A Problem Set 1

A.1 Geometric Sums

  1. Calculate the following geometric sum, when \(x \neq 1\), as a function of \(x^{n+1}\) in particular: \[\sum_{i=0}^n x^i = 1+x+x^2+...+x^n\]

  2. State a condition on \(x\) such that the infinite geometric sum (when \(n \to \infty\)) has a finite value.

  3. Assuming that this geometric sum is finite, calculate: \[\sum_{i=0}^{\infty} x^i = 1+x+x^2+...+x^n+...\]

  4. More generally, for \(m\) a positive integer, calculate: \[\sum_{i=m}^n x^i = x^m+x^{m+1}+...+x^n\]

  5. What is the present discounted value of an infinite stream of incomes, which grows at rate \(g=2\) %, starts at \(y_0=90000\), if the interest rate is \(i=3\) %?

A.2 Taylor Approximations

  1. If \(x\) is small, and \(n\) is an integer, show that: \[(1+x)^n \approx 1+nx.\] Compute the two sides of the equation exactly for \(x=0.02\), and \(n=3\).

  2. If \(x\) and \(y\) are small, then show that: \[(1+x)(1+y) \approx 1+x+y.\]

  3. If \(x\) and \(y\) are small, then show that: \[\frac{1+x}{1+y} \approx 1+x-y.\]

  4. Your savings account offers a nominal interest rate of 1%. Meanwhile, annual inflation is 1.5%. What is an exact value for the real interest rate, defined as the rate of increase in your purchasing power if you leave your money in the bank? What is an approximate value for this real interest rate, using one of the previous formulas?

A.3 Growth Rates

  1. If \(y_t\) grows at a constant rate \(g\) during a given period, then show that the growth \(G\) of \(y_t\) after \(T\) periods is: \[G = \frac{y_T}{y_0}-1=(1+g)^T-1.\]

  2. Conversely, if the growth rate of \(y_t\) after \(T\) periods is \(G\), then show that the average growth rate of \(y_t\) per period is: \[g = \frac{y_{t+1}}{y_{t}}-1=(1+G)^{1/T}-1.\]

  3. If your annual return on your savings rate is 1%, what is your daily return (assuming a year is 365 days)? How much do you then give up each day by leaving $100K on a zero interest checking account?

A.4 Replicating Figures 1.1, 1.2, and 1.3

In this exercise, we use Google Sheets in order to replicate Figures 1.1, 1.2, 1.3. from Lecture 1. In doing so, we learn more about trends and fluctuations in GDP, and how hard it is to distinguish between the 2.

  1. Download Tables 1.1.6 (or Table 1.2.6) from the National Income and Product Accounts (NIPA) of the Bureau of Economic Analysis (BEA) https://apps.bea.gov/iTable/index_nipa.cfm, named NIPA Table 1.1.6 for short. (download all years)

  2. Create a new sheet, and copy-paste a column with Year, and another one with Real GDP numbers.

  3. Create a new column with the log of Real GDP. Plot Real GDP.

  4. Replicate Figure 1.1 using the FORECAST function, which helps fit a linear trend through the data, until 1971.

  5. Replicate Figure 1.2.

  6. Replicate Figure 1.3, computing Real GDP as a percentage of Trend GDP.