Question

(Long-Term Productivity Growth) Suppose that two nations start out in 2016 with identical levels of output per work hour—say, $100 per hour. In the first nation, labor productivity grows by 1 percent per year. In the second, it grows by 2 percent per year. Use a calculator or a spreadsheet to determine how much output per hour each nation will be producing 20 years later, assuming that labor productivity growth rates do not change. Then, determine how much each will be producing per hour 100 years later. What do your results tell you about the effects of small differences in productivity growth rates?

Answer 1

The two nations start at the productivity level of $100 of output that can be produced per hour employing the existing resources. Now the labor productivity growth rate is the rate at which the productivity of the labor increases leading to an increase in output. Now to calculate the output and the change in it after a) 20 years and b) 100 years, where in both cases the growth rate of out put remain constant, i.e. fixed at the given levels.

Therefore, Initial Productivity rate = 100/hour

Growth rate of labor= 1%/ year in first nation and

= 2%/ year in the second nation

Now the formula for calculation of the changes in output is,

Y _time= Productivity rate*(1+growth rate) ^time
or

^{GDP _time =} Productivity rate*(1+growth rate) ^time

It is form the calculation for the value of output that can be obtained with a certain growth rate in productivity of labor or workers following the basic formula of Future value^{_{at t}}= Present value*(1+g)^t

Intuitively, (1+g) ^time is the calculation for the growth rate and is multiplied by current or initial productivity rate to arrive at the total output at a time point per hour. Hence using the formula we can calculate the output for 20 years and 100 years as follow,

For nation A,

Y ₂₀ = 100*(1+0.1) ²⁰

Y ₂₀ =122.01900399479668244827490915526

Similarly,

Y ₁₀₀ = 100*(1+0.1) ¹⁰⁰ = 100*(1.01¹*1.01²*1.01³*.....*1.01¹⁰⁰)

Implies, Y ₁₀₀ = 270.48138294215260932671947108075

For Nation B,

Y ₂₀ = 100*(1+0.2) ²⁰

Y ₂₀ =148.59473959783543420355740092833

Similarly,

Y ₁₀₀ = 100*(1+0.2) ¹⁰⁰ = 100*(1.02¹*1.02²*1.02³*.....*1.02¹⁰⁰)

Implies, Y ₁₀₀ = 724.46461182523356351557281920319

Hence the differences in growth rate across the nation and are,   

	Y 20	Y 100
A	122.019004	270.4813829
B	148.5947396	724.4646118
B-A	26.5757356	453.9832289

The chart above shows the actual and the difference in per hour rate of productivity between country A with growth rate of 1% productivity and country B with growth rate 2% productivity per year. In A after 20 years productivity increase to 122 (approximate) per hour and after 100 years increases to 148 per hour. However in the second country, B, with only 1% more growth rate i.e. at 2% constant per year shows an increase of productivity to around 270 in only 20 years and a leap to 724 per hour in 100 years. When compared, the 1% extra productivity growth led to a huge increase in physical terms of productivity growth. The calculation would place the figures in trillions.

production here is considered to be conducted using, Financial, Physical and Human capital along with technological innovation. The technological innovation is considered to enhance labor productivity significantly. Thus, the above calculation shows that in long term if the labor productivity can be increased by even 1% point, would lead to a huge increase in the output per hour and thus the total gross domestic product too. In the real world, the weak USA economy observed a growth rate of around 1% per year as in country A, and high growth rates at around 3%, also countries like China and Japan have experienced higher growth rates in labor productivity.