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Section A: Answer ALL questions
1. Stochastic Permanent Income and Tax Shocks
Consider a problem of choosing a consumption sequence {ct} to maximise
U = Et
∞X
t=0
β
tu(ct)
subject to the budget constraint
ct + At+1 = (1 + r)At + Wt
,
where ct
is real consumption, β ∈ (0, 1), At
is financial wealth and Wt
is a (stochastic) real
disposable income. The real interest rate r is strictly positive and constant. Et
is the
conditional expectations operator. The instantaneous utility u(ct) is given by
u(ct) = α0ct
α1
2
c
2
t
,
where α0 and α1 are strictly positive and constant. In addition, assume that β(1 + r) = 1.
1
(Question 1 continued overleaf)
EC9D50
(a) Write down Bellman’s equation for this problem. (5 marks)
(b) Using first-order and envelope conditions derive the consumption Euler equation and
explain its meaning. (5 marks)
(c) Determine the consumption function, i.e. express the current consumption as a linear
function of financial wealth and non-financial wealth. (5 marks)
Now assume that the disposable income is given by Wt = ˉW(1 τt), where ˉW > 0 and τt
is
a (stochastic) income tax rate. The process of τt
is given by
τt = ρτt 1 + εt
where | ρ |≤ 1. The random term, εt satisfies: the mean E(εt) = 0 and variance V (εt) = σε
2
.
The shock is iid and unpredictable, i.e. Etεt+i = 0 for all i ≥ 1.
(d) Determine the consumption function. (5 marks)
(e) Discuss the effects of the parameter ρ on the consumption function. (5 marks)
2. Consider the following Solow model:
Output per worker is:
yt = f(kt) = Akt
1/2 + Bkt
,
where, yt
is output per worker and kt
is capital per worker, both in period t. B > 0 and
A > 0.
The saving rate is s ∈ (0, 1), the depreciation rate is δ ∈ (0, 1), and population growth is
n ≥ 0.
(a) Find kt+1 = φ(kt). (5 marks)
(b) Find φ
0 (kt) and a restriction on B such that a non-trivial steady state ˉk exists.
(5 marks)
(c) Suppose now that B < n + δ. Find the Golden Rule captal labour ratio k GR. (5 marks) Suppose now that A = 3, s = 1/2, n = 0 and δ = 1. In addition, the productivity parameter B is a function of kt . In particualr, B = BL if kt < 4, and BH if kt ≥ 4. (d) Find conditions on BL and BH such that the φ function has two localy stable steady states. (5 marks) 2 (Question 2 continued overleaf) EC9D50 (e) Find ALL the steady states for BL = 0 and BH = 1. For each steady state mention if it is stable or unstable. (5 marks) Section B: Answer ONE question 3. Additive Technology Consider an economy in which an homogenous good Yt can be either consumed, Ct , or invested, It : Yt = Ct + It . The good Yt is produced using the following technology Yt = Zt + aKt , where a > 0 and Kt
is the capital stock in period t. Zt
is a stochastic variable that evolves
according to
Zt = ρZt 1 + εt
where | ρ |< 1. The random variable εt satisfies Et 1εt = 0, where Et 1 is the expectation operator conditional on the information set given in t 1. The capital stock evolves according to the following law of motion: Kt+1 = (1 δ)Kt + It where δ ∈ (0, 1) is a constant capital depreciation rate. The representative household seeks to maximise the following intertemporal expected utility function: Et ∞X i=0 β iu(Ct+i) where u(Ct) = α0Ct α1 2 Ct 2 and α0 and α1 are positive real numbers. The parameter β ∈ (0, 1) is a subjective discount factor. Et is the expectation operator conditional on the information set in period t (i.e. when consumption decisions are made). (a) Determine the FOCs of the social planner optimisation problem. (5 marks) 3 (Question 3 continued overleaf) EC9D50 (b) We impose β(1 + a δ) = 1. Interpret this restriction. Show that the Euler equation on consumption has the following form (5 marks) Et Ct+1 = 0. (c) Compute the solution, i.e. express the choice variable Ct in terms of the pre-determined variable Kt and the exogenous variable Zt . (7 marks) (d) Compute the dynamic response of consumption, capital and output after a positive shock to the technology. Plot the responses. (4 marks) (e) Determine the value of the ratio σ c σ y when ρ = 1, where σ’s are the standard deviations. (4 marks) 4. Real Business-Cycle Model Consider the Real Business Cycle (RBC) model in which the representative household maximise the lifetime utility over consumption and leisure, given by max {ct,ht}∞ t=0 ∞X t=0 β t " c 1 σ t 1 1 σ + ψ (1 ht) 1 η 1 1 η # where ct is consumption at time t and ht is hours worked at time t. The budget constraint of the household is given by Kt+1 = (1 + rt δ)Kt + wtht ct . The firm produces output using Cobb-Douglas production function given by: Yt = AtKt αh 1 α t , (1) taking as given wage, wt , and rental rate of capital, rt . Assume that the initial capital stock, K0, is given and that exogenous technology process follows an AR(1) process given by At = ρaAt 1 + εt with ρa ∈ (0, 1) and εt ～ N(0, σε 2 ). (a) Explain the properties of the representative household’s preferences. What do parameters σ and η represent (3 marks) (b) Write down the Bellman equation for this problem. List the state and choice variables of this model. Explain. (3 marks) (c) Solve for the equilibrium conditions of this model. Carefully explain the intuition behind them. (3 marks) (d) Solve for the steady state of this model. Assume that β = 1/(1 + r). (4 marks) 4 (Question 4 continued overleaf) EC9D50 (e) Explain the effects of a one-time positive technology shock on consumption, hours, investment and output in this model. How does your answer depend on the value of the persistence parameter, ρa (4 marks) (f) Explain how you would simulate this model and compare it with the data. How would you then judge the performance of the model (4 marks) (g) Suppose that agents receive news that there will be a one-time positive technology shock several periods from now. Explain why these preferences cannot reproduce a comovement between consumption, investment, output and hours when the news is announced (4 marks) Section C: Answer ONE question 5. Consider an Endogenous R&D model. The economy exists one period and consists of N identical individuals. Each individual has one unit of time which is allocated to one of two sectors: the R&D sector producing innovations or to the production of the final good. Within the period, first innovations take place and then the final good is produced using the innovations. The fraction of individuals in the R&D sector is λ and the fraction in the production of the final good in 1 λ. Productivity in the production of the final good is a function of innovations. In particular, output produced by each individual is: y = A, where A is productivity, which is the sum of innovations used in production. The production of innovations is μ per worker, and thus: A = λNμ. (a) Find the efficient allocation λ and the resulting income per worker y. (5 marks) Suppose now that innovations are traded. In particular, workers in the R&D sector sell the right to use all innovation to all workers in the final good sector for the price of β per innovation per final good sector worker. In equilibrium income is the same in both sector. (b) Find the equilibrium fraction of workers in the R&D sector, λ, and the equilibrium income. (4 marks) (c) Is there a ”scale effect” in this model Explain briefly. (4 marks) (d) Explain the effect of β on income. In particular, explain what happens and why, when β → 0 and when β → 1. What is the optimal β (3 marks) (Question 5 continued overleaf) 5 EC9D50 Suppose now that the economy operates over many periods and that the production of innovations by each R&D worker is μAt 1, where At 1 is productivity in the previous period. New innovations replace old innovations and therefore productivity in t, is At = λNμAt 1. That is, output per final good worker, is: At = λNμAt 1. (e) Find the equilibrium fraction of workers in the R&D sector, λ, and the equilibrium income yt+1 in each period. Find the growth rate of the economy. (3 marks) (f) Explain the effect of many periods, in comparison to one period, on the efficient allocation in the economy. (3 marks) Suppose now that the production function of innovation is : A = √ λN, where λN is the number of R&D workers. The income of each R&D worker is the average income in the R&D sector. The price of using an innovation is β. (g) Find the equilibrium allocation λ and the income y. (3 marks) 6. Consider the following OLG model. Individuals live for two periods as in the Galor-Zeira model: when young they could invest in skills and in physical capital. When adults they work, and allocate a fraction β of their income as a transfer to their one offspring. Human capital, h i t+1, is an increasing function of investment, e i t . The production of human capital is subject to diminishing marginal product. In particular: h i t+1 = h(e i t ) = (e i t ) γ , where, h i t+1 is the number of efficiency units of labour supplied by individual i in t + 1, ei t is the investment in education by i in t, and γ ∈ (0, 1). The wage per unit of labour is w. Thus, the labour income of i in t + 1 is whi t+1. The return to physical capital is R. There is no borrowing by individuals for the sake of investment. Individuals invest in human capital and in physical capital so as to maximize their income as adults. (a) Find the investment in human capital by individual i as a function of b i t . First find the highest investment in human capital and denote it by ˉe. (5 marks) (b) Find income b i t+1 as a function of b i t (4 marks) (c) Find a restriction on the parameters that prevent endless growth. Under this assumption, how many steady states exist in the economy (4 marks) (d) Discuss the effect of wealth inequality on efficiency and on economic growth during the convergence to the steady state, within the specific model of this question. (4 marks) (e) Suppose now that the government taxes at a rate τ the income from physical capital. The revenue is equally allocated to all young individuals. Discuss the effect of the tax on efficiency and growth. Suggest an alternative more efficient tax policy. (4 marks) 6 (Question 6 continued overleaf) EC9D50 (f) Discuss the effect of taxing capital income on efficiency and growth, when a perfect loan market exists. What is the effect of the tax on aggregate welfare (4 marks)