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EC9D50
UNIVERSITY OF WARWICK
January Examinations 2024
Macroeconomics B
Time Allowed: 3 Hours, plus 15 minutes reading time
Read all instructions carefully – and read through the entire paper at least once before you start
entering your answers. (Please also note, further exam instructions are also available at this link:
Examinations Instructions).
There are THREE sections in this paper. Answer ALL questions in Section A, ONE question in
Section B and ONE question in Section C. All questions carry equal weight (25 marks each).
The number of marks available for a question will be stated at the end of each question.
Approved pocket calculators are allowed.
You should not submit answers to more than the required number of questions. If you do, we will
mark the questions in the order that they appear, up to the required number of questions in each
section.
Section A: Answer ALL questions
1. Consider the Real Business Cycle (RBC) model in which the representative firm produces
output using Cobb-Douglas production function given by:
Yt = At(utKt)
αh
1 α
t
taking as given wage, wt
, and rental rate of capital, rt
. The novelty in this exercise is the
utilisation rate of capital, denoted by ut
.
Assume that the initial capital stock, K0, is given and that exogenous technology process
follows an AR(1) process given by log At = ρa log At 1 + εt with ρa ∈ (0, 1) and
εt ～ N(0, σε
2
).
Household maximises the lifetime utility over consumption and leisure, with instantaneous
utility function given by:
U(ct
, ht) = ln ct + ψ ln (1 ht),
1
(Continued overleaf)
EC9D50
where ct
is consumption at time t and ht
is hours worked at time t. Households discount
future at the rate β, where 0 < β < 1. Assume that households own capital stock and choose the level of utilisation, and then rent capital services (product of utilisation and physical capital) to the firms at rate rt . More utilisation of capital comes at a cost of greater depreciation, which is captured by the following depreciation function: δ(ut) = δ0 + φ1(ut 1) + φ2 2 (ut 1)2 , where δ0 is the depreciation rate when capital is fully utilised, and value of the parameters φ1 and φ2 is such that it ensures utilisation is 1 in the steady state. (a) Write down the maximisation problem of the household for this model. (3 marks) (b) List the choice and state variables in this model. (5 marks) (c) Derive the equilibrium conditions for this model. (7 marks) (d) Suppose that in period t there is a one-time positive technology shock. Explain how hours worked, consumption, investment, capital, utilisation, output, real wage and real rate of return react to this unanticipated shock. In particular, how do these responses differ from those in the model without capital utilisation (7 marks) (e) What is the data counterpart of At in this case (3 marks) 2. Consider a Solow economy with the following production function of income per worker as a function of capital per worker: yt = f(kt) = A ln(1 + kt) The saving rate is s, population growth is n, and depreciation is δ. The number of workers in each period is Lt , where Lt+1 = (1 + n)Lt . It is a market economy in which factors are paid their marginal products. (a) Find the wage rate in the economy, wt , and the rental rate of capital Rt . (Remember that the derivative of ln x = 1/x). (4 marks) (b) Find the function governing the evolution of capital over time: kt+1 = φ(kt), and its properties: φ 0 (kt), limk→0 φ 0 (kt) and limk→∞ φ 0 (kt). (4 marks) 2 (Continued overleaf) EC9D50 (c) Which of the following is the one correct statement (As an answer just provide the i, ii, iii or iv) (i) The model predicts conditional convergence because the φ function is concave. (ii) The model predicts β global convergence, but not σ convergence, because limk→∞ φ 0 (kt) < 1. (iii) The model predicts σ conditional convergences, but not β convergence, because poor countries can overtake rich countries. (iv) The model predicts global convergence because limk→0 φ 0 (kt) > 1,
limk→∞ φ
0 (kt) < 1 and φ is concave. (4 marks) (d) Find the two necessary and sufficient conditions on the parameters such that there exists a steady state ˉk > 0. (4 marks)
Suppose now that δ = 1 and that n is endogenous: nt = γyt 1,
(e) Find ˉk. Find the growth rate of aggregate income in the steady state. (3 marks)
(f) Which of the following is the one correct statement regarding the equilibrium in part
(e) (As an answer just provide the i, ii, iii or iv)
(i) The equilibrium is Malthusian because income has a positive effect on population
growth, and the size of the population has a negative effect on income per capita.
(ii) The equilibrium is Malthusian because income has a positive effect on population
growth, and population growth has a negative effect on income per capita.
(iii) The equilibrium is Malthusian because income has a positive effect on population
growth, and the size of the population is endogenously determined by available
resources such that income is at subsistence.
(iv) Income has a positive effect on population growth, and the size of the population
has a negative effect on income per capita, but it is not a Malthusian model as
there is no constant factor of production.
(3 marks)
(g) Suppose now that individuals save all their income from capital and consume all their
income from labour. Suppose n = 0 and δ = 1. Find kt+1 = φ(kt), φ0 (kt),
limk→0 φ
0 (kt), limk→∞ φ
0 (kt). Find a sufficient and necessary condition such that there
exists a steady state ˉk > 0. (3 marks)
3
(Continued overleaf)
EC9D50
Section B: Answer ONE question
3. Consider the Real Business Cycle (RBC) model in which the representative household
maximises the lifetime utility over consumption and leisure, where instantaneous utility is
given by:
u(ct
, ht) = ln ct + ψ ln (1 ht),
ct
is consumption at time t and ht
is hours work at time t. Households discount future at the
rate β, where 0 < β < 1. 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 log At = ρa log At 1 + εt with ρa ∈ (0, 1) and εt ～ N(0, σε 2 ). Finally, assume that households own capital stock and choose the level of utilisation, and then rent capital to the firms at rate rt . (a) Write down household’s Bellman equation for this model. Explain what the two parts of the Bellman equation are. (2 marks) (b) Starting from the Bellman equation, derive household’s equilibrium conditions. (2 marks) (c) Explain the role of At in this model. (3 marks) (d) What is the data equivalent of At How would you go about calculating At from the available data (3 marks) (e) Explain how is the propagation introduced into this model. (4 marks) (f) Clarify the reasons why this model fails to replicate the observed low correlation between labor productivity and wages in the data. (5 marks) (g) Suppose that in period t there is a one-time positive technology shock. Explain how hours worked, consumption, investment, capital, output, real wage and real rate of return react to this unanticipated shock when ρ = 0 and when ρ = 0.9. (6 marks) 4 (Continued overleaf) EC9D50 4. A consumer lives for two periods and derives utility from consuming consumption ct , given by u(ct). The utility function is well behaved, in a sense that u 0 (c) > 0, u
00 (c) < 0 and limc→0 u 0 (c) = ∞. Do not assume any functional form unless explicitly told so. This consumer discounts future at the rate β ∈ (0, 1), and is endowed with the current and future income, W1 and W2. They can borrow and lend at the interest rate r. (a) Write down the consumer’s problem and explain how you would obtain consumer’s demand functions. (4 marks) (b) Represent this problem graphically and then, on the same graph, represent the effect of an increase in the interest rate. (4 marks) Now consider the case where income in the first period remains the same but income in the second period is uncertain in that it takes values W2 ε or W2 + ε with equal probabilities. (c) Write down the consumer’s problem and derive the consumer’s demand functions. (4 marks) (d) Explain under which conditions precautionary savings would arise in this model. (4 marks) Finally, consider the case where consumer lives infinitely, and where income is an exogenous stochastic process with the following law of motion: Wt = ρWt 1 + εt , where ρ ∈ (0, 1) is the persistence of the income process, and εt is an iid stochastic process with mean zero and variance σε 2 . (e) Write down the consumer’s problem and explain how you would derive the consumer’s demand functions. (4 marks) (f) Write down the demand functions assuming quadratic utility function and that β(1 + r) = 1. Does the process for consumption implied by this model hold in the data How would you test it (5 marks) (Continued overleaf) 5 EC9D50 Section C: Answer ONE question 5. Consider an economy in which income per worker is: yt = f(kt , ht) = Akt αh 1 a t , where kt is physical capital per worker in t, ht is human capital per worker in t, and α ∈ (0, 1). Physical capital fully depreciates after one period of production (δ = 1). Individuals live two periods in OLG (as in the Galor-Zeira model). In particular: in the first period of life they are young. They receive a financial transfer from their parent denoted bt (in period t). They do not consume, and invest bt in physical and/or human capital so as to maximize their income in the second period of their life (t + 1). Borrowing is not possible. In the second period they are adults. They allocate their income (from both physical and human capital) as follows: a fraction β to their one child and a fraction 1 β to their own consumption. All individuals are identical in each generation, so income per worker yt is equal to the income of each worker (adults). Suppose that each individual has an endowment of one unit of human capital: ht = 1, and investment in schooling to increase human capital isn’t possible. Investment is only in physical capital. (a) Find the equation governing yt over time: yt+1 = φ(yt). (3 marks) (b) List ALL the correct statements from the list below (just the letters of each chosen statements): (i) The model predicts no growth in the steady state because the marginal product of capital is decreasing to zero and δ > 0.
(ii) The model predicts no growth in the steady state because δ = 1. For δ < 1 there is endless growth. (iii) The model predicts endless growth because population growth is zero. (iv) A necessary condition for the model to predict any growth is that as kt approaches zero the marginal product of physical capital approaches infinity, for ht = 1. (3 marks) 6 (Continued overleaf) EC9D50 Suppose now that the young allocate a fraction γ ∈ (0, 1) of bt to investment in human capital and a fraction 1 γ to investment in physical capital. The production function of human capital, is the same as the production function of physical capital: every unit invested turns into one unit of capital: ht+1 = γbt and kt+1 = (1 γ)bt . (c) Find the equation governing yt over time: yt+1 = φ(yt), and find a condition on the parameters for which the model’s prediction is sustained economic growth. (3 marks) (d) Below is a list of the model’s assumptions. List ALL the assumptions that play an important role in the prediction of sustained economic growth at a constant rate (under some condition on the parameters). (i) All factors are accumulated (there is no fixed factor). (ii) The production function is CRS. (iii) Both the production of physical and human capital are not subject to diminishing marginal products. (iv) The allocation of investment between physical and human capital is a constant exogenous parameter (not optimally chosen by the young agent). (v) The young do not consume. (4 marks) Suppose now that γ is a choice variable by the young (remember that the young invest to maximize their income in their adulthood). (e) For α = 1/2, find γ. What is the economic condition for this maximum (4 marks) (f) Find the minimum level of productivity, A, as a function of β, that allows for sustained economic growth. (3 marks) (g) Suppose now that the production of human capital is subject to diminishing marginal product: ht+1 = (γbt) λ , where λ ∈ (0, 1). Assume here that γ is an exogenous parameter. Find the function yt+1 = φ(yt). Is there a set of parameters that allows for sustained economic growth Illustrate your answer with an equation. (3 marks) (Continued overleaf) 7 EC9D50 (h) Suppose now that bt is the average transfer but it is not the same for all the young. There is a perfect loan market: the young could lend to each other, repaying the loan when they are adults. The young optimally invest in human and physical capital (unlike above where γ is exogenous). Which of the following is the only correct statement (As an answer choose from i, ii, iii or iv): (i) Because of the inequality in the wealth of the young and the concavity of the production function of human capital, investment will be allocated inefficiently. (ii) Investment will be efficient and the gross interest rate will be equal to the marginal productivity of investment in human capital: Rt+1 = h 0 (et). (iii) Investment will be efficient and the gross interest rate will be equal to the marginal productivity of investment in human capital multiplied by the marginal productivity of human capital in final output: Rt+1 = f(kt+1, ht+1) ht+1 h 0 (et) (iv) The marginal product of physical capital will be smaller than that of human capital: f(kt+1, ht+1) ht+1 >
f(kt+1, ht+1)
kt+1
(2 marks)
6. Consider the Quality Ladder Model. The economy operates only one period. In that period,
first the R&D workers produce innovations and then the production of the final good takes
place.
Output produced by each worker in the final good sector is:
y = A,
where A is knowledge (inventions) purchased by the worker. Note that unlike the model in
the lecture, there are no existing innovations: without investment in R&D A = 0.
Thus, the income of a worker in the final good sector is output net of the cost of the licence
to use the innovations: βA, where β is the price paid to use each innovation.
The number of workers is N. They can work in the final good sector or in the R&D sector.
In the R&D sector, each worker produces μ innovations. So the total number of innovations
produced is μH, where H is the number of R&D workers, and L = N H is the number of
producers of the final good.
The equilibrium allocation of H and L is determined such that the income of workers is equal
in both sectors.
8
(Continued overleaf)
EC9D50
(a) Suppose the price of an innovation, β, is sufficiently low such that all innovations are
purchased by all producers of the final good. Find the equilibrium allocation of workers
to the R&D sector: H, the productivity A, and output per worker Y/N. (4 marks)
(b) Find the optimal H and the price β that would lead to an optimal allocation. (4 marks)
Suppose now that
(AL)
1/2
y =
L
(c) Find the efficient allocation of workers in the economy – the optimal H. (4 marks)
(d) What is the property of production (in the final good sector and/or in the R&D sector)
that explains your finding from part (c) above regarding the effect of μ on H
Choose one of the statements from the list below (as an answer provide the letter of the
correct statement; i, ii, iii or iv).
(i) There are no innovations that are not protected by property rights and production
in the R&D sector isn’t subject to diminishing marginal products.
(ii) There are no innovations from the past, and the elasticity of output with respect to
innovations is constant.
(iii) The linearity of the production of innovations and the lack of existing knowledge
from the past.
(iv) The linearity of the production of innovations, the constant elasticity of output with
respect to innovations, and the lack of existing knowledge from the past.
(4 marks)
Suppose now that
y = A
1/2
(e) Suppose the price of an innovation, β, is sufficiently low such that all innovations are
purchased by all producers of the final good. Find the equilibrium allocation of workers
to the R&D sector: H, and productivity A. (4 marks)
(f) For y = A1/2
, find the highest number of innovations Amax each worker in the final
good sector would purchase as a function of β.
Find the highest β, denoted β
max
, such that all innovations are purchased in equilibrium
by all workers (the allocation of workers to the two sectors is determined endogenously
by β
max, N and μ). Is the allocation of H efficient under β
max (5 marks)
(End)
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