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ECON 6002/6702 capital and to labour

创建日期：2024-05-13 14:55:17

所属分类：经济Economics

标签： ECON 6002

capital and to labour

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Problem Set 1 (Solow-Swan, Ramsey, Diamond Models)

ECON 6002/6702

NOTE: To receive full marks, show your workings for algebraic manipulations.
1. Suppose the aggregate production function in the Solow-Swan model is Cobb-Douglas, y =
kα, with α = 0.3. Assume population growth n = 2%, technology growth g = 3%, and
depreciation δ = 10%.
(a) Derive k∗, y∗, and c∗ as functions of the model parameters and determine their values
when the saving rate s = 15%.
(b) Assume both labour and capital are paid their marginal products and the economy is
on a balanced growth path at time t = 0:
i) What is the real wage w(0) if A(0) = 1?
ii) What is the growth rate of wages w˙/w?
iii) What is the return to “working” capital r? (working refers to capital minus depre-
ciation)
iv) What are the shares of income going to (both “working” and “dead/depreciated”)
capital and to labour?
(c) How would your answers to part (b) change if the saving rate were s′ = 30% and the
economy was on the balanced growth path?
(d) Draw the transition given a change in the saving rate from s = 15% to s′ = 30% in the
basic diagram for the Solow model (You may use MATLAB to make this diagram if you
like).
(e) What are the growth rates of real wages, w˙/w, and the return on working capital, r˙/r
at the beginning of the transition when k = 1? What do these results predict about real
wage growth and the return on working capital as an economy with a high saving rate
such as China gets closer to the new steady state?
(f) Can the economy achieve a higher c∗ than for s = 30%? Why or why not?
2. Now consider the Ramsey model with a Cobb-Douglas aggregate production function, y = kα
and α = 0.3. Assume the discount rate ρ = 5%, population growth n = 2%, technology
growth g = 3%, and there is no capital depreciation.
(a) Derive k∗, y∗, and c∗ as functions of the model parameters and determine their values
when the coefficient of relative risk aversion θ = 5.
(b) How do your answers to part (a) change if the coefficient of relative risk aversion changes
to θ = 2 (i.e., the intertemporal elasticity of substitution rises from 0.2 to 0.5)?
(c) Draw the transition given the change in the coefficient of relative risk aversion in the
phase diagram for the Ramsey model.
(d) For this economy, what is the impact of a permanent fall in the growth rate of technology
on the saving rate along the balanced growth path? How does your answer depend on the
intertemporal elasticity of substitution? Hint: calculate ∂s∗/∂g, where s∗ = 1− c∗/y∗.
(e) What happens to s∗ given a permanent fall in the growth rate of technology to g = 2%
under both scenarios of θ = 5 and θ = 2?
3. Consider the Diamond model with logarithmic utility (ln(Ct)) and Cobb-Douglas production.
Describe in words and using equations how the following affects kt+1 as a function of kt:
(a) A fall in n.
(b) A downward shift in the production function (that is, f(k) takes the form Bkα, and B
falls).
(c) A rise in α
4. The Covid Economic Slump: Let’s consider the economic impact of the pandemic on the
economy. Assume that the pandemic evolves according to the basic SIR model:
S˙ = −βS(t)I(t) (1)
I˙ = βS(t)I(t)− γI(t) (2)
R˙ = γI(t) (3)
with initial conditions S(0), I(0), and R(0), satisfying
S(0) + I(0) +R(0) = 1
and
S˙ + I˙ + R˙ = 0⇒ S(t) + I(t) +R(t) = 1
for all t ≥ 0. Assume that some fraction 0 < ϕ ≤ 1 of infected people are too sick to work.
This implies that during the pandemic the effective labour force E(t) is given by
E(t) = (1− ϕI(t))L¯,
where we assume the labour (L¯) is fixed and does change over time. Finally, assume that
production is Cobb-Douglas such that
Y (t) = K¯α(A¯E(t))1−α,
where capital is fixed (K¯) and technology (A¯) are fixed.
(a) Find an expression for the growth rate of output in terms of the growth rate of infections
gI = I˙/I(t). Is output growth positively or negatively related to infection growth?
(b) Let K¯ = A¯ = L¯ = 1, α = 0.33, ϕ = 0.5, S(0) = 1, and the solution path for infections
be
I(t) = 1− S(t) + 1R0 ln(S(t)/S(0)).
Create a MATLAB graph that plots output against the percentage of the population
that is susceptible. Compare R0 = 2.5 to R0 = 1.4, where the latter is the average R0
of seasonal influenza. Assuming ϕ is the same for Covid and influenza, is Covid’s impact
on the economy similar to influenza in this model? (Tip: set the axes to go no higher
than 1, e.g. axis([0 1 0.5 1]) so that only values of I(t) ≥ 0 are shown.)
(c) Using your MATLAB program, consider whether it is possible to save lives and the
economy by practicing social distancing. Assume that you can lower R0 (lower β) by
lowering L¯. Create some graphs that show that it is possible to actually lose less output
to the pandemic by forcing some people to stay home. For example, show that if we
reduce L¯ to 0.95, i.e., 5% of the labour force is made to stay home, but that by eliminating
those jobs the R0 falls to 1.4, it is possible that the recession is less and we save lives.
(d) Critique your answer to (c). What assumptions does it rely on?

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