Question: to obtain a somewhat deeper understanding of the results in...
To obtain a somewhat deeper understanding of the results in Jones and Klenow’s analysis of welfare across countries, let us consider how they actually compute welfare of the citizens in a country, to be distinguished from mere GDP/capita. Jones and Klenow take into account life expectancy, the consumption/output ratio, leisure, and inequality; we will illustrate how they deal with three of these four. The general idea is to use a utility function—an idea borrowed from microeconomics— and to assume that all consumers in the country have a specific such function u(c,l),where c is consumption and l is leisure. The empirical literature has offered estimates of what such u functions look like in real life: functions that well approximate how people actually behave.
(a) We begin with consumption only and the function that is most often used: u(c) = log c (again we encounter the log!). First plot this function in a diagram. With this function, consider the following data: total GDP in countries 1 and 2 is 100 and 140, respectively, the population size is 8 and 10, respectively, and the consumption-output ratios are 0.25 and 0.2, respectively. Which country’s citizens have higher utility per person?
(b) Now people enjoy leisure too so let us extend the utility function above to u(c,l) = log c + ψlog l. The parameter ψ can be estimated from individual data; let us say ψ= 0.5. Now add to the information above about the two countries that country 1 on average has 6 hours of leisure per day whereas country 2 has 10. Which country has higher utility per citizen?
(c) To understand the role of inequality, let us abstract from leisure and just consider consumption. The idea here is that a country with higher inequality will be “riskier” to live in for an average citizen if this citizen is rich with some probability and poor with some other probability. So suppose that there are only two levels of consumption in each country and that the ratio of the high to the low consumption level is 5 in country 1 and 10 in country 2. Moreover, suppose that you become poor or rich as a result of an even coin flip; i.e., half the population is rich and half is poor in each country. Using the data on average consumption per capita above, and the information here, in which country would you prefer to live? Hint: To answer this question you need to calculate expected utility (Eu):