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Assignment 1
(Due no later than August 23, Friday, 4pm)
Assignment Overview
This assignment asks you to study the implications of the life-cycle model with growth and to document and explain updated facts on cross-country differences in long-run growth.
Please type the solutions to your assignment (e.g., in Word or LaTeX) and convert them to a PDF file for online submission through Gradescope on the LMS. Handwritten submissions can also be scanned and submitted through Gradescope as a PDF, but they must be legible for marking.
Please note this is a group assignment with up to 4 students per group. You can form your groups directly in Gradescope. You don’t need to be in the same tutorial to form a group and you can submit your assignment individually if you prefer.
All students within the same group will receive the same mark and no two groups may submit the same assignment. You can collaborate with members of your own group (and all group members must provide input), but not with other groups. Please list all members of your group clearly on the first page of your submission and make sure you keep draft copies of your own working (for each member of the group).
This assignment should reflect your own work and ideas (see also the section: Artificial Intelligence Software in the Preparation of Material for Assess- ment in the Subject Guide). AI assistance tools are not required for this assignment, but if you do use them it should be for editing and proofing of your work only. Any use of AI tools for editing purposes needs to be clearly acknowledged at the start of your assignment. Further information on the acceptable use of these tools can be found here.
Question 1: The Lifecycle Model and Convergence (25 marks in total)
Consider the life-cycle model discussed in Lectures 3-5, but now suppose that each young individual born in period t has the following preferences:
where cyt is consumption when working in period t, cot+1 is consumption when retired in period t + 1, γ > 0, and ρ > 0 are positive parameters and exp(.) is the exponen- tial function. As in the lectures, assume that individuals supply one unit of labour inelastically and receive real wage wt per unit of labour supplied.
(a) (4 marks) Show that these preferences are well behaved. That is, the marginal utility of consuming in each period is positive, but diminishing.
(b) (4 marks) Write down the individual lifetime budget constraint. Write down the individuals’ utility maximisation problem and use the first-order conditions of this problem to derive the individual Euler equation. Explain the intuition underlying this equation and the role of the parameter ρ .
(c) (4 marks) Now using your Euler equation and the lifetime budget constraint, solve for optimal consumption when working (cyt) and saving (at+1) as functions of the real wage (wt) and real net return to saving (rt+1).
(d) (5 marks) Assuming firms use a Cobb-Douglas production function in labour and capital and operate under perfect competition (as we assumed in the lecture), derive an implicit transition equation for the aggregate capital-labour ratio over time in kt+1 and kt in this economy. By implicit, here we mean a transition equation of the form F(kt+1, kt ) = 0.
(e) (4 marks) Use the Implicit Function Theorem to show under what conditions that kt+1 is increasing in kt , that is the transition equation is upward slowing with > 0.
(f) (4 marks) Assuming or taking the limit as ρ approaches 1, graph your transition equation. Briefly discuss how your results compare those discussed in the lectures.
Question 2: Empirical Facts on LR Growth Across Countries
(25 marks in total)
Download the latest version of the Penn World Tables (version 10.01) from:
https://www.rug.nl/ggdc/productivity/pwt/ in either Excel or as a Stata Data file. In this question, feel free to use your favourite software or coding language to answer these questions (R, Python, Stata, Eviews, Excel etc.). However, please include your final answers in a PDF as part of your submission through Gradescope (via the LMS).
(a) (5 marks) Compute the average long-run growth rate in output per capita (defined as the annual average (compound) growth rate in output per capita from 1960 to 2019) for all countries where data are available. For output, use the measure “redgpe” that denotes expenditure-side real GDP at chained PPPs (in millions 2017 $US) and for population use “pop” that denotes population (in millions).
(b) (5 marks) Graph a histogram of the average annual per capita country growth rates across countries for 1960–2019 (using approximately 20 bins) and report the mean and standard deviation (again for all countries where data for 1960 and 2019 are available). How does the average long-run per capita growth across countries compare with that of the US as we discussed in the lecture?
(c) (5 marks) Graph a scatter plot of the average annual growth rate in output per capita for each country (as computed in (a)) on the x-axis against the level of real output per capita in 1960 on the y-axis (again for all countries where data are available), together with a line of best fit. Using the information in your scatter plot together with a regression of average annual growth in output per capita from 1960 to 2019, on output per capita in 1960, is there evidence of convergence in the level of output per capita across countries?
(d) (5 marks) Now restrict the sample to current OECD member countries only. Repeat the steps in parts (b) and (c). Explain any differences or similarities in your answers in terms of the mean and standard deviation of growth rates across the samples, and in the scatter plot and the growth regression.
(e) (5 marks) Relate your findings to the simple life-cycle model of growth discussed in the lectures. In particular, briefly explain whether that model predicts conver- gence in output per capita across countries (or not) and discuss the factors that explain any similarities (or differences) between your empirical findings and the predictions of the simple life-cycle model.