Econ problem set

Homework 2

April 20, 2022

Overview Due date: Friday, April 29

In this exercise, you will make use of the spreadsheet provided on the class website.

The spreadsheet features data with different tabs, corresponding to different exercises in the homework. We have constructed some simulated data by choosing parameter values and following the Solow model structure. At the top of tabs 1a and 2a we have specified parameters of the Solow model: depreciation (δ), the Cobb-Douglas technology parameter (α), the savings rate (s) along with exogenous growth rates for TFP (x) and population (n). In addition, we have specified initial conditions for the level of technology (A0), population (L0), and capital (K0) in the initial period (the year 1955). For tabs 1b and 2b you are invited to adjust the parameter values and compare the model-simulated growth to real life data (which you will find online). For tabs 3 and 4 we supply growth rate patterns for population and technology that don’t follow the simple constant growth rate structure, while the other aspects of the Solow model are left unchanged.

Recall that the Solow model specifies a production function of the form:

Yt = AtKαt L 1−α t

where t indexes the time period in which production takes place.

When asked to fill in values for real wages, rental rates, and income shares you may assume that these values are pinned down by cost minimization behavior and that all available inputs (labor and capital) are fully utilized.

Question 1 - Solow Model for US (25 points) For this questions please use sheets 1a and 1b.

(a) (10 points) In sheet 1a in columns G through M, please construct spreadsheet formulas to calculate the requested values using the Solow model. Specifically, please calculate: output per efficiency units (ŷ = Y/ÃL), output per capita, capital-labor ratio, the labor share of income, real wages, the growth rate of GDP per capita, and the real rental rate of capital.

(b) (10 points) In sheet 1a in cells E2 and E3, please calculate the long-run stable level of capital per effective worker (k̂ = K/ÃL) and output per effective worker (ŷ = Y/ÃL). In cell H2 enter the first year in our simulated data when the capital per effective worker is within 1% of its long-run stable level. For the year in cell H2, how large is the gap between the growth of output per worker in the simulated data and the growth rate that would prevail on the balanced growth path?

(c) (5 points) Download data on US real GDP based from the Penn World Tables at the link here. Enter US real GDP (PPP), series rgdpe, into column G of sheet 1b and population, series pop, into column H, and calculate real GDP per capita growth in column I. How does the simulated data in sheet 1a compare to this data? Try adjusting any model parameters or initial conditions to better match the simulated and observed data.

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Question 2 - Solow Model for China (35 points) For this question, please use sheets 2a and 2b. Relative to the scenario for the US (on sheet 1a) we allow for faster population growth, a larger initial population, and a smaller initial capital stock. In addition, we freeze China’s TFP growth for the first 20 years (1955-1975) and then have China’s TFP match the US level from 20 years prior (e.g. Chinese TFP in 2020 equals the US level in 2000).

(a) (10 points) In sheet 2a in columns G through M, please construct spreadsheet formulas to calculate the requested values using the Solow model using the given data and parameters for the China model.

(b) (10 points) In sheet 2a in cells E2 and E3, please calculate the long-run stable level of capital per effective worker (k̂ = K/ÃL) and output per effective worker (ŷ = Y/ÃL) for the China model. Are these values higher or lower than the corresponding values for the US? Explain why the two values differ. In cells H2 and H3, calculate the long-run stable level of capital per effective worker and output per effective worker if there is no TFP growth (as we assumed for 1955-1975). Are these values higher or lower than the values for the scenario with TFP growth? Explain your answer.

(c) (10 points) Given the assumptions in sheet 2a, what is the long-run growth rate of GDP per capita in China? In our model, is growth in China increasing or decreasing over time? How does growth change between 1974 and 1976? How does growth evolve after 1976? Explain why the model and other assumptions generate this pattern.

(d) (5 points) Download data on Chinese real GDP based from the Penn World Tables at the link here. Enter Chinese real GDP (PPP), series rgdpe, into column G of sheet 2b and population, series pop, into column H, and calculate real GDP per capita growth in column I. How does the simulated data in sheet 2a compare to this data? Try adjusting any model parameters or initial conditions to better match the simulated and observed data.

Question 3 - One Child Policy (20 points) For this question, please use sheet 3. In addition to liberalizing its economy under the Reform and Opening Up policies of Deng Xiaoping, China also experienced a profound change in its demographic patterns in part driven by the imposition of the One Child policy. In this section, we adjust the model from question 2 to account for these demographic changes.

(a) (5 points) In sheet 3, construct an alternative path for Chinese population (column B, rows 15 to 110) as follows. For 1955 to 2019, use the population growth data from Penn World tables. Then, from 2019 to 2050, assume a constant growth rate of -1% (n = -0.01). Based on the Penn World tables data what was the average growth rate of China’s population each of the 20-year periods 1955-1975, 1975-1995, and 1995-2015? (Note: columns C, D, E, and F should update automatically when you fill in column B).

(b) (5 points) Construct the deduced variables in columns G through M using the new path for population growth. How does the new path for population growth affect the growth rate of total output? What about the growth rate of output per worker? For both variables, is there a permanent (i.e. on the balanced growth path) effect to the growth rate from this change in population growth?

(c) (10 points) Using the growth accounting procedure, what is the contribution of capital, labor, and technology growth to Chinese growth in questions 2 and 3. Please fill in these values in columns O, P, and Q for sheets 2a and 3. Explain the differences.

Question 4 - Technology Catch-Up (20 points) For this question, please use sheet 4. In addition to China beginning a period of TFP growth, we might expect that China begins to close the gap in technology relative to the US. In this question we construct an

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alternative path for TFP that allows for a non-constant growth rate of TFP over time in China. Note that this sheet will automatically pull in the population path you built in sheet 3.

(a) (5 points) Inspect the formula in column C for years 1976 onward (row 36 on). Describe in words what you think this formula is doing. In questions 2 and 3 what happened to the relative level (ratio) of TFP in the US and China over time? With this new formula what happens to the relative level of TFP over time? Will this ratio tend to a particular value in the long run?

(b) (5 points) Construct the deduced variables in columns G through M. How does Chinese growth in this scenario compare to what you saw in questions 2 and 3? Does this do a better or worse job of matching the observed data?

(c) (10 points) Given the new path for TFP, in the long-run (e.g. 2100 onward) will Chinese living standards grow faster, slower, or at the same rate as in the US? Will the level of Chinese living standards be higher, lower, or the same as in the US? Explain your answer.

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