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Econ 100B: Economic Analysis – Macroeconomics

Problem Set #2

1. Reproducing results is a critical component of the scientific process and was the subject of some controversy in macroeconomics recently. The goal of this exercise is for you to analyze the data of Cobb and Douglas and to see if you can reproduce the coefficients they give for their production function. For this exercise you should use Excel or your favorite software package. Referring to their paper:

(a) Calculate the indices for capital and labor using the data given in Tables II and III of the paper. With your calculated indices and the index for manufacturing given in Table IV of the paper create a graph similar to the one shown in the upper panel on slide 9 of lecture 4. Annotate this graph in the same way that you did for the inflation graph in the first problem set.

(b) Create your version of the graph on slide 12 of lecture 4 as follows:

i. With these indices calculate the log-ratios shown in the final equation on slide 11 of lecture 4. These log-ratios are the data points shown in the graph on slide 12 of lecture 4.

ii. Perform a linear regression on these data and show the regression result as a line on your graph: the intercept is ln(A) and the slope is α. An easy way to do this in Excel is to add a linear trendline to the graph and (in the Trendline options) show the equation of the Trendline and the R2 on the graph.

Annotate this graph in the same way that you did for the inflation graph in the first problem set. Be sure that your graph shows your individual effort on this problem: identical graphs will not be accepted.

(c) In Section 6 of their paper, Cobb and Douglas express their production function as

which implies – in our notation – that α = 0.25 and A = 1.01. Do your regression results agree with those of Cobb & Douglas when rounded to 2 decimal places?

2. Use the calculation framework you developed in question 1 above to estimate the coefficients A and α for the Lao People’s Democratic Republic (Laos) using annual data from 1994 to 2017. The needed data is available in the Penn World Table Database as follows:

● cgdpe: Expenditure-side real GDP at current PPPs (in mil. 2017US$).

● cn: Capital stock at current PPPs (in mil. 2017US$).

● emp: Number of persons engaged (in millions).

where each of the codes (e.g., cgdpe) is a column header in the Excel version of the database. To answer this question download the database and use the data to create and submit the following:

(a) The Laotian version of the log-ratio graph you did in question 1 above.

(b) Your estimates of A and α for Laos together with a brief comparison with the values you obtained in question 1 above.

3. Prove that if the more general form of the Cobb-Douglas production function

exhibits constant returns to scale then β = 1 − α.

4. Given the equation for the Cobb-Douglas production function

where 0 < α < 1, A ≥ 0, K ≥ 0, and L ≥ 0:

(a) Derive the equation for the marginal product of capital. Show your work.

(b) Can the marginal product of capital be negative in an economy described by the Cobb-Douglas production function? Briefly explain.

5. Before the COVID-19 outbreak California – the 5th-largest economy in the world – is in equilibrium with a fixed supply of labor.

(a) Draw the graphs for (i) output as a function of labor, (ii) the marginal product of labor as a function of labor and identify the equilibrium levels of output and the real wage.

(b) The COVID-19 pandemic resulted in lockdowns that required people to work from home and parents to supervise their children who were no longer in school. This caused a downward shock to total factor productivity. On the graphs from Question (5(a)) above, add the curves and equilibrium conditions for the post-pandemic situation. Use a different color and/or line type to distinguish these post-pandemic curves from the pre-pandemic curves. Identify the equilibrium levels of output and the real wage.

(c) Briefly explain the pre- and post-pandemic economic conditions illustrated by your graphs.