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

penalised.

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

effects.

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

𝐴𝑡

=

∆𝐴𝑡+1

𝐴𝑡

and population

(workers) growth 𝑛 =

𝐿𝑡+1−𝐿𝑡

𝐿𝑡

=

∆𝐿𝑡+1

𝐿𝑡

. 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

, 𝐴𝑡𝐿𝑡

) = 𝐾𝑡

𝛼

(𝐴𝑡𝐿𝑡)

1−𝛼

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

for capital:

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

𝑥𝑡

= 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:

max

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

,𝐴𝑡𝐿𝑡

) = 𝐾𝑡

𝛼[(1 − 𝑢)𝐴𝑡𝐿𝑡

]

1−𝛼

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

𝑘̃

𝑡

, 𝑦̃𝑡

, 𝑎𝑛𝑑 𝑐̃𝑡 𝑓𝑜𝑟 𝑡 = 0, 1, … , 300. Also, provide a brief explanation of

your findings.

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

worker (𝑦𝑡

), 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

𝑘̃

𝑡

, 𝑦̃𝑡

, 𝑐̃𝑡

, ln(𝑘𝑡

) , 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

variables.

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

these steps:

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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