  Show that this 3­equation model can be reduced to a system of two difference equations in 北t  and Tt  and then check, using either of these, that the solutions for 北t  and Tt  are given by:

北t     =   − ut

Tt     =   T + ut

16 marks

Equation (7) below represents a (log­linearised) aggregate consumption Euler equation (af­ ter imposing goods market equilibrium) for a representative agent model (RANK) where all households have access to financial markets and engage in consumption smoothing. In con­ trast, equation (8) represents the equivalent condition for a model with two types of house­ holds (TANK): a fraction Λ of households are hand­to­mouth, whereas the remaining house­ holds engage in consumption smoothing as in the representative agent model.

RANK :         北t = Et北t+1 − a(rt − r )n                                                      (7)

TANK :         北t = Et 北t+1 − 1 − XΛa(rt − r )n

(8)