Professor Xavier Presents... PROBLEM SET 2
ECON UN3213 - PROF. XAVIER SALA-I-MARTIN
THIS PROBLEM SET IS DUE ON WEDNESDAY 11th OCTOBER, BY 11:59 PM. YOU
SHOULD SUBMIT YOUR ANSWERS ON CANVAS BY THE DEADLINE. YOU CAN
EITHER TYPE YOUR ANSWERS OR SCAN THEM. MAKE SURE TO UPLOAD A
UNIQUE FILE. LATE PROBLEMS WILL NOT BE ACCEPTED.
Problem 1
Consider the Solow-Swan growth model, with a savings rate, s, a depreciation
rate,δ, and a population growth rate, n. The production function is given by:
𝒀 = 𝑨𝑲 + 𝑩𝑲𝟏/𝟐 𝑯𝟏/𝟒 𝑳𝟏/𝟒
where A and B are positive constants. Note that this production is a mixture of
Romer’s AK model and the neoclassical Cobb-Douglas production function.
(a) Express output per person, 𝑦 =𝑌/𝐿, as a function of capital per person, 𝑘 =𝐾/𝐿.
(b) Write down an expression for 𝑦/𝑘 as a function of 𝑘 and graph. (Hint: as 𝑘
goes to infinity, does the ratio 𝑦/𝑘 approach zero?)
(c) Use the production function in per capita terms to write the fundamental
equation of the Solow-Swan model.
(d) Suppose first that 𝑠𝐴 < d+ n. Draw the savings curve and the depreciation
curve. What number does the savings curve approach as 𝑘 goes to zero? As 𝑘
goes to infinity, the savings curve approaches a number: what number is that? Is
it zero?
(e) Under these parameters, will there be positive growth in the long run?
(Remember that A and B are constants). Why?
(f) What will be the growth rate of consumption and investment under these
Parameters in the long term?
g) Imagine that we have two countries with the same parameters (same A, B, s,
d, and n). One of them is rich and the other is poor. Which one of the two will
grow faster? Why? Will those two countries eventually catch up?
(h) Suppose now that 𝑠𝐴 > d+ n. Draw the savings and depreciation curves.
Under these circumstances, will there be positive growth in the long run? Why?
(i) If s = 0.52, A = 1.5, B = 1, d= 0.15, h*= 10000 and n = 0.2, the growth rate
converges to some value as time goes to infinity. What is this value?
(j) For these specific parameters, say that this country has a GDP (Y) value of $
300 billion. What is the level of consumption?
(k) What will be the GDP next year?
Problem 2
In class, we argued that if people could accumulate human as well as physical
capital, the production function would look like the “AK” production function.
(a) If the production function is AK and the savings rate is constant at rate “s”,
and the rates of depreciation and population growth are δ and n respectively, what
would the growth rate of the economy be?
(b) What would be the macroeconomic consequences of increasing the savings
rate in this economy? Explain using the model and intuitively.
(c) What would be the consequences of an increase in fertility in this economy?
Are these consequences good or bad? Is this answer unambiguous?
(d) How are human and physical capital different from one another in the way
they evolve from period to period?
(e) Does human capital have an upper limit? If it does, what is the resulting
production function when this is reached and the growth rate of the economy? If
it doesn’t have a limit, what is the resulting growth rate of output as it increases?
(f) Assume country A’s economic growth can be modelled by a neoclassical
growth model (i.e., all the neoclassical assumptions hold). Country A’s
government is looking to grow the economy. They claim that they can make it
grow forever by accumulating both human and physical capital at the same time
as this combination will allow them to avoid diminishing returns. Is country A’s
government correct? If so, why? If not, what is a better alternative? Make sure to
reference the Neoclassical assumptions to answer this question.
Problem 3
Consider the production function modelled by the following equation:
Y6 = A𝐾89:; (𝐵𝐻8 ); 𝐿@8
where Kt is capital, Ht is human capital, Lt is the number of workers, B is a scalar
larger than 0, A is the (constant) level of technology, 0 < a <1 and 0 < ε<1.
a) Does this production function satisfy all the neoclassical properties? Discuss
the meaning of each property intuitively and mathematically.
Imagine that parents invest in the human capital of their children up to the point
where the marginal product of physical capital, Kt, is equal to the marginal
product of human capital, Ht.
b) What is the relation between Kt and Ht? Use this relation to write down total
output as a function of Kt only.
In this economy, we will relax the assumption that every person is in the labour
force. Thus, the number of people, Nt, is different from the number of workers
(some people do not work). Let 𝑙𝑡 = 𝐿𝑡/𝑁𝑡 be the number of workers per capita
(the fraction of the population that works). Let 𝑦𝑡 = 𝑌𝑡/𝑁𝑡 be output per capita
and 𝑘𝑡 = 𝐾𝑡/𝑁𝑡 be capital per capita. Finally, let n be the rate of population growth
and γL be the growth rate of labour.
c) Using the “effective production function” you derived in (b), write down output
per capita, 𝑦𝑡, as a function of capital per capita, k𝑡, labour per capita, lt, the level
of population Nt, and the level of technology, A.
Following Solow and Swan, assume there is no government and no net exports,
that the depreciation rate of capital is the constant δ > 0 and the savings rate is
constant 0 < s < 1.
d) Derive the fundamental equation of Solow-Swan. How does the growth rate of
capital depend on employment per person, lt? Explain intuitively.
e) Does the equation of the growth rate of capital depend on the growth rate of
population, n, or the growth rate of employment? Explain intuitively.
Problem 4
a) In the lecture, Professor Xavier described steady states as “horrible places to
be.” Was this claim absolute or relative? Do countries having steady states that
are lower in the capital have worse living qualities?
b) Imagine a world where there are “n” countries with different levels of capital,
k. They all share the same saving level, birth rate, and the same GDP per capita
function. The only thing that differs is their depreciation rate.
In this world, how does the distribution of income in the short term and long term
look like? Think about the shape it would take.