Question

In macroeconomics there are two key questions: first, how to measure (real) output growth? and second, how is growth connected to wellbeing?

As for the first question, there are three ways to measure real GDP growth. It starts with the definition of a percentage difference between two numbers, say x1 compared to x0. When there is only one good (e.g. apples) there are no prices to worry about for the calculation. But for the economy as a whole, we need prices to be able to add up quantities (of apples, bananas, and everything else) in dollar terms. So for real GDP growth, which holds prices constant on a given base year to control for inflation, then the trick is to apply those prices before, during, and after that base year in order to calculate the total dollar value of all goods and services produced and track growth that is not affected by inflation.

In this exercise you will work out the connection between the three ways to measure GDP growth: A) As a percentage change holding prices constant. B) Expressing growth in terms of a common item (converting everything into a particular good (e.g. apples). And C) as the weighted average of the growth in each of the goods weighted by their corresponding expenditure shares.

The second question in macroeconomics refers to a mapping from goods and services to subjective well-being. That is, why do we care about economic growth, after all?

Instructions

Work out the algebraic steps that:

1. take you from the first to the second equation below
2. take you from that same top equation below; and

   . Fix a base year price

   ???? ????+????? ????−?=??,?∗??+?+??,?∗??+???,?∗??+??,?∗??−?〖real GDP〗_(t+1)/〖real GDP〗_t -1=(p_(a,t)∗a_(t+1)+p_(b,t)∗b_(t+1))/(p_(a,t)∗a_t+p_(b,t)∗b_t )-1

   2. Or in terms of one good, say, apples (divide numerator and denominator by ??,?p_(a,t))

   ???? ????+????? ????−?=??+?+??,???,?∗??+???+??,???,?∗??−?〖real GDP〗_(t+1)/〖real GDP〗_t -1=(a_(t+1)+(p_(b,t)/p_(a,t) )∗b_(t+1))/(a_t+(p_(b,t)/p_(a,t) )∗b_t )-1

   In the first, real GDP is in constant year t dollars, in the second, real GDP is measured in units of apples.

   An equivalent representation is a weighted average of the growth in each of the goods based on their expenditure shares.

   ???? ????+????? ????−?=????+???+?−????+???−?〖real GDP〗_(t+1)/〖real GDP〗_t -1=(θ_t ) ̂(a_(t+1)/a_t )+(1-(θ_t ) ̂ )(b_(t+1)/b_t )-1

   where ??(θ_t ) ̂ is the fraction of nominal GDP accounted for by purchases of apples, and ?−??(1-(θ_t ) ̂ ) the share accounted for by purchases of bananas

3. prove the relationship Why does real GDP growth matter? Well, suppose that households get utility from apples and bananas every period t:

   ??=?????+?−?????u_t=θln(a_t )+(1-θ)ln(b_t )

   ??+?=?????+?+?−?????+?u_(t+1)=θln(a_(t+1) )+(1-θ)ln(b_(t+1) )

   Then,

   ??+?−??=???+???+?−???+???−?u_(t+1)-u_t=θ(a_(t+1)/a_t )+(1-θ)(b_(t+1)/b_t )-1

   So, if ?θ ̂ = ?θ, then utility increases whenever real GDP growth is positive.

You will find all the algebraic steps within the body of Chapter 1. Your task is to write up a two page ‘study guide’ that compiles the derivations of points (1)-(3) above. Given that the answers are in the book, this exercise will be graded as either submitted (100) or not (zero).

Answer 1