class: center, middle, inverse, title-slide # Mankiw, Romer and Weil (1992) ## Macroeconomics (2017Q2) ### Kenji Sato ### 2017-06-26 --- ## Homework Assignment HW03 on convergence Visit [kjst.jp/ma17q2hw](kjst.jp/ma17q2hw) > Due 2017-07-02 18:00. > Hand in by Pull Request. --- ## Conditional convergence Recall the following relationship beteween the growth rates of `\(K\)` and `\(k\)`. `$$\frac{\dot K}{K} = \frac{sf(k)}{k} - \delta$$` It holds that `$$\frac{d}{dk} \left( \frac{\dot K}{K} \right) = s\left(\frac{kf'(k) - f(k)}{k^2}\right) < 0$$` since, by strict concavity of `\(f\)`, `$$f'(k) < \frac{f(k)}{k}, \quad k > 0$$` --- ## Conditional convergence (cont'd) <img src="fig/fig.001.jpeg" style="display: block; margin: auto;" /> --- class: center, middle # Mankiw, Romer and Weil 1992 --- ## MRW In the mid-term report, you will conduct an exercise similar to [Mankiw, Romer and Weil (1992, QJE)](https://doi.org/10.2307/2118477) (henceforth, MRW) with updated data. In today's class and the next, we study the theory underlying the paper. This paper is extremely influential. Please read it. --- ## Cobb-Douglas production function We consider the following function form: `$$F(K, L) = K^\alpha L^{1 - \alpha}$$` The intensive-form: `$$f(k) = k^\alpha$$` --- ## Steady state Verify that the steady state is given by `$$k^* = \left(\frac{ s }{ \delta + g + n }\right)^{\frac{1}{1 - \alpha}}$$` and that output per worker is given by `$$\frac{Y}{L} = A \left(\frac{ s }{ \delta + g + n }\right)^{\frac{\alpha}{1 - \alpha}}$$` --- ## Elasticity of output per capita wrt saving rate `$$\ln \left( \frac{Y}{L} \right) = \log A(0) + gt + \frac{\alpha}{1 - \alpha} \ln (s) -\frac{\alpha}{1 - \alpha} \ln (\delta + g + n)$$` Data suggests that `$$\alpha \simeq \frac{1}{3} \Longrightarrow \frac{\alpha}{1 - \alpha} \simeq \frac{1}{2}$$` The Solow model, therefore, suggests that 1% increase of `\(s\)` implies 0.5% increase of `\(Y/L\)`. Increment of `\(\delta + g +n\)` by 1% decreases `\(Y/L\)` by 0.5%. --- ## Question Do data suppor these predictions of the Solow model concerning the determinant of standard of living? --- ## Assumptions MRW assumes the following: * `\(g\)` and `\(\delta\)` are constant across countries. * advancement of knowledge, which `\(g\)` reflects, is not country-specific * no strong reason to expect `\(\delta\)` to vary across countries * A(0) is decomposed into `\(A(0) = a + \epsilon_j\)`, where `\(a\)` is common to all countries and `\(\epsilon_j\)` is country-specific shock. --- ## Empirical specification Consider `\(t = 0\)`. MRW's first empirical specification: `$$\ln \left( \frac{Y_j}{L_j} \right) = a + \frac{\alpha}{1 - \alpha} \ln (s_j) -\frac{\alpha}{1 - \alpha} \ln (\delta + g + n_j) + \epsilon_j$$` Under the assumption that `\(s\)` and `\(n\)` are independent of `\(\epsilon\)` we can estimate the above with the ordinary least squares. `$$\ln \left( \frac{Y_j}{L_j} \right) = \beta_0 + \beta_s \ln (s_j) + \beta_n \ln (\delta + g + n_j) + \epsilon_j$$` --- ## Empirical specification (cont'd) `$$\ln \left( \frac{Y_j}{L_j} \right) = \beta_0 + \beta_s \ln (s_j) + \beta_n \ln (\delta + g + n_j) + \epsilon_j$$` If the Solow model is a good model of the economy, then the OLS would predict `$$\beta_s \simeq 0.5$$` `$$\beta_n \simeq - 0.5$$` To perform the empirical exercise, we need to determine `\(Y_j/L_j\)`, `\(s_j\)`, and `\(\delta + g + n_j\)` from data. --- ## Empirical specification (restricted regression) Since the model predicts that `\(\beta_s\)` and `\(\beta_n\)` are the same in maginitude and opposite in sign, the following specification should work. `$$\ln \left( \frac{Y_j}{L_j} \right) = \beta_0 + \beta_1 \left( \ln (s_j) - \ln (\delta + g + n_j) \right) + \epsilon_j$$` --- ## Data MRW uses data set constructed by Summers and Heston (1988), an earlier version of the Penn World Table that you are already familiar with. Their dataset convers the period of 1960 to 1985. --- ## Data (cont'd) They measure * `\(n_j\)` as the **average rate of growth of the working-age population**. (Unlike PWT v9.0, the earlier version does not contain total number of the employed.) * `\(s_j\)` as the average share of real private and public investment. (**average share of gross capital formation**) * `\(Y_j/L_j\)` as **real GDP devided by the working-age population** in 1985. (1985 is the latest sample in their dataset.) They simply assume that `\(\delta + g = 0.05\)`. --- ## Non-oil, Intermediate, OECD They used three sample for the empirical study. * Non-oil countries exclude countries that are heavily reliant on oil production. * Intermediate countries exclude countries whose population in 1960 is less than 1 million. * OECD countries consist of OECD countries with populations greater than 1 million. .small[ According to a recent data by World Bank, Venezuela and Chad are more heavily reliant on oil than Iran but MRW includes those two countries while excluding Iran. ] --- Table 1 on p. 414 (emphasis added) <img src="images/mrw_table1.png" width="600px" style="display: block; margin: auto;" /> --- ## Implied `\(\alpha\)` The restricted regression predicts higher `\(\alpha \simeq 0.59\)` from the coefficient on ln(I/GDP); `$$\frac{\alpha}{1- \alpha} \simeq 1.43$$` implies `$$\alpha \simeq 0.588..$$` which is much higher than the common wisdom `\(\alpha \simeq 1/3\)`. --- ## Exercise (A part of the mid-term) Do the above two regression analyses * non-restricted, bivariate OLS * restricted, simple OLS with PWT v9.0 dataset with * countries the populations of which in 1960 are more than 1 million, Report on what you observe. --- ## What's wrong? MRW's strategy to fix the bad prediction: **Add human capital to the Solow model** --- ## Production function Let `\(0 < \alpha + \beta < 1\)`, `\(\alpha, \beta > 1\)`. The output is given by `$$Y = K^\alpha H^\beta (AL)^{1 - \alpha - \beta}$$` * `\(H =\)` stock of human capital * `\(K =\)` stock of physical capital * `\(A =\)` knoledge * `\(L =\)` labor `$$\begin{aligned} k = \frac{K}{AL},\quad h = \frac{H}{AL},\quad y = \frac{Y}{AL} = k^\alpha h^\beta \end{aligned}$$` --- ## Investment They assume that the constant investment rate for both capitals. * `\(s_k =\)` saving rate for physical capital * `\(s_h =\)` saving rate for physical capital Verify that `$$\begin{aligned} \dot k &= s_k y - (\delta + g + n) k\\ \dot h &= s_h y - (\delta + g + n) h \end{aligned}$$` --- ## Takeaway Compute the steady state `$$k^* \quad \text{and} \quad h^*$$` and verify that `$$\begin{multline} \ln \left( \frac{Y}{L} \right) = \log A(0) + gt + \frac{\alpha}{1 - \alpha - \beta} \ln (s_k) \\ + \frac{\beta}{1 - \alpha - \beta} \ln (s_h) - \frac{\alpha + \beta}{1 - \alpha - \beta} \ln (\delta + g + n) \end{multline}$$`