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ECON-UA 12 — Intermediate Macroeconomics
Professor: Corina Boar
TA: Benjamin Castiglione
Spring 2024
Problem Set 5
Due Friday, March 8
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Competitive equilibrium with a fixed cost
In this question, we analyze the market for a new product that requires research and development before it can be introduced.
Consider the Cobb–Douglas production technology
Y = AK α L1−α .
This technology is available to each firm in the market but only after one of the firms pays
a fixed cost F that is spent on research and development (R&D). We will call the firm that
spends F on R&D the innovator. The remaining firms that utilize the outcomes of R&D
in production without paying the fixed cost are called immitators.
Question 1.1 Consider the a competitive, profit-maximizing immitator who rents capital
at rate r and hires workers at wage L.
Setup the profit maximization problem of the immitator. Take the first-order conditions
with respect to labor and capital. Verify that the equations take the form
MPL = w
MPK = r
Multiply the first equation by L and the second equation by K, and use the results to show
that the immitator earns zero economic profit.
Question 1.2 Now consider the problem of the innovator. The innovator also rents
capital at rate r and hires workers at wage L but on top of that also has to pay the fixed
cost of innovation F .
Setup the profit maximization problem of the innovator. Take the first-order conditions
with respcet to labor and capital and verify that they are identical to those of the immitator.
Repeat the procedure from the preceding question to show that the innovator actually
earns a negative profit.
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Question 1.3 Will the innovator ever spend the resources on innovation? What does it
imply for the competitive product market?
Question 1.4 Imagine that the innovator decides to pay workers less than the competitive
wage. Can he in this way recoup the cost of innovation and at least break even?
Question 1.5 What are the ways in which a government intervention can help innovation?
What are the potential disadvantages connected to such interventions?
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Preparation for the Romer model
The core of the Romer model consists of two equations. The first equation is the production
function. We will assume that there is no capital in the model and output is produced using
only labor. Denote Lyt the number of workers working in the production sector at time
t. Then output is given by
Yt = At Lyt
where At is the total factor productivity.
We will understand total factor productivity as the stock of productive ideas. Ideas are
accumulated in the research sector. There are Lat researchers in the research sector, who
produce ∆At+1 = At+1 − At new ideas in period t
∆At+1 = z̄At Lat
where z̄ captures the productivity of the research sector.
Question 2.1 Show that the growth rate of the stock of ideas depends on the productivity
of the resaerch sector and the number of researchers in the economy.
To close the model, we need to decide how are people in the economy allocated to the
research sector and to the production sector. First, there are L̄ people in the economy
and each of them has to decide whether to be a researcher or a worker. The resource
constraint therefore is
Lyt + Lat = L̄.
Finally, we will assume that a fraction ¯l of the population decides to work in the research
sector:
Lat = ¯lL̄.
This completes the model.
Question 2.2 Derive the growth rate of the stock of ideas in terms of the parameters of
the model.
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Question 2.3 Show that the growth rate of output is equal to the growth rate of ideas.
Now plug in the same numbers as those in Problem 7 in Chapter 6 of the textbook (page
1
160). Let ¯l = 0.06, z̄ = 3,000
, L̄ = 1, 000. Also, let the initial stock of ideas (in period 0) be
Ā0 = 100.
Question 2.4 Using these numbers, what is the growth rate of output per capita in the
economy?
Question 2.5 What is the initial level of output per peson? What is the level of output
per person after 100 years?
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Experiments in the Romer model
This problem assumes the baseline Romer model without capital.
Question 3.1 Assume that the economy is on the balanced growth path. Assume that there
is a technology breakthrough that suddenly increases the amount of knowledge (measured
by At ) by 10%.
Describe in words what happens immediately after this breakthrough?
What will be the growth rate of the economy in the years after this breakthrough?
Plot the trajectory of output per capita in the economy before and after the breakthrough (similar to that in Figures 6.3 and 6.4 in the textbook).
Question 3.2 Repeat question 3.1 with the following experiments:
1. Smart people become more interested in finance than in research, and the share of
reseachers ¯l decreases.
2. The government allows an inflow of educated people from abroad that keeps the
number of workers Lyt the same but increases the number of researchers, Lat (notice
that from the perspective of the model parameters, there are two effects, one on L̄
and one on l).
3. The government decides that part of the tax revenue will be invested in research and
development rather than spent on government consumption. This increased R&D
spending increases the productivity of the researchers z̄.
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Solow and Romer models combined
In this section, we will combine the Solow and Romer models together and derive a model
that will exhibit both transition dynamics (catching up of poor countries) as well as persistent growth. The derivation of the model closely follows the textbook and the slides.
Time is discrete and indexed by t = 0, 1, 2, . . .. The economy starts at time 0 with
an initial stock of ideas A0 and stock of capital K0 , and is described by the following five
equations, with variables that we defined in class.
1−α
1 : Yt = At Ktα Lyt
2 : ∆At+1 = z̄At Lat
3 : ∆Kt+1 = sYt − δKt
4 : L̄ = Lyt + Lat
5 : Lat = ¯lL̄
Question 4.1 Give names to each of the five equations, representing their economic meaning.
Question 4.2 Solve for the growth rate of ideas gA,t ≡ ∆AAt+1
.
t
We now introduce a method how to solve separately for the transition dynamics. The
method will consist of rescaling output Yt and capital Kt by the level of technology. Define
so-called scaled variables
Ỹt =
Yt
(At )
1
1−α
and K̃t =
Kt
1
.
(1)
(At ) 1−α
We can interpret Ỹt and K̃t as output and capital levels relative to the level of technology.
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You may wonder why At has the exponent 1−α
in this scaling, but for now taking it as a
‘good guess’ which will become clearer very soon.
Question 4.3 Express Yt and Kt from equations (1) and plug them into the output
production function (first equation of Solow–Romer model above). Show that you can
simplify the equation so that the technology At drops out of the equation.
Question 4.4 In the same way, plug in Yt and Kt from equations (1) into the capital accumulation equation (third equation of Solow–Romer model above). Simplify the equations
and show that the production function will now not depend on At directly, and the capital
accumulation equation will only contain At as a ratio AAt+1
, which is equal to 1 + gA,t .
t
Hint: This is very similar to the scaling procedure in the Solow growth model with
population growth. Before you start, it is advantageous to write the capital accumulation
equation as
Kt+1 = sYt + (1 − δ) Kt .
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You should obtain
K̃t+1 =
s
(1 + gA,t )
1
1−α
Ỹt +
1−δ
1
K̃t .
(2)
(1 + gA,t ) 1−α
Question 4.5 Show that we can write this capital accumulation equation as
∆K̃t+1 = s̃Ỹt − δ̃ K̃t
where s̃ and δ̃ are parameters that can be called ‘adjusted saving rate’ and ‘adjusted depreciation rate’. Express these parameters as a function of the original parameters of the
model and the growth rate gA,t .
Observe that we can now write the production function and the capital accumulation
equation as
Ỹt = K̃tα L1−α
yt
K̃t+1 − K̃t = s̃Ỹt − δ̃ K̃t
Question 4.6 Express Lyt in terms of the parameters of the model. Now determine the
‘steady state’ in variables Ỹt , K̃t .
Question 4.7 When K̃t and Ỹt are in steady state, at what rate do capital Kt and output
Yt grow? Notice that you will obtain that Kt and Yt grow at constant rates — such a
situation is called the balanced growth path.
Question 4.8 Verify that gY > gA . Why does output and capital grow at a faster rate
than technology?
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An experiment in the Solow–Romer model
This problem assumes the combined Solow–Romer model with capital. It is very useful to
solve Problem 4 first.
Assume two countries in the Solow–Romer model that are completely identical, they
differ only in their saving rate s. In particular, the ‘poor’ country P has a lower saving rate
than the ‘rich’ country R, i.e., sP < sR .
Knowledge is global, so that at every time t, both countries have the same stock of ideas
At . The rest of the model for each country is as in Problem 4.
Also assume that both countries are on their balanced growth paths.
Question 5.1 Argue that these two economies have the same growth rates of output but
they differ in their level of output. Plot the trajectories of their output over time (time on
horizontal axis, output in logs on vertical axis).
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Hint: Recall that in Problem 4, we defined so called scaled variables Ỹt and K̃t such
that
Yt
Kt
Ỹt =
and K̃t =
1
1 .
(At ) 1−α
(At ) 1−α
When the countries are on a balanced growth path, it must be that scaled output in the
poor country is in steady state, denoted Ỹ P ∗ , while scaled output in the rich country is
potentially in a different steady state, Ỹ R∗ .
Knowing this, you can compute the growth rates of output in both countries, and argue
that the levels are indeed different.
Question 5.2 Assume now that the the poor country increases the saving rate to sP = sR .
Describe what will happen in the poor economy and continue the plot of the trajectories for
the output in the poor and in the rich country. Why do the trajectories have to converge
to each other?
Question 5.3 Interpret the result from the previous country as the poor country catching
up with the rich country through capital accumulation, while the rich country continues to
grow along the balanced growth path generated by technological progress.
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