Econ 200H: David Reiley
Due Thursday, 16 November 2006

## Problem Set #5

As before, the following exercises come from the "problems" at the end of each chapter of Taylor. For this problem set, I recommend not spending more than about 15 minutes thinking about any single problem. (The extra-credit problems are an exception - I expect them to take longer, but they are an opportunity for those who are interested in learning more.)

1. (5 points) Taylor, problem 17.1.
2. (10 points) Taylor, problem 17.3.
3. (10 points) Taylor, problem 17.6.
4. (5 points) Taylor, problem 17.7.
5. (10 points) Taylor, problem 18.5.
6. (5 points) Taylor, problem 18.6.
7. (10 points) Taylor, problem 18.8.
8. (10 points) Taylor, problem 18.10.
9. (10 points) Taylor, problem 20.1.
10. (10 points) Taylor, problem 20.3.
11. (10 points) Taylor, problem 20.7.
12. (10 points) Taylor, problem 20.8.
13. (10 points) Taylor, problem 20.11.
14. (10 points) Taylor, problem 20.12.
15. (10 points) Taylor, problem 21.2.
16. (10 points) Taylor, problem 21.6.
17. (5 points) Taylor, problem 21.7.
18. (10 points) Taylor, problem 21.9.
19. (10 points) Taylor, problem 21.10.
20. (10 points) Taylor, problem 21A.1.
21. (10 points) Taylor, problem 22.4.
22. (5 points) Taylor, problem 22.6.
23. (15 points) Taylor, problem 22.7.
24. (15 points) Taylor, problem 22.8.
25. (10 points) Taylor, problem 23.2.
26. (15 points) Taylor, problem 23.3.
27. (5 points) Taylor, problem 23.4.
28. (10 points) Taylor, problem 23A.4.
29. (5 points) Taylor, problem 23A.6.
30. (15 points) Consider an economy with the consumption function C = 120 + 0.80Y. Suppose that investment (I) equals 320, government expenditures (G) equal 480, and net exports (X) equal -80, with no dependence on income.
1. Write down the equation for total income, and solve this equation algebraically to find the equilibrium level of GDP.
2. Suppose that government expenditures fall to 400. Find the new equilibrium level of GDP.
3. What is the marginal propensity to consume?
4. What is the marginal propensity to import?
5. Use your answers to the previous two subquestions to find the value of the multiplier.
6. Show that the multiplier correctly explains the difference between parts (a) and (b).
31. (Optional - 20 extra-credit points) Visit the site for the Index of Economic Freedom, published by the Heritage Foundation and the Wall Street Journal. Look up the freedom category (Free, Mostly Free, Mostly Unfree, or Repressed) for each of the following countries. In addition, look up per-capita GDP and the GDP growth rate for each country.

Once you have looked up the data, explain why you think the different countries compare the way they do in terms of the GDP measures. Describe any general relationship you find between "economic freedom" and the measures of GDP.
1. Sweden
2. Finland
3. Poland
4. Belarus
5. Ukraine
6. Russia
32. (Optional - 10 extra-credit points). Provide a newspaper article that illustrates a reason why GDP is not a perfect measure of productive output in the economy. Describe why GDP either overestimates or underestimates productive output in this case.
33. (Optional - 20 extra-credit ponts) In this problem, you will derive the "rule of 70," which says that to find out how many years it will take to double an amount of money earning at a given interest rate, you should take the interest rate percentage and divide it into 70 to obtain the answer.
1. For computational simplicity, we will assume that interest compounds continuously (rather than annually or daily). Let x(t) equal the amount of the investment balance at time t. For example, if we start with \$1000 at time 0, then x(0)=1000. Let r equal the rate of interest (for example, a 7% annual rate of interest would have r=0.07 and t measured in years). Recall that at any moment in time, the cash balance is increasing by the amount of interest being earned, which equals the interest rate times the current cash balance. Write down an equation for (dx/dt) that expresses this relationship.
2. Integrate both sides of the equation to get a solution for x(t), after rearranging your equation so that x appears only on the left side, and t appears only on the right side. You should end up an answer showing that the natural logarithm of x is a linear function of t.
3. Solve this equation for x by exponentiating both sides. Solve for the constant of integration using the symbol x(0) to represent the initial cash balance x at time t=0.
4. To figure out how many years it will take for the cash balance to double, we need to find the value of t where the cash balance has doubled, or where x(t)=2x(0). Solve for this value of t as a function of r.
5. Demonstrate that your result in (d) proves the "rule of 70." Note that 70 is not the exact number derived in our equations, but that it is the closest round number. The book uses 72, which is conveniently divisible by 2,3,6,8,9,12, and so on. Also, your answer in (d) should be a slight underestimate if interest is compounded only annually or monthly instead of continuously.