Development Accounting

• Some Log Calculus

One of the essential tools of development and growth accounting is the log dif- ference to measure relative “distances” across space (i.e. relative incomes across countries at a particular point in time) and time (e.g. growth rates of GDP per capita over time in a particular country).

For now, we’re interested in development accounting – the decomposition of rela- tive income differences into their individual components – and we’ll tackle growth accounting a little later.

To fix ideas, consider two countries, r (for “rich”) and p (for “poor”), with incomes yr = 2 × yp. Put differently, GDP per capita in r is two times higher than in p. The most common way to measure relative distances is in percentage terms and, using the standard formula, we find that:

yr − yp

= 100%.

yp

However, we can also use the rich country as our reference point and we would find that:

y p − yr

yr

= −50%.

In other words, the relative size of the income gap depends on the country we choose as our reference and that’s not very convenient.

Alternatively, we can use the log of relative incomes, or equivalently the log differ- ence, to measure the relative income gap:

ln ( yr \ = ln(yr ) − ln(yp ) = 0.6931 . . .

Conveniently, the absolute value of this log difference does not depend on our reference country (i.e. the GDP per capita we select for the the denominator) since:

ln ( yp \ = ln(yp ) − ln(yr ) = −0.6931 . . .

Now that we have this new log tool, we’re putting it to work in a development accounting exercise.

To start with, we need some data!

2          The Penn World Tables (PWT), Version 9.0

For the development accounting exercise we’re going to use the Penn World Ta- bles, the most widely used dataset for income and output comparisons across countries and/or across time. The reasons for this are somewhat technical and I’ll spare you the details for now.

For this particular exercise, we need only a subset of the variables in the tables and they are saved in a separate EXCEL spreadsheet (see link in the Canvas as- signment).

Before we can decompose the income differences into their individual compo- nents, you’ll have to take care of some preliminary steps.

2.1        Country Selection

For this exercise you need to select six countries from the Penn World Tables. To start with, choose a reference country. Traditionally, we use the U.S. as a reference point since it has been in steady state for virtually the entire post-WW II period. You may want to follow this convention.

In addition, select five countries that interest you for one reason or another. My only recommendation is that you select countries with “enough” data. For in- stance, countries that emerged from the former Soviet Union have a relatively short time series (typically starting in the early 1990s) and that can be problematic for reason that I’ll explain below. Apart from that, anything goes.

Once you’ve selected your countries, you want to copy your data into separate spreadsheet tabs, one for each country. In each tab, you’ll have the years in the rows and the different variables in the columns. It is good practice to create a sep- arate document for this subset of the data. If you ever mess up your spreadsheet (e.g. you accidentally delete some of your data), you can always go back to the original spreadsheet and import the raw data again.

Once you’re done with this, we need to construct capital stocks.

2.2        Capital Stocks using Perpetual Inventory Method

Recall that the National Income Accounts keep track of flow variables like con- sumption (in expenditure approach) or labor income (in income approach). Phys- ical capital, however, is a stock and we need to build it from the flow data in the accounts.

We build the capital stock using investment and depreciation data from the Penn World Tables together with the capital accumulation equation from the Solow

Growth Model. To see how this can be done, let’s use Yt to denote GDP, Nt is pop- ulation, Kt is the aggregate capital stock, It is aggregate investment, yt = Yt /Nt is GDP per capita, δt is the depreciation rate (which we’re allowing to vary by year and country) and kt = Kt /Nt is the capital stock per capita. We would like to con- struct the capital stock per capita kt from the information given in the Penn World Tables. We know that the law of motion for capital is given by:

Kt+1 = (1 − δt)Kt + It.

Given that we are interested in the capital stock per per capita, we divide by Nt on both sides to get:

Kt+1

= (1 − δ ) Kt  + It .

Nt

Dividing both sides by Nt+1 gives:

Kt+1      =    1

t   Nt       Nt

/(1 − δ ) Kt  + It \ .

or:

Nt+1 Nt

Nt+1

t   Nt       Nt

kt+1 =

Nt

Nt+1

/(1 − δ )k

+ It \ .

Nt

This equation describes how the change in capital per capita kt depends on invest- ment per capita It/Nt and on population growth. Intuitively, if the population is increasing quickly (Nt+1 is much larger than Nt), then there will be little capital per person in the next period.

We still need to link the law of motion to the quantities that we observe in the Penn World Tables. Multiplying and dividing by Yt in the last term gives:

kt+1 =

Nt Nt+1

/(1 − δ )k

+ It Yt

Yt \ .

Nt

Thus, investment per capita is given by the investment-GDP ratio It/Yt times GDP per capita yt = Yt/Nt . The Penn World Table data give us the following variables:

• popt= Nt (Population).

• csh it= It

(investment share in GDP).

• cgdpot= yt (GDP per capita).
• deltat= δt (depreciation rate)

Using the Penn World Table variable names, we can write the law of motion as:

  po pt    

kt+1 =

pop

t+1

((1 − deltat )kt + csh it × cgdpot ) .

With this equation, we can construct a time series for kt, provided that we have an initial value k0. Since this is not given in the data, we are simply going to guess this initial value. We will assume that:

k0 = y0,

where time 0 corresponds to the first year that you have data for.

Clearly, this procedure gives only a rough estimate of the capital stock; the initial value is chosen fairly arbitrarily but has only a small effect on our estimate of cap- ital stocks in the long run. For the purpose of comparing countries this procedure will yield acceptable results. Official estimates of capital stocks are constructed using similar methods, albeit with more attention to detail.

Using the method described above, construct estimated time series of capital per capita for your set of countries (once you have “built” the formulae for one coun- try, you can copy and paste them to the other tabs in your spreadsheet).

3          . . . and finally, Development Accounting

In this final step, you can assume that the production function has the familiar Cobb-Douglas form:

α     1−α

Yt = AtKt Nt       .

It’s straightforward to transform the production function into intensive form (i.e. into per capita terms). Once you’ve done that you have two of the three “ingredients”   to compute the residual At, namely yt and kt. What’s left to do is to pick a value for the capital income share α.

Luckily, the Penn World Tables compute labor income shares by year and country (labsh). To compute αt use the value for the labor income share, year-by-year, for each country:

αt = 1 − labsht .

All that’s left to do is to rearrange the production function (in per capita terms) to compute TFP. To do so, pick the most recent year for which you have data for all six countries and compute At for that year for each country.

Now we’re off to the races! We can finally compute the decomposition of pair- wise income differences (with respect to the reference country) into contributions from the capital-labor ratio and TFP.

To do so, you need to take the log difference of income per capita for each country with respect to the reference country. For example, if you’re comparing Argentina

to the USA in 2013 you would compute:

ln(yArgentina,2013) − ln(yUSA,2013 ) = ln(AArgentina,2013) − ln(AUSA,2013)

+ αArgentina,2013 ln(kArgentina,2013 )

− αUSA,2013 ln(kUSA,2013 )

In the last step, we calculate the contribution of A to the GDP per capita gap:

ln(AArgentina,2013) − ln(AUSA,2013)

contribution of TFP =

ln(yArgentina,2013) − ln(yUSA,2013)

Once you’ve done this for the five non-reference countries in your dataset, you can compute the average contribution of A by calculating the sample mean:

average contribution of TFP = 1 ∑ ln(Ai,t) − ln(AUSA,t)

5 i=1

ln(yi,t) − ln(yUSA,t)

where i is the index for the five countries in your sample and t is the most recent year for which you have data for all your countries.

In the textbook, Jones found that TFP accounts for roughly 75 percent of the in- come differences and the capital-labor ratio for the remaining 25 percent. Do you find similar numbers? If not, what is a potential explanation for your results? For instance, could it be that the labor income shares are very different from our

standard value of 1 − α = 2 ?

4          Assignment Grade

Last but not least, let’s talk about grading!

The assignment is worth 10 points in total and you will receive credit for the fol- lowing steps:

1. Compute the capital stock ktfor each country correctly (4 points)
2. For the most recent year with complete data for all countries, compute the TFPs correctly (3 points)
3. For that same year, compute the pair-wise and average contributions of TFP (2 points)
4. Brief discussion of your results in a MS Word document or any other text editor (1 point)

Submit your spreadsheet and the discussion of your results to the Canvas website.

For this assignment, you can collaborate with a classmate but you have to submit your own spreadsheet and your own answers for full credit.

Good luck!