ECON 0028: The Economics of Growth Term 1 (Fall 2022) ASSIGNMENT 2
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ECON 0028: The Economics of Growth
Term 1 (Fall 2022)
ASSIGNMENT 2
Due on November 11 (Fri), 9am
1. A warm-up question: How is population growth incorporated into the Solow model? Why does the model predict that countries with higher population growth rates will have lower steady-state income per capita? Speculate on the following question (max one sentence answers please): how might these conclusions change once you incorporate the fact that more people means more brains means more ideas?
2. For each of the following scenarios, identify which variable or parameter in the Malthusian model changes as a result. Use the graphical depic- tion of the Malthusian model to illustrate what happens to a country’s population size and per-capita income in the short run and in the long run.
(a) Scientists discover a new strain of wheat that can produce twice as much grain per acre.
(b) New strains improve yields by a given percentage each year (so there is continual growth in agricultural technology). At some point scientists discover how to run a scientific research lab and learn to collaborate, which increases the pace of invention (the percentage improvement that happens each year is now higher).
(c) A war kills half of the population.
(d) A volcanic eruption kills half the people and destroys half the land.
(e) Antibiotics get discovered (hint: there is no explicit variable or a parameter in the model that stands for health technology or medicine standards. But there is a parameter that gives the level of income per capita at which people will have roughly 2 surviving children. Which parameter is it? How will discovery of antibi- otics change this parameter, i. e. will people need more or less income per capita to sustain 2 surviving children? Once you an- swer this question, trace the effects on population size and income per capita.)
3. Derive the formula for the steady state level of income per capita in the
Solow model with population growth and no exogenous technological progress. Try to do this without looking at the slides / book. Make sure to start your derivations from the capital accumulation equation K˙ = sY _ δK, which you want to convert to a differential equation in a variable that will be constant in steady state (e.g. capital per worker).
Suppose that there are two countries, X and Y, that differ in both their rates of investment and their population growth rates. In Country X, investment is 20% of GDP and the population grows at 0% per year. In Country Y, investment is 5% of GDP, and the population grows at 4% per year. The two countries have the same levels of productivity A. In both countries, the rate of depreciation, δ, is 5%. Using your formulas, calculate the ratio of their steady-state levels of income per capita, assuming that α = 1/3. Think about which countries in the world roughly fit the description given above (if any).
4. Consider the Solow model with population growth. Assume that pop- ulation can grow at two different rates n1 and n2 , where n1 > n2 . The population growth rate depends on the level of output per capita (and
therefore the level of capital per capita). Specifically, population grows at rate n1 when k < k and slows down to rate n2 when k > k .
Draw a diagram for this model. Assume that (n1 + δ)k > sf (k) and that (n2 + δ)k < sf (k). Explain what the diagram says about the steady state of the model.
2023-04-28