Gross Domestic Product (GDP) per capita is the abbreviation for Gross Domestic Product (GDP) per capita (per person). It is calculated by simply dividing total GDP (see definition of GDP) by the population. In international markets, per capita GDP is usually stated in local current currency, local constant currency, or a standard unit of currency, such as the US dollar (USD).
GDP per capita is a key metric of economic success and a helpful unit for comparing average living standards and economic well-being across countries. However, GDP per capita is not a measure of personal income, and it has certain well-known flaws when used for cross-country comparisons. GDP per capita, in particular, does not account for a country’s income distribution. Furthermore, cross-country comparisons based on the US dollar might be skewed by exchange rate movements and don’t always reflect the purchasing power of the countries under consideration.
For the last five years, the table below illustrates GDP per capita in current US dollars (USD) by country.
Are you looking for a forecast? The FocusEconomics Consensus Forecasts for each country cover over 30 macroeconomic indicators over a 5-year projection period, as well as quarterly forecasts for the most important economic variables. Find out more.
What does GDP’s natural log mean?
This semester, I’m preparing for my undergraduate growth course (which uses an AWESOME book, by the way). Because our econ majors at UH are not required to take calc, I have to ease them into a few topics throughout the course. Using logs to show and then assess economic progress is one of them. This is a collection of notes I started writing to help students understand the ideas.
This could be useful if you’re a teacher. This might be useful if you’re interested in growth but are afraid of arithmetic. It will kill around 10 minutes and/or help you fall asleep if you have nothing else to do.
We’re interested in real GDP per capita as a gauge of living standards. The absolute value of real GDP per capita isn’t something we worry about because it depends on how we calculate price indices, base years, and other factors. What we’re interested in is how quickly economies grow. This is equivalent to saying that whether you start at 150 pounds or 68 kilos, if your weight increases by 5% per year, you will gain weight.
We always plot real GDP per capita in logs to view those growth rates and do some quick analysis. When I say log, I’m referring to the natural kind. There are a lot of fascinating ways to explain how natural logs work, but this isn’t one of them. For the time being, we’ll take it on faith that natural logs work as I claim.
Calculate the natural log of GDP per capita for each year and plot it against the year. The image below is an example of real GDP per capita plotted using the grey line for India.
Natural logs have a number of advantages that are ideal for our needs. When they’re used, each step up the y-axis corresponds to a 1% change in real GDP per capita. For example, increasing real GDP per capita from 7.0 to 7.5 is a 65 percent increase. The increase in real GDP per capita from 7.5 to 8.0 is likewise a 65 percent increase. This is true even when the GDP per capita increases from $1,096 to $1,800 from 7.0 to 7.5, and from $1,800 to $2,980 from 7.5 to 8.0. We can visualize % increases rather than absolute increases by plotting things in natural logs.
Even more entertaining is the fact that, thanks to this attribute, we can simply determine the average growth rate between two years. The slope of the straight line connecting the two end points is the average growth rate over two years. I’ve overlaid the computation of the growth rate for numerous sub-periods, as well as the average growth rate from 1950 to 2010, in the chart for India.
Connect the two end points, for example, from 1950 to 2010, then get the slope of the line. This can be done mathematically as follows:
You repeat the process for the other sub-periods, but this time utilizing the end points from those time periods. So, from 1950 to 1975, growth was 2.0 percent on average, -3.0 percent from 1975 to 1985, and 5.2 percent from 1985 to 2010.
Why did you choose certain sub-periods over others? They were chosen only because they appeared to be obvious break points. There isn’t a formula for determining which sub-periods should be calculated. It just seems to make sense “Around 1985, for example, “something distinct” happened that distinguished that time apart from the others. If you wanted to, you could calculate the growth rate from 1972 to 2003. You want to dispute with me that the period from 1985 to 2010 should be divided into pre-2003 and post-2003 sub-periods again? Okay. I’m afraid I can’t tell you that you’re mistaken.
The average growth of the three sub-periods I calculated appears to be quite similar to the real growth. That is, the real path of GDP per capita does not deviate significantly from the straight line we used to compute average growth from 1950 to 1975. What exactly do I mean by that? “A long way away”? I’m not trying to be technical; I’m simply guessing.
However, compare those three sub-periods to the entire period from 1950 to 2010. We calculate average growth at 2.5 percent, but for practically the entire period, the straight line that gives us that result is well above the actual real GDP per capita. Average growth from 1950 to 2010 does not provide a particularly clear picture of India’s growth experience throughout history, as there appear to be three distinct periods.
That isn’t to say that 2.5 percent is incorrect. It’s the same as the average growth rate between 1950 and 2010. If you started with $798 in real GDP per capita in 1950 and added 2.5 percent growth, you’d end up with $798.
which is only a few cents short of the actual worth of $3,596. However, the route from 1950 to 2010 appears to be considerably different, with an average annual growth rate of 2.5 percent, when compared to the actual path of real GDP per capita in India during this time period.
Regardless, periods when the average growth rate is very close to the actual growth rate of real GDP per capita will pique our interest (the straight line is close to the gray line). As previously said, 1950 to 1975, or arguably 1985 to 2010. These times may indicate what we’ll refer to as “BGPs stand for “balanced growth paths.”
We’re interested in BGPs because our growth theories imply that in the absence of substantial reform, a country will inevitably end up on one. That is, in the absence of a large shock to a basic feature of the economy, an economy’s real growth will likely to be near to average growth. Furthermore, if we notice distinct BGPs, it means that something essential has changed.
India’s picture shows that the time from 1950 to 2010 was not a BGP for the country. From 1950 through 1975, one could say one thing, and then in 1975, something fundamentally different happened. What was it, exactly? We can’t tell based on this graph. To make a decision, we’d have to look at other statistics on India, and our theory might point us in the right direction. Something appears to have fundamentally altered in 1985, when India switched to what appears to be a new BGP. We’d have to look at more data to figure out what had changed.
To be clear, what we see in the graph is required but insufficient to prove that India was on a BGP. That is, there are other characteristics that must be met in order to identify a period of time as being historical “There could be other reasons why India was not on a BGP from 1975 to 1985, for example, despite real growth being close to average. However, looking at a statistic like this indicates where we should begin our search.
To be even more thorough, simply because the majority of our ideas predict that countries will end up on a BGP in the absence of a huge shock does not guarantee they are correct. It’s possible that our theories are utterly incorrect. Perhaps nothing essential changed in 1975 or 1985, and all that happened was that India had a string of bad luck. Perhaps we’re exaggerating the significance of these long periods of equal growth rates.
In economics, what does log mean?
Let me start by going over compound interest rates quickly. If you invest an amount for one year at an interest rate of r percent, you will have an amount at the end of the year. For example, if you get a 4% return, your interest may be compounded, meaning you get your money back plus 2% after six months, and then you can earn 2% interest on both your original investment and the first six months’ interest: Your money would have increased 4.04 percent at the end of the year if you had an interest rate of r = 4% compounded twice a year like this. Quarterly compounded, or 4.06 percent at the end of the year. As the frequency of compounding n increases, the formula converges to a certain function, with the limit beingwhere e is a peculiar number (roughly equal to 2.72) that possesses this and a number of other astonishing qualities. Thus, if you earn r percent interest that is compounded annually, your money will have increased by at the end of the year. If r = 4%, for example, continuous compounding would yield 4.08 percent at the end of the year, somewhat higher than the 4.06 percent from quarterly compounding or the 4.0 percent from no compounding.
Taking natural logarithms is the inverse of the preceding operation: or, because the log of a ratio is the difference of the logs,
In other words, calculating a rate of return on a holding by subtracting the log of the price in year 2 from the log of the price in year 1 is the same as calculating a rate of return on the holding expressed as a continuously compounded rate.
The continuously compounded return is nearly equal to the noncompounded return for low values of r, hence the log difference is nearly equal to the percentage change. The percentage change in the above example is, whereas the log change is. So, if you see a graph that is measured in logs, a change of 0.01 on that scale equates to a 1 percent rise quite precisely. When plotted in logs, a graph that is a straight line through time correlates to annual growth at a constant percentage rate.
When compared to looking at simple percent changes, using logs, or summarizing changes in terms of continuous compounding, provides a number of advantages. For example, if your portfolio increases by 50% (from $100 to $150) and subsequently drops by 50% (from $150 to $75), you aren’t back where you started. When you compute your average percentage return (in this case, 0%), you’ll see that it’s not a very useful summary of the fact that you ended up 25% below where you started. In contrast, if your portfolio increases by 0.5 in logarithmic terms and subsequently decreases by 0.5 in logarithmic terms, you are back to square one. The difference in log price between the time you acquired it and the time you sold it, divided by the number of years you kept it, is the average log return on your portfolio.
When it comes to analyzing economic data, logarithms are typically far more beneficial. Here’s a graph showing the overall U.S. stock price index, which dates back to 1871. Nothing can be seen in the first century when plotted on this scale, yet the most recent decade looks to be madly erratic.
Is GDP calculated per capita?
The Gross Domestic Product (GDP) per capita is calculated by dividing a country’s GDP by its total population. The table below ranks countries throughout the world by GDP per capita in Purchasing Power Parity (PPP), as well as nominal GDP per capita. Rather to relying solely on exchange rates, PPP considers the relative cost of living, offering a more realistic depiction of real income disparities.
What is the difference between logs?
Regrettably, this method does not work “Calculating the percent change going forward from a former period to a subsequent period is not the same as calculating the percent change going backward from the following period to the prior period, which is symmetrical. As an example, imagine the previous period’s value is 100 and the subsequent period’s value is 125. The percent change from one period to the next is 25% (i.e. the quantity grows by 25% from its base value of 100), whereas the percent change from one period to the next is -20%. (i.e. the quantity decreases by 20 percent from its base value of 125). Because of the absence of symmetry, more attention must be taken while determining the suitable basis for this metric’s calculation.
If you don’t want to use percent change, there is another option “differences in logs”
The logarithm of one quantity is subtracted from the logarithm of another quantity in this procedure. The advantage of this method of evaluating change is that the computations are symmetrical in both directions. The additive inverse of the log difference in one direction is the log difference in the other direction. (A number equals zero when added to its additive inverse.) Furthermore, the log difference between the last observation and the first is the same as the sum of the log differences throughout a complete sequence of integers.
Consider the following table, which shows an example of a series with five x observations and their natural logarithms, as well as change, percent change, and log difference from the previous period calculations:
Why do we keep a data log?
In charts and graphs, there are two basic reasons to employ logarithmic scales. The first is to respond to large-value skewness, which occurs when one or a few points are significantly larger than the rest of the data. The second option is to display % change or multiplying factors. I’ll start by explaining what logarithms are. Then I’ll go over each of these reasons in greater depth and provide instances.
Logs are just another way of writing exponential equations that allows you to separate the exponent on one side of the equation, if you remember your school algebra.
The second equation is
What exactly is the distinction between log and ln?
The distinction between log and ln is that log refers to base 10 whereas ln refers to base e. For instance, the log of base 2 is represented as log2 and the log of base e is expressed as loge = ln (natural log).
A natural logarithm is the power to which the base ‘e’ must be raised in order to acquire a number known as its log number. The exponential function is denoted by the letter e. John Napier, who discovered and developed the notion of logarithms, was the first to discover it in the 17th century. Let’s look at the definitions of log and ln before diving into the fundamental differences between the two.
Is log the same as the growth rate?
The estimator becomes non-linear when logarithms are introduced to the equation of (1) by altering the functional form. But first, let’s look at what logarithms can do to the x variable. Assume that x is a time variable with a strong rising trend and is strongly heteroscedastic, as seen in the graph below.
Variablex has a positive trend and has departures from his mean over time, which may be seen graphically. A logarithm transformation can be used to reduce the amount of HT in the series. The following graph depicts the behavior using natural logarithms.
The units have shifted dramatically, and the logarithm of x is now somewhere between 2 and 5. Previously, it ranged from 10 to 120. (the range has been reduced). Because logarithms are specified as monotonic transformations, the natural logarithm reduces HT (Sikstar, s.f.). When we employ this type of transformation in econometrics, we get something like this:
The coefficient B is no longer the marginal effect; instead, it must be divided by 100 to be understood (Rodrguez Revilla, 2014). As a result, the final result should be written as follows: A one-unit increase in x results in a B/100 change in y.
The elasticity in this situation is simplyB, which is expressed as a percentage. If B=0.8, for example. A 1% rise in x would result in a 0.8 percent increase in y.
B must be multiplied by 100 in this situation, and it can be regarded as a growth rate in average per unit of x increases.
If x=t denotes years, then B denotes y’s average annual growth rate.
Equation (5) states that the difference of logarithms is approximately equivalent to the growth rates of a variable (left hand of the equation). Returning to our x variable in the previous graph, we can see that the growth rates are similar in both calculations.
The monotonic transformation has a significant impact; the growth rate formula has more upper (positive) spikes than the difference of logarithms formula. The lower spikes, on the other hand, are caused by the difference of logarithms. Nonetheless, both are approximate growth rates, indicating how our x variable has changed over time.
Let’s say you want to put the date of the tenth year on the aforementioned graphic.
Between year 9 and present, the difference in logarithms shows a -0.38 percent growth rate, while the growth rate formula indicates a -0.41 percent growth rate.
Between these years, there has been a 0.4 percent decline in growth.
When we utilize logarithms in those kinds of changes, we obtain something like this mathematically:
Some authors simply do so for the sake of normalizing the data (i.e., lowering the HT), but would the meaning be the same? What are the ramifications of doing so? Is it anything good or something bad?
It depends, as is customary. What happens if we look at the years 9 and 10 of our initial x variable again? We can see that the change is negative, and hence the growth rate is negative. When the value is negative, we usually can’t estimate a logarithm.
With this exercise, we can observe that the first consequence of overusing logarithms (both differenced logarithms and general growth rates) is that the calculus becomes undefined if we get negative values, resulting in missing data. If we plot the outcomes of such an experiment, we get something like this:
In Excel, the undefined values (result of the logarithm of negative numbers) are treated as 0 at this point; other software may not even place a point.
We got negative growth rate figures (as planned), but now we have a worthless set of data. This is dangerous since we’re erasing important data from previous time periods.
Let’s put the x variable out of our minds for the time being.
Let’s pretend we have a square function now.
Consider that if z=0, the first log would be undefined, and hence the second could not be calculated. This is evident in various calculations, as seen in the table below.
The logarithm of 0 is unknown, and the double logarithm of that is unknown as well. The natural logarithm is 0 for z=1, and the second transformation is also undefinable. Another issue that we can see here is when certain authors use logarithms arbitrarily to normalize data. When the data values are zero, the monotonic transformation could result in a potential missing data problem.
Finally, if the data ranges between 0 and 1, the logarithm transformation will result in a negative number for the calculus. As a result, the second logarithm transformation is meaningless because the entire range of data is now undefined.
The article’s conclusions are that when we employ logarithms in growth rates, one thing is certain to happen: 1) If we use logarithms to calculate potential negative values in the initial growth rate, the value becomes undefined, resulting in missing data. As time goes on, the interpretation becomes more difficult. If we apply a double log transformation to the data, the zero and negative values will become undefined, and the missing data problem will reappear. Econometricians should think about this because it’s a common question during research, and assessing the original data before applying logarithms should be a step before completing any econometric technique in order to make the best inferences.
Bibliography
R. Nau (2019). The logarithm transformation is a mathematical transformation. Data ideas obtained The logarithm transformation is a mathematical transformation. Source: https://people.duke.edu/rnau/411log.htm
R. Rodrguez Revilla, R. Rodrguez Revilla, R. Rodrguez Revilla (2014). Econometria I and II are two different types of economics. Universidad Los Libertadores, Bogot.