Magic money tree

See if you can answer the question below set by St John’s Economics Tutor Dr Kate Doornik

You have been given some magic money.  If you plant the money then it will grow into a magic money tree, which will grow more money for you to pick the following year. To be precise, if you plant £Xnow then next year you will have £2√X growing on the tree.

This magic money is your only income, so you need to spend some of the money each year. You decide to spend three quarters of the newly grown money and plant the remaining one quarter. You also replant all the old money from the previous year, but unfortunately 1/10 of this is taken away by mice each year.

For example, if you start with £4, then in Year 1 you:

  • Grow £4 of new money
  • Spend £3 and plant the remaining £1
  • Lose 40p of the old money to mice (don’t worry about how the mice do this!)
  • Plant (or replant) a total of £1+£4-40p = £4.60 for Year 2.

You want to know if and when you are going to get rich and also how this depends on your spending and planting strategy. Try answering the following questions.

  1. Suppose you start with £4 as in the example, so you plant £4.60 for Year 2. How much new money will you grow in Year 2? How much will you spend? How much money will you plant in total at the end of Year 2? What happens in year 3? How is the amount of money that you have changing over time?
  1. What would happen if you started with £100 instead of £4?
  1. If you keep repeating the process of growing, spending and planting money, what happens in the long run to the amount of money that you have? Does it depend on how much you had to start with?
  1. What would happen if you changed the fraction of new money that you plant from one quarter to something else?
  1. Suppose that the only thing you care about is the amount of money that you get to spend each year in the long run. What fraction of your new money should you plant each year?

NOTE: This problem is based on the Solow Economic Growth Model. Robert Solow developed the neo-classical theory of economic growth and he won the Nobel Prize in Economics in 1987. He has made a huge contribution to our understanding of the factors that determine the rate of economic growth for different countries.  The savings rate that maximizes the level of long run spending is known as the “Golden Rule” savings rate.