Recitation 6.pdf - Intermediate Macroeconomics Recitation 6...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Intermediate Macroeconomics Recitation 6 Topics: Growth rate and growth series on a log scale, Malthus model (economics and equa- tions, the steady-state, graphical solutions for population and wage dynamics after shocks). 1 Growth rate and growth series on a log scale 1.1 Log-log scale plot of growth series A growth rate g of y between t - 1 and t is g = y t - y t - 1 y t - 1 . Note: (1) The growth rate g is a percentage change : y t = y t - 1 (1 + g ) (2) Constant growth rate yields exponential growth: y t = y 0 (1 + g ) t Why plotting growth series on a log scale instead of in ”levels”? Because we are interested in whether the growth rate is constant, increasing, or decreasing. Plotting in levels makes it difficult to visualize. From the graph, we observe that the level of per capita GDP, y t , has been rising by more and more since 1870 (as the curve gets steeper), but the growth rate g is roughly proportional to per capita GDP levels. y 2005 - y 2004 = g × y 2004 y 2005 - y 2004 y 2004 = g 1
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Taking logs on both sides of the equation makes visualizing the growth rate easier. logy t = logy t 0 + tlog (1 + g ) The log scale is what Jones calls the ratio scale . Jones often uses log 2 . Alternatives are the natural log and log10. We usually use the natural log (to base e ). 1.2 Rule of 70 We want to estimate the time for a variable with a constant growth rate (or exponential growth) to double . For example, what is the time t that it takes for y t to double, if y t grows at a rate of g per year? According to the rule of 70 , t 70 g Starting with the equation that characterizes the constant growth rate, y t = y 0 (1 + g ) t 2 y 0 = y 0 (1 + g ) t log 2 = tlog (1 + g ) Note that log 2 0 . 7. Then, recall the Taylor series approximation of f ( x ) around x 0 f ( x ) = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) + 1 2 f 00 ( x 0 )( x - x 0 ) 2 + ... For log (1 + g ) at g = 0, log 1 = 0, d dg ( log (1 + g )) = 1 2
Image of page 2
Thus in the case of the natural log, we have log (1 + g ) g for g close to 0. t = log 2 log (1 + g ) = 0 . 7 g = 70 100 g What does the rule of 70 tell us about the doubling time of income or that of a country’s GDP per capita ? In China, GDP per capita doubles around every 7 years. In the U.S., GDP per capita doubles around every 35 years. Growth Rate (%) Doubling Time (years) 1 70 2 35 3 23 5 14 10 7 2 The Malthus Model 2.1 Economics and equations in Malthus In the Malthus model, there are two equations. The first is the wage as a function of popu- lation, which comes from firms’ optimal behavior: w t = (1 - a ) A t D HN t a , (1) where D is land, H is hours, N t is population, A t is technology. The second equation is the evolution of population as a function of the wage: N t +1 N t = w t w s ξ t , (2) where w s is the subsistence wage (say, $ 2 per day for food) and ξ t is an exogenous shock to population. Note that this equation is given to you, it does not come from an economic behavior or optimality conditions.
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern