Growth Calculations

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Purposes:

• Understand relationship among nominal GDP, real GDP, and inflation
• Become comfortable with calculating processes that involve compound growth
• Understand how ratios, such as output per worker, are affected by different rates of growth of numerator and denominator

Interest on a bank deposit

• Y₀ is the money you deposit
• n is the number of years you put the money on deposit
• r is the annual interest rate, expressed as a decimal
• Y_n is the balance you end up with

[1] Y_n = Y₀(1+r)ⁿ

Can solve for any variable, as long as we know the other three.

Example of solving for the ending balance. Suppose that:

• we deposit $1000; Y₀ = 1000
• interest rate is 10 percent; r = .10
• 4 years; n = 4

ending balance, Y₄ = $1464.10

In a spreadsheet, such as Microsoft Excel, you use the formula +1000*(1+.10)^4

Next example--Solve for beginning balance. Same example as before, except we want to know how much to deposit now in order to have $1600 in four years.

1600 = Y₀(1+0.10)⁴

Divide 1600 by (1.1)⁴ to get $1092.82

Next example--Solve for interest rate. Suppose you start with $1000 and end up with $1600 after four years. What was the interest rate?

1600 = 1000(1+r)⁴

Three steps:

1. Divide both sides by 1000
   1.6 = (1+r)⁴
2. Get rid of the exponent by taking both sides to the 1/n power
   (1.6)^(1/4) = (1+r) = 1.125
3. Solve for r and convert to percent: r = .125 = 12.5 percent

Final example--Solving for the number of years. Suppose you start with $1000, the interest rate is 10 percent, and you want to know how long it will take until you have $2000. Here, we use logs (you can use either regular logs or natural logs).

logY_t = logY₀ + t[log(1+r)]

Solving for t gives

t = (logY_t - logY₀)/log(1+r)

In our example

t = (log[2000] - log[1000])/log[1+.10] = 7.27 years

For problem set, note that:

• real GDP = (nominal GDP)/(price index)
• real GDP per capita = (real GDP)/(population)

In general, a ratio increases when the numerator grows faster than the denominator, and vice-versa.