Question 1
Growthistan has a very technologically advanced neighboring country, Techistan, whose initial level of technology is A*=10. Suppose that the economy operates with a production function of the form Y(t) = A x [K(t)]0.5.
Let's compare the growth of both countries from 2005 to 2100.
(a) Suppose that both Techistan and Growthistan have s=0.1, d=.05, and initial level of capital K0=2. Growthistan's level of technology is 2. Plot Y(t), ln[Y(t)] and the growth rate for both countries. What happens to the level of Y(2100), and what happens to the growth rate of the economy toward the year 2100? Compare the difference.
Techistan has just decided to start trading liberally with Growthistan. As those two countries trade over time, technology diffuses from Techistan to Growthistan. The rate of diffusion depends on the proportionate technology gap between these two countries, and is governed by this equation:
A(t+1) = A(t) + g x [A* - A(t)]
Where A(t) is Growthistan's technology level, and A* is Techistan's technology level. The starting value of Growthistan's technology, A(2005) = 2.
g is a constant of technological diffusion whose value is 0.05
The production function is Y(t) = A(t) * [K(t)] 0.5
s = 0.2; d = 0.05; g = 0.05; K0 = 5
(b) Graph Growthistan's GNP for 2005 to 2100.
(c) Compare the two graphs for GNP for Techistan and Growthistan. What is the difference in the final value of GNP for each country? (1 paragraph)
(d) Plot the growth rate for Techistan and Growthistan on one plot. How do these rates compare? Why? (1 paragraph)
(e) Can you think of two real-world countries whose growth paths have been similar to this? Who and why? (1 paragraph)
Question 2
Following the example in section, we will look at an intertemporal problem of nonrenewable resource depletion.
Assume that the remaining oil supply in the world is
S = 6,000 units
The energy demand during each period is
D = 100,000 / P
And the annual interest rate is 3.5% so that R = 1 over 20 years (like in the lecture slides).
a) We are now interested in three periods: today (period 1), 20 years from now (period 2), and 40 years from now (period 3). Assume that there is a backstop technology currently available, and its price is $100. Solve for the price of energy and the consumption of energy in the three periods, as well as how much of it in every period comes from oil versus the backstop technology.
b) We will now compare scenario (a) with one where today's society can forego some consumption in the present to develop alternative technologies for the future. Assume that solar power can be made available for period 2 at a price of $50 only if we choose today to pay the equivalent of 1000 energy units into R&D (assume that consumption is equal to the amount of energy consumed minus the amount put into R&D). Under this scenario, when will we begin to use solar power? Is any time period better off (in terms of consumption) compared to time period (a)? Will the present generation choose to invest in the R&D, assuming it cares about its own well-being much more than about the future?
c) Now assume that the price of solar technology made available by R&D for period 2 is $35. What is the new energy use for the three periods? Will today's generation invest in the R&D, assuming it cares about its own well-being much more than about the future?