assumption. Use these two equations and your earlier result to derive an expression for consumption per worker in steady state.

Transcription

1 Tutorial sheet 2 for UBC Macroeconomics Martin Ellison, 2018 Exercise on consumption in the Solow growth model The Solow growth model is in steady-state when investment ss YY tt is exactly offset by depreciation dd KK tt. i) Show that, together with the production function YY tt = AA KK tt 1/3 LL 2/3, this implies that steady-state output per worker is given by: yy tt YY tt LL = AA 3/2 ss dd ii) Consumption is subject to the resource constraint CC tt = YY tt II tt, where II tt = ss YY tt by assumption. Use these two equations and your earlier result to derive an expression for consumption per worker in steady state. iii) Sketch a plot of how consumption per worker varies in steady state with the investment rate ss. Explain the intuition for why the plot has the shape that it does. iv) What is the level of ss that maximises steady-state consumption per worker? Work graphically with the Solow diagram to identify the optimal investment rate and then use basic calculus on the expression you derived in part iii) to calculate its precise value. 1/2 v) Here is data on investment rates in a selection of countries Country % Country % Country % Argentina 19.0 Germany 20.9 Spain 24.1 Australia 24.5 India 23.8 Sweden 18.8 Brazil 22.5 Israel 20.5 UK 17.4 Canada 20.6 Italy 21.1 US 18.5 Chile 20.8 Japan 27.1 Zambia 15.4 China 33.7 Mexico 19.9 Low income 17.9 Congo 11.3 Russia 20.5 Middle income 24.0 Egypt 22.0 Singapore 33.9 High income 21.2 France 19.9 South Africa 19.0 World 21.7 Identify which countries appear to be investing too little and which might be investing too much. What counterarguments could be made against someone advising a country to increase its investment rate? vi) Suppose that everyone agrees that increasing the investment rate would raise consumption per worker in a given country. Assuming that the country is currently at steady-state, make a plot of what happens to investment, output, capital stock and consumption over time after an increase in the investment rate. With special reference to the dynamics of consumption, assess whether you think increasing the investment rate is likely to garner political support. Exercise with a simplified Romer model Consider a two period version of the Romer model. The technology at the beginning of period 1 is at a given exogenous level AA 1. In period 1, agents spend proportion 1 ll 1 of their time in productive activities, with production function:

2 YY 1 = AA 1(1 ll 1)LL where LL 1 is the total labour supply in the economy. Agents spend the remaining proportion ll 1 of their time in period 1 producing ideas, according to the production function: AA 2 AA 1 = zz AA 1ll 1LL with zz 1 a parameter measuring the productivity of workers producing ideas. In the final period 2, workers spend proportion 1 ll 2 of their time in productive activities according to: YY 2 = AA 2 (1 ll 2)LL i) Assuming that agents only value output produced in periods 1 and 2, what would be the best proportion of time to spend producing ideas in period 2? ii) Make a plot of the combinations of yy 1 and yy 2 that are possible when agents choose different proportions of their time to spend on production in period 1. To start you off, what happens to yy 1 and yy 2 if ll 1 = 0 so that all time in period 1 is spent on production? What happens if ll 1 = 1 so that all time in period 1 is spent producing ideas? What about intermediate cases where ll 1 is between 0 and 1, for example ½? iii) Agents ideally want to consume production goods in both periods 1 and 2. One way to model this is through a multiplicative utility function: UU = yy 1 yy 2 Use the three production functions given in the introduction to this question to solve out for output per head yy 1 and yy 2 as functions of ll 1 and AA 1, zz, LL alone. What amount of time should be spent producing ideas in period 1 to maximise utility? Add indifference curves to your plot in part ii) to illustrate your answer. iv) How does the optimal allocation of time in period 1 change as the productivity zz of hours spent producing ideas increases? v) Without doing any extra calculations, discuss what you would expect to happen if agents start valuing goods produced in period 1 more that those produced in period 2 (why might they do this?). For example, their utility function might be UU = yy φφ 1 yy 2, where φφ > 1. Answer intuitively, using a diagram where that would be helpful.

3 Exercise on explaining cross-country income differences The table below shows data from the World Bank and OECD on aggregate GDP and population in a selection of countries in The GDP figures are given in Purchasing Power Parity dollars, a way of comparing GDP across countries that accounts for the purchasing power of different currencies in different countries. Country GDP $PPP Total population US 16,768,053,000, ,498,000 Japan 4,612,630,249, ,298,000 Korea 1,661,722,646,620 50,220,000 Denmark 245,833,879,589 5,614,932 France 2,478,250,840,694 63,786,140 Germany 3,553,605,420,066 80,611,000 Italy 2,109,333,808,260 60,224,730 Netherlands 785,388,563,147 16,804,430 Norway 333,427,279,945 5,080,000 Sweden 428,042,210,295 9,609,000 UK 2,483,661,038,301 63,237,940 i) Calculate GDP per capita for each country and comment on the ranking of countries. Which country would you prefer to live in? You will find it helpful to use a spreadsheet as it will make calculations easier in the next part. GDP per capita is an incomplete measure of labour productivity since it does not take into account how participation in the labour market and hours worked differ across countries. To address this, consider the following decomposition: GGGGGG pppppp cccccccccccc = GGGGGG PPPPPPPPllllllllllll and the following definitions: pppppp pppppppppppp = rrrrrrrr = = 1 UUUUUUUUUUUUUUUUUUUUUUUU rrrrrrrr PPPPPPPPPPPPPPPPPPPPPPPPPP rrrrrrrr = PPPPPPPPPPPPPPPPPPPP The decomposition allows us to answer the question of why GDP per capita is so high in some countries. Is it because these countries have a high percentage of their population participating in the labour market? Is it because a high proportion of those participating in the labour market are employed? Is it because those employed work a lot of hours? Finally, is it simply that workers are very productive in the hours that they do work?

4 ii) Use additional data from the OECD below to further decompose GDP per capita in each country. Country Hours worked Labour Employment force US 256,492,502, ,043, ,259,000 Japan 109,952,940,000 65,970,000 63,410,000 Korea 52,625,727,000 26,108,000 25,313,000 Denmark 3,856,716,000 2,880,000 2,682,000 France 37,994,185,300 28,559,000 25,771,000 Germany 53,877,337,500 41,720,000 39,543,000 Italy 38,423,591,700 25,261,000 22,173,000 Netherlands 11,718,987,000 8,937,000 8,247,000 Norway 3,675,141,000 2,703,000 2,610,000 Sweden 7,594,682,000 5,137,000 4,726,000 UK 50,407,138,000 32,522,000 30,202,000 Your calculations should enable you to complete the following table: Country GDP per capita GGGGGG PPPPPPPPPPPPPPPPPPPP US Japan Korea Denmark France Germany Italy Netherlands Norway Sweden UK iii) Comment on what the decomposition reveals about why different countries have differing levels of GDP per capita. Has your view of which country you would like to live in changed? Why?

5 Short essay questions 1. In the Solow growth model will an economy that increases its savings rate experience faster growth? How does your answer depend on the properties of the production function? 2. During the 1950s and 1960s, Germany and Japan had much faster rates of economic growth than did the United States. What might account for these differences in growth rates? Did these differences occur because of a fundamental defect in the U.S. economy? 3. Consider the Solow growth model in a steady-state in which the technology growth rate and the capital depreciation rate are both equal to 1% per period. Suppose that the technology growth rate increases to 2% per period. Sketch the path followed over time by output per worker, in the transition from the old steady-state to the new steadystate. How is this adjustment paths related to the marginal product of capital? Long essay questions 4. The front cover of The Economist on 12th January 2013 showed Rodin's statue The Thinker, sitting on a toilet asking himself Will we ever invent anything this useful again? Is the answer to this question crucial for the credibility of endogenous growth theory? 5. Suppose that technological improvements in a production originate from research and development centres located in a small group of developing countries. What are the implications for economic convergence between countries? What policies could the governments of developing countries adopt to achieve the same long run level of per capita GDP as developed countries?