Professor Debraj Ray
19 West 4th Street, Room 608
Office Hours: Mondays 2.30–5.00pm
email: debraj.ray@nyu.edu, homepage: http://www.econ.nyu.edu/user/debraj/
Webpage for course: click on “Teaching”, and then on the course name.
Econ-UA 323
Answers to Sample Midterm Examination
What follows is an outline of the answers. The main theme is that I need my answers brief
and simple, and to the point. Just writing a lot won’t necessarily get you more credit, but
also write enough to make your point clearly.
(1[a]) The income of poor countries is often understated when using the exchange-rate
method of measuring per-capita income, as opposed to the PPP method.
True. In general, the exchange rate equates the prices of traded goods across two countries,
and in particular nontraded goods are left out. But there is a systematic reason why the
nontraded prices should be higher in a rich country: people have more money to spend and
this should drive up nontraded prices relative to a poor country. That means that a PPP
estimate, which tries to control for the overall price level, will generally favor poorer countries.
[You could add examples of nontraded goods, or countries which are scaled up or down as a
result of the PPP correction.]
[b] The Solow model predicts that a higher savings rate is incapable of leading to sustained
long-run per-capita economic growth, in the absence of technical progress.
True. The Solow model assumes diminishing returns to the capital input, so that per-capita
capital and per-capita output converge to a steady state.
(At this point, a diagram of the Solow model and/or the writing of the equations would get
credit.)
Therefore an change in the savings rate will increase output, but sustained long-term growth
would be zero (per-capita) or at the rate of population (overall), with per-capita output at
a permanently higher level.
[c] A production function that exhibits diminishing returns to every input cannot exhibit
increasing returns to scale.
False. A counterexample using Cobb-Douglas production technology in which the coefficients
α and β are both less than one, but their sum α + β exceeds 1, gets perfect credit. [You
should write down exactly what the production function is for full credit.]
[d] If a country with population growth saves a positive amount from its income, and there
is no depreciation of capital, then the per-capita capital stock must rise over time.
1
2
False. Write the equation for per-capita capital stock accumulation, and show how the percapita capital stock might fall when the population growth rate is positive.
[e] The example of complementarities discussed in class (QWERTY versus Dvorak) has a
unique equilibrium, because everybody takes the same action in the end.
False. Draw a diagram (label axes) and show that the QWERTY-Dvorak example has three
equilibria. In two of these all agents take the same actions as one another, but that has
nothing to do with the uniqueness of equilibrium.
[f ] The Gini is Lorenz-consistent, so if the Gini coefficient for country A exceeds that for
country B, the Lorenz curve for A must lie below and to the right of that for B.
False. The Lorenz-consistency of the Gini means that if the Lorenz curve for A lies below
and to the right of that for B, then the Gini for A is higher. But the converse is not true!
After all, the Gini provides a ranking even when the Lorenz curves cross.
[If you could give an explicit numerical example that would get you extra credit, but this is
not necessary.]
(2a) [5 points] The success and jail probabilities have been written as specific functions of
n. Explain what intuitive features these functions are trying to capture.
The success probability function is one way of showing that the probability of success is increasing in the fraction of the population which participates in the uprising. The jail function
does the opposite: your chances of incarceration go down with the number of participants.
The perfect answer should also pay attention to the end-points: when n = 0 the success
probability is zero, and the chances of going to jail are perfect. When n = 1 the rebellion
wins for sure, while the chances of going to jail (even conditional on the rebellion itself failing), are zero: there are too many participants out there and the despot can only get an
infinitesimal fraction of them.
(b) [10 points] Assuming that each person tries to maximize her expected return by choosing
whether to participate or not, describe the equilibria of this model.
Write down the net payoff from participation as a function of n. It is
f (n) = n − 2(1 − n)2
and this needs to be nonnegative to induce participation. (The (1 − n)2 comes from the fact
that the revolution has to fail first, and conditional on that there is a 1 − n probability of
being jailed.)
One equilibrium is where n = 0, another where n = 1 (you should show that the implied
payoffs from participation are negative and positive respectively). You should argue that the
function f (n) is increasing. Extra credit if you argue that there is an interior equilibrium
at n∗ satisfying f (n∗ ) = 0: at this point people are indifferent between participating and
not. Any more participants creates a move towards n = 1, any less towards n = 0. You can
calculate n∗ = 1/2.
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(3a) [4 points] All countries converge to some common level of income by 2020. Describe
the possible mobility matrices consistent with this.
This will be the matrix with the middle column populated by 100 everywhere, and zeros
everywhere else.
(b) [5 points] In 2020, no country stays in the same category from which it started. Describe
the possible mobility matrices consistent with this.
This will be the set of all matrices with zeros along the principal diagonal. (You should write
down a particular example as well.)
(c) [6 points] Can the following be valid mobility matrices for the three categories described
above? Explain why or why not:






100 0 0
25 75 0
25 60 15
[I]  100 0 0 , (II)  0 100 0 , (III)  0 100 0 .
100 0 0
0 100 0
20 40 60
(I): No, because all countries cannot have less than half the world average. (II): Yes, it is
true that some countries are less than half the world average but that can be balanced by the
other countries being (for instance) somewhat above the world average. (III): No, because
the entries in the bottom row sum to over 100%.
The Two Extra Questions
(4) The economy of Lorenzia has a population of three individuals with wealth levels W1 ,
W2 , and W3 , where W1 < W2 < W3 .
(a) [3 points] Suppose that the person with wealth level Wi earns an annual income of yi , and
saves a fraction si of it. If the rate of interest on asset holdings is r, write down a formula
for this person’s wealth next year.
Wealth next year is
Wi0 = (Wi + si yi )(1 + r).
(b) [4 points] Assume that the income of each individual bears a constant ratio to wealth
(that is, yi /Wi is the same for all i), and that the savings rate is the same across individuals.
How does the Lorenz curve for wealth next year compare with its counterpart for the current
year?
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Take any two persons i and j, and suppose that Wi = λWj . Then the ratio of wealth next
year is given by
Wi0
Wj0
=
(Wi + si yi )(1 + r)
(Wj + sj yj )(1 + r)
W i + s i yi
Wj + sj yj
λWj + sj (λyj )
=
Wj + sj yj
= λ,
=
which means that the ratio of the two wealths next year is the same as the ratio of the two
wealths this year. But the Lorenz curve is insensitive to the uniform scaling of all wealths
and must therefore be unchanged.
(c) [4 points] Retain the same assumptions on income and wealth as in part (b) but now
suppose that the savings rates satisfy s1 < s2 < s3 . Now how does the Lorenz curve for
wealth next year compare with its counterpart for the current year?
Try the same logic as above. Take any two people i and j with i poorer than j. The same
chain as above holds, but there is one inequality when we replace si by sj :
Wi0
Wj0
=
(Wi + si yi )(1 + r)
(Wj + sj yj )(1 + r)
W i + s i yi
Wj + sj yj
λWj + si (λyj )
=
W j + s j yj
λWj + sj (λyj )
<
Wj + sj yj
= λ,
=
which means that the ratio of i’s to j’s income has worsened if i was poorer than j to begin
with.
So the Lorenz curve must be more unequal next period. You can draw a diagram and show
that the poorest third and the poorest 2/3 own a smaller fraction of total wealth in the next
period than they did before.
(d) [4 points] Now assume that all wages are equal and so are savings rates. Once again,
compare the Lorenz curves for wealth this year and the next.
5
We can again take the same approach of comparing ratios. Take any two people i and j with
i poorer than j. Write their common savings rate as s and their common income as y. Then
Wi0
Wj0
=
(Wi + si yi )(1 + r)
(Wj + sj yj )(1 + r)
W i + s i yi
Wj + sj yj
λWj + sy
=
Wj + sy
λWj
>
Wj
= λ,
=
where the above inequality follows from the fact that if you add the same positive number to
the numerator and denominator of a fraction that is smaller than 1, the value of the fraction
must go up.
Now the Lorenz curve must be more equal next period. You can draw a diagram and show
that the poorest third and the poorest 2/3 own a larger fraction of total wealth in the next
period than they did to begin with.
(5) This is right out of problem set 2.
(a) [5 points] The idea here is that the production function is linear so when you draw the
Solow diagram that we did in class, we have two straight lines. If the sy + (1 − δ)k line lies
above the (1 + n)k line, output will grow forever. If it lies below, then output will constantly
shrink over time to zero. If you point out the knife-edge case of equality, you get a point of
extra credit.
(b) [4 points] Take the equation
(1 + n)k(t + 1) = sy(t) + (1 − δ)k(t),
and divide through by k(t); then we see immediately that
(1 + n)(1 + g(t)) = sA + (1 − δ),
where g(t) is exactly as defined in the question. Since nothing else in the equation depends
on time, nor can g(t). Transposing terms, we see that
sA = (1 + n)(1 + g) − (1 − δ).
(c) [3 points] Here the savings rate has a persistent effect on growth rates. The reason is
that the Harrod Domar model does not have any diminishing returns, so that a higher rate
of savings feeds into a higher rate of economic growth. In contrast, in the Solow model, that
effect gets dampened by diminishing returns and ultimately, even though the new steadystate level of capital and income per capita are higher, there is no effect on the rate of
growth.
6
You can give other differences, but they will all have to rely somehow on this fact that the
Harrod-Domar model allows for persistent growth even in the absence of technical progress,
whereas the Solow model does not.