FIX: fix titles in long-run-growth

Author: mmcky

lectures/long_run_growth.md

@@ -370,7 +370,8 @@ plt.legend(loc='center left', bbox_to_anchor=(1.0, 0.5))
 
 +++ {"user_expressions": []}
 
-# Prior Work using World Bank Data (@aakash)
+
+## Prior Work using World Bank Data
 
 +++
 
@@ -382,7 +383,7 @@ wbi = pd.read_csv("datasets/GDP_per_capita_world_bank.csv")
 
 +++ {"user_expressions": []}
 
-## Comparison of GDP between different Income Groups
+### Comparison of GDP between different Income Groups
 
 A few countries from different Income Groups where chosen to compare the GDP at a more targeted level. GDP of countries from all income groups were compared. 
 
@@ -411,7 +412,7 @@ A few countries from different Income Groups where chosen to compare the GDP at
   - Afghanistan
 ```
 
-### Plot for all countries
+#### Plot for all countries
 
 We compare time series graphs of all the countries in the list. The clear separation between high income countries and other groups are clearly seen. It seems at first glance that other income groups probably have similar economies. Let's look at that further in the following sections.
 
@@ -439,7 +440,7 @@ ax.set_ylabel("GDP per capita (current US$) ")
 
 +++ {"user_expressions": []}
 
-### Plot for Upper middle and lower middle income groups
+#### Plot for Upper middle and lower middle income groups
 
 Now, we compare the time-series graphs of GDP per capita for  upper middle and lower middle income group countries, taking one country from each group. China and Pakistan was chosen as they are from the same region. 
 On analysing the graph, the difference is quite striking from 90s onwards. But also expected, as during that time China opened up for trade and labour. 
@@ -454,7 +455,7 @@ ax.set_xlabel("year")
 ax.set_ylabel("GDP per capita (current US$) ")
 ```
 
-### Plot for lower middle income
+#### Plot for lower middle income
 
 Here, we compare the time-series graphs of GDP per capita for two lower middle income group countries. Keeping Pakistan fixed in our set, we chose Vietnam as the second country. Apart from its turbulent past, its comeback from it and a steady growing economy qualifies it to be in this set. 
 From the graph, we can see that Vietnam has done quite well from around 1990 onwards, and has quite surpassed Pakistan. We can also conclude that countries in the same income groups can be also be quite different.
@@ -468,7 +469,7 @@ ax.set_xlabel("year")
 ax.set_ylabel("GDP per capita (current US$) ")
 ```
 
-### Plot for lower middle income and low income
+#### Plot for lower middle income and low income
 
 Finally, we compare time-series graphs of GDP per capita between a lower middle income country and a low income country. Again, keeping Pakistan fixed in our set as a lower middle income country, we choose Democratic Republic of Congo as our second country from a low income group. Congo is chosen for no particular reason apart from its unstable political atmosphere and a dwindling economy. 
 On comparing we see quite a bit of difference between these countries. With Pakistan's GDP per capita being almost four times as much. Further strengthening our assumption that countries from different income groups can be quite different.
@@ -484,7 +485,7 @@ ax.set_ylabel("GDP per capita (current US$) ")
 
 +++ {"user_expressions": []}
 
-## Histogram comparison between 1960, 1990, 2020
+### Histogram comparison between 1960, 1990, 2020
 
 We compare histograms of the **log** of GDP per capita for the years 1960, 1990 and 2020 for around 170 countries. The years have been chosen to give sufficient time gap between the histograms. We see that the overall plot is shifting towards right, denoting the upward trend in GDP per capita worldwide. And also, the overall distribution is becoming more Gaussian. Which indicates that the economies have gotten more uniform over the years. Economic disparities are getting lesser possibly because of globalisation, technological advancements, better use of resources etc.