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ECON3102 Tutorial 03 - Week 4
1. The Speed of Convergence:
Suppose the economy is described by the Solow growth model with a Cobb-Douglas pro-duction function, no population growth and no technological progress. The economy is not in steady state. GDP capita in period t is yt = θyss, with θ ∈ (0, 1).
(a) How far away from steady state is the capital stock? Find an expression for the level of kss/ks that would be consistent with yt = θyss, in terms of θ and parameters.
(b) Find an expression for kss, the steady state level of capital, in terms of parame-ters.
(c) Use your results from parts (a) and (b) and equation 1 to find an expression for gy, the rate of growth of GDP, as a function of θ and parameters.
gy = sf′ (kt) − δα (1)
(d) Let α = 0.35 and δ = 0.04. Compute gy for an economy whose GDP is θ = 0.8 times its steady-state level? Compute 1−θ/gy. What fraction of the gap between yt and yss does the economy close in one year? Repeat this calculation for θ= 0.9 and θ= 0.99.
2. Measurement of the Capital Stock:
The capital stock is not easy to measure. A common way to try to do this is to:
• guess a value for K0 at some date in the past (maybe when statistics were first collected),
• measure investment every year thereafter (which is also not that easy), and
• use the equation
Kt+1 = (1 − δ)Kt + It (2)
to compute the capital stock for every year thereafter.
Consider an economy that is in a steady state without technological progress (and has been there for an infinitely long time). Every year the level of investment is:
It = 0.2
and the depreciation rate is:
δ = 0.1
(a) What is the capital stock?
(b) The statistical office only started measuring investment in year 0. By then the economy was already in steady state. They guessed (incorrectly) that the capital stock in year 0 was K0 = 1. Use equation 2 to compute how the estimate of the capital stock changed over time after year 0 and plot your answer. (You don’t need to provide a closed form solution, a nice graph made with a spreadsheet is enough). Compute the ratio:
KTrue/KEstimate
for the years 5, 10, and 50.(Again, you can just take the numbers from a spreadsheet).
(c) Suppose now that the statistics office started collecting statistics a long time ago (an infinite time ago) so we don’t have the problem of making the wrong initial guess. However, instead of using the correct value of δ, the statistical office incorrectly believes that δ = 0.05. What is their estimate of the capital stock? Why does it differ from the truth?
(d) The production function in this economy is
Y = AKαL1−α (3)
and there are accurate data on GDP and on total hours of labor supply. Using the inaccurate estimate of K from part (c), plus accurate data on total labor and GDP, an economist is trying to measure the economy’s productivity, i.e. to solve for A in equation
3. Find an expression for
ATrue/AEstimate
as a function of
KTrue/KEstimate
Given the numbers from part (c) and using α= 0.35, how far would the economist’s es-timate be from the truth?
3. GDP Accounting, Interest Rates and Growth Accounting:
Proletaria is a mythical country in Central Asia, where the currency is the ruble. The exchange rate is 10 rubles per euro.
Suppose the production function in Proletaria is given by:
Y = KαL1−α (4)
(a) Find expressions for the output-to-capital ratio K/Y and the marginal product of capital FK as functions of K and L. During 2014, the following events took place in Proletaria.
• Kapitas, the main manufacturer in the country, imported an industrial welder made in Germany, for which it paid 1,000 euro.
• By the end of the year, the industrial welder was no longer new. Its estimated value in the resale market was 960 euro.
• 100 workers worked for Kapitas the entire year making screws and nails, using the new welder. Each of them received wages for 350 rubles.
• Each of the Kapitas workers paid 100 rubles in income taxes.
• The total output of Kapitas consisted of 1 million screws and 1 million nails. All of it was exported to Austria for a total of 4,000 euro.
• The government of Proletaria employed one of the 100 workers as Chief of the Secret Police (in addition to his factory job) to maintain law and order, and paid her a salary of 10,000 rubles.
• The workers ate beef imported from France, which cost a total of 2,500 euro.
(b) Construct GDP accounts (in rubles) for Proletaria by production, income and expen-diture.
(c) What were the income shares of labor and capital?
(d) Suppose that we know that the capital stock in Proletaria is 100,000 rubles and that the depreciation rate of the industrial welder is typical for this country’s capital stock. What interest rate should we expect to observe?
(e) During 2015, additional investment in Proletaria has exceeded depreciation so that the capital stock now stands at 120,000 rubles. The labor force also grew thanks to im-migration, and now consists of 105 workers instead of 100. GDP during 2015 was 55,000 rubles (prices were constant). How much of the growth in GDP between 2014 and 2015 can be attributed to growth in Total Factor Productivity?