Elasticities and percentage change

Problem

Data: http://media.nek.lu.se/data/elasticities.xlsx

In this problem, we will confirm that \(β_2=\left( ∂y/∂x_2 \right)(x_2/y)\) is approximately equal to the percentage increase in y when \(x_2\) increases by 1%. We also confirm that \(β_2=∂log y/∂log x_2\) which is approximately equal to the change in log y over the change in log x2.

In the excel file Part 6 you have the following (see previous problem for notation)

Column D/E calculates y, log x2, log x3 and log y. You set start values x2 and x3 (check formulas in E3)

Column G/H calculates y, log x2, log x3 and log y. You set end values x2 and x3.

B7 has the percentage change in y (start in E2 end in H2)

B8/B9/10 has the change in log x2, log x3 and log y

B11/B12 has the ratios (set to zero if infinite)

  1. Increase end value of x2 by 1%, that is, enter 1.01 in cell H1. What is the percentage change in y? Confirm that it is close to beta2. Do the same for x3.
  2. What is the change in log y over the change in log x2? Confirm that it is close to beta2.
  3. Do the same thing, increasing x2 by 0.1%. What is the percentage change in y? What is the change in log y over the change in log x2.
  4. Now change beta2 to -0.3 and do a-c again.