This assignment is worth 90 marks and contributes 30% of the total mark for this subject. The assignment is due at midnight 26 October 2018 and must be submitted using the link that will be provided on LMS. Late submissions will be subject to penalties as per university policy: https://policies.latrobe.edu.au/download.php?id=148&version=1
Your report should provide concise and relevant answers to all questions below,
as well as the corresponding computer outputs. Your report should be typed or
hand written clearly on A4 pages, double-spaced. Unclear hand-writing may be
In this assignment you will:
Develop understanding of discrete-time Solow Growth Model. In
particular, you will solve the model analytically for the steady-state and
Golden-Rule steady-state levels of the key variables involved.
Explore the effect of a change in a policy parameter on the dynamics of
the model by simulating time paths of key variables in an Excel
Spreadsheet (use attached Excel file). This will help you to interpret
transition dynamics and identify the difference between levels vs. growth
Read, analyse and interpret the main results of the academic research
paper “A Contribution to the Empirics of Economic Growth” by Mankiw,
Romer and Weil (1992).
Part A: Analytical Work (2+4+3+2+5+4=20 Marks)
Consider the Solow economy in discrete time framework with exogenous
technology that grows at a constant rate of g =𝐴𝑡+1−𝐴𝑡
(workers) growth 𝑛 =
. The stocks of capital depreciate each year
with a constant rate of 𝛿, where 0 < 𝛿 < 1. The economy produces aggregate
output by using the following Cobb-Douglas production function:
𝑌𝑡 = 𝐹(𝐾𝑡
) = 𝐾𝑡
Where 0 < 𝛼 < 1,𝑡 = 0, 1, 2, …
a. Find output per effective worker (𝑦̃𝑡 = 𝑌𝑡⁄𝐴𝑡𝐿𝑡
) as a function of capital
per effective worker (𝑘̃
𝑡 = 𝐾𝑡⁄𝐴𝑡𝐿𝑡
b. Capital stocks evolve over time according to the following law of motion
𝐾𝑡+1 − 𝐾𝑡 = 𝑠𝑌𝑡 − 𝛿𝐾𝑡
Transform this law of motion into that of capital per effective worker.
Hint: Divide both sides by 𝐴𝑡𝐿𝑡 and simplify. Also, use 𝑥𝑡+1
= 1 + 𝑔𝑥 and
(1 + 𝑔)(1 + 𝑛) ≈ 1 + 𝑛 + 𝑔.
c. Find the steady-state level of capital per effective worker, output per
effective worker, and consumption per effective worker.
d. Provide the equation for output per worker at the steady state. Also, find
the growth rate of output per worker and capital per worker. Hint: from
the definition of capital per effective worker and output per effective
worker derive the growth rates assuming continues variables so apply
logs and derivatives w.r.t. time.
Golden-Rule Steady State
The golden-rule steady state is the level of capital per effective worker at steady
state that maximises the steady-state level of consumption per effective worker.
Consider the following consumption maximisation problem (Note that at steady
state consumption is equal to income per effective worker 𝑓(𝑘̃) minus savings
per effective worker at steady state (𝑛 + 𝑔 + 𝛿)𝑘̃:
𝑐̃= 𝑓(𝑘̃) − (𝑛 + 𝑔 + 𝛿)𝑘̃
e. Find the values of 𝑘̃, 𝑦̃ 𝑎𝑛𝑑 𝑐̃at the golden-rule steady state by solving the
above maximisation problem.
Unemployment in the Solow Model
f. Now suppose the aggregate production function is given by:
𝑌𝑡 = 𝐹(𝐾𝑡
) = 𝐾𝑡
𝛼[(1 − 𝑢)𝐴𝑡𝐿𝑡
Where 𝑢 is the natural rate of unemployment. Find the output per effective
worker 𝑦̃𝑡 as a function of 𝑘̃
𝑡 and 𝑢. Obtain the steady-state level of output per
effective worker 𝑦̃
and explain how unemployment affects this economy. (Hint:
after obtaining output per effective worker plug this expression into the law of
motion of capital per effective worker and solve for 𝑘̃ at steady state and then 𝑦̃
at steady state).
Part B: Computational Work (10+10+15=35 Marks)
In this section of the assignment you will setup simulations in MS-Excel to
analyse the dynamics of the discrete version of the Solow Model with technology
over time. You will illustrate the dynamics by drawing the time paths of key
variables and will investigate the effects of changes in the equilibrium of the
model due to to some policy intervention.
Consider the following table of values of the parameters of the model:
Table 1: Benchmark values of Model Parameters
g. Use parameters values reported in table 1 to obtain the steady-state
levels of 𝑘̃, 𝑦̃, 𝑐̃, 𝑘̃
𝑔𝑜𝑙𝑑, 𝑦̃𝑔𝑜𝑙𝑑 and 𝑐𝑔𝑜𝑙𝑑 ̃ (these first calculations can be done
on paper using calculator). Compute in Excel and plot the time paths of
, 𝑎𝑛𝑑 𝑐̃𝑡 𝑓𝑜𝑟 𝑡 = 0, 1, … , 300. Also, provide a brief explanation of
Now suppose that the economy is at 𝑡
∗ = 110 and the government implements a
policy that results in a permanent increase in the saving rate from 12% to 20%.
All other parameters remain unchanged.
h. Given this change, obtain the new steady-state levels of 𝑘̃, 𝑦̃, 𝑐̃. Would this
policy change the growth rates of capital per worker (𝑘𝑡
); output per
), and consumption per worker (𝑐𝑡
)? Why or why not? What
would be the short-run impact of this policy change on ̃𝑐𝑡? Briefly explain.
i. Calculate and plot the time paths of
) , ln(𝑦𝑡
) 𝑎𝑛𝑑 ln(𝑐𝑡
) 𝑓𝑜𝑟 𝑡 = 101, 102, … , 170. Also, describe
the short-run and long-run impact of this policy change on these
Part C: A Contribution to the Empirics of Economic Growth. Solow Model
with Human Capital (6+3+6+3+6+5+6=35 Marks)
Read Mankiw, N. Gregory, Romer, David and Weil, David N., “A Contribution to
the Empirics of Economic Growth”, The Quarterly Journal of Economics, 1992.
Answer the following questions:
j. Using the necessary equations from Eq. (1) to (5) show the derivation of
the empirical Eq. (6) and then (7). Include all the steps and make sure you
identify all variables expressed in “per effective worker terms” with a
tilde as we did in class.
k. What are the Solow model predicted elasticities of income per capita
with respect to the saving rate 𝑠 and (𝑛 + 𝑔 + 𝛿)?
l. Consider the estimation results for the saving rate ln(𝐼⁄𝐺𝐷𝑃) and ln(𝑛 +
𝑔 + 𝛿) reported in Table 1 of the paper. Briefly explain why the authors
believe the estimation of these coefficients supports the Solow model
predictions and why the model, as estimated, is not completely successful.
m. What do the authors propose to improve the model?
n. The Mankiw, Romer and Weil (1992) original model differs slightly from
the version of the model covered in class. The key difference is the
treatment of human capital. Mankiw, Romer, and Weil assume that human
capital is accumulated just like physical capital, so that it is measured in
units of output instead of years of time.
This problem asks you to solve the model presented in the original paper. Follow
i. Transform the production function, Eq. (8), in per-effective-worker
terms (where 𝑦̃ = 𝑌
𝐴𝐿 ⁄ , 𝑘̃ = 𝐾
𝐴𝐿 ⁄ and ℎ̃ = 𝐻
𝐴𝐿 ⁄ ).
ii. Transform the following laws of motion for physical and human
capital into per-effective-worker terms, Eq. (9a) and Eq. (9b).
Human capital accumulation is governed by:
𝐻̇ = 𝑠𝐻𝑌 − 𝛿𝐻
Where 𝑠𝐻 is the constant share of income invested in human capital. Hint:
apply logs and derivatives w.r.t. to ℎ̃ = 𝐻
𝐴𝐿 ⁄ and use the resulting
equation in your derivation. Physical capital accumulation is governed by:
𝐾̇ = 𝑠𝐾𝑌 − 𝛿𝐾
Where 𝑠𝐾 is the constant share of income invested in physical capital.
Hint: apply logs and derivatives w.r.t. to 𝑘̃ = 𝐾
𝐴𝐿 ⁄ and use the resulting
equation in your derivation.
o. Substitute equation (10) into the production function and derive
equation (11). Show the steps.
p. Mankiw, Romer and Weil (1992) estimate equation (11), but in this equation
there is a new variable to estimate: 𝑠𝐻. Briefly describe the proxy the authors
used to account for this variable and summarise the results of estimating Eq.
(11) making sure you mention why the authors claim that this equation is a
better fit of the data.
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