Is anyone availabe to do this before 10 p.m CET?
CEU – Macroeconomic Theory 1 Hand-in problem set 1 Taking the Solow Model to the Data We are now going to solve the canonical Solow model in discrete time and try to see if there is hope to explain the income distribution across countries with the model. We know that without population growth, there will be no growth in the long-run. Hence, we will not be able to see anything about the time-series behavior (as most countries did grow in the last 50 years). Hence, we focus on the cross-section of countries in the year 2010. Consider the standard Solow environment. Time is discreet, the saving rate is s, pro- duction in country i is given by: Yi(t) = F (Ki(t), ZiLi) where F is a neoclassical production function and Zi is a labor-augmenting technology term. Note that Li and Zi are assumed to be constant over time but allowed to be different across countries. Capital depreciates at rate δ. Both the saving rate and the rate of depreciation is assumed to be equal across countries. a. State the accumulation equation for capital per capita ki(t) = Ki(t) Li (i.e. the equa- tion for ki(t + 1)) and express the level of output per worker yi(t), wages wi(t) and capital returns Ri(t) as a function of ki(t) and Zi. How do (yi(t), wi(t), Ri(t)) depend on Zi and ki(t)? b. Derive the condition for the steady state capital per capita k∗i . How does k∗i de- pend on Zi? Taking k∗i as a function of Zi, how does y∗i , w∗i , R∗i depend on Zi? c. Now suppose that F takes the Cobb-Douglas form: Yi(t) = F (Ki(t), ZiLi) = Ki(t) α(ZiLi) 1−α Derive the expression for the steady state capital-labor ratio k∗i . Derive an ex- pression for ln yi(t) as a function of lnZi, ln ki(t). Derive an expression of ln y∗i as 1 a function of parameters. Now let’s go to the data. The most important cross-country dataset are the Penn World Tables. You can find this on the internet, and use the newest release, PWT 10.0. In par- ticular, download the data on real GDP (output-side and expenditure-side real GDP at chained PPPs (in mil. 2017US$)), on real capital stock (capital stock at constant 2017 national prices (in mil. 2017US$)), on population and on employment for the year 2010 for all countries available (these can be found in the growth accounting (GA) section). From these compute output and capital per person and output and capital per worker. Use the measure which provides the most number of observations. d. To get a rough idea about the magnitude of income differences across the world, calculate per-capita income relative to the US for the following countries: China, India, France, Vietnam, Nigeria. Report your results or plot them in a graph. e. Now we are going to test how well the Solow model does in explaining the dif- ferences in income across the world. We still assume that Y (t) = Ki(t)α(ZiLi)1−α. Let ln yi = φ(Zi, ln ki) be the relationship between ln yi and Zi, ln ki(t) derived in part c. Suppose that α = 1/3. (i) Assume that technologies are equal across the world, i.e. Zi = ZSolow for all countries i. Then variation in income across countries is fully driven by the variation in the capital-labor ratio. Let ZSolow satisfy: (1− α) lnZSolow = 1 N N∑ i=1 ln yi − α 1 N N∑ i=1 ln ki where N is the number of countries. The predicted income by the model is ln ŷSolowi = φ(Z Solow, ln ki) Plot ln ŷSolowi against ln yi. Does the model do a good job in predicting in- come differences across the world? Does it over- or underestimate the in- 2 equality across countries? What does this tell you about the assumption that Zi = Z, i.e. technology is the same across countries? (ii) Now suppose that we wanted to make the model consistent with the data. Given the data on (ln yi, ln ki) find the implied values of log productivity lnZi such that the model fits the data perfectly. Plot lnZi against ln yi. How does productivity in rich countries compare to productivity in poor coun- tries? Did you expect this result from your answer in part (i)? (iii) Up to now we have taken the data on ki as given, i.e. we have not used the model’s formula for the steady-state. The steady-state capital-labor ratio you found above follows a relationship: k∗i = ψ ((s δ ) 1 1−α , Zi ) i.e. the variation of capital across countries is informative about productiv- ity differences. Let ln (( s δ ) 1 1−α ) satisfy: ln ((s δ ) 1 1−α ) = 1 N N∑ i=1 ln ki − lnZSolow, Use part a. to find Z̃i such that k∗i = ki, i.e. that the observed capital-labor ratios are consistent with a steady state. Now predict per capita income by ln ŷFullmodeli = φ ( Z̃i, ln ki ) . How well does this model do? What does this suggest about the Solow model? 3