Problem
This is an exercise to understand the Solow model
(a) The economy of Ping Pong produces its GNP using capital and labor. The labor force is growing at 2% per year. At the same time, there is "labor-augmenting" technical progress at the rate of 3% per year, so that each unit of labor is becoming more productive. How fast is the effective labor force growing?
(b) Now let's look at the production possibilities in Ping Pong. We are going to plot a graph with capital per unit of effective labor (k) on the horizontal axis and output per effective unit of labor (y) on the vertical axis. Here is a description of the "production function" that relates y to k. As long as k is between 0 and 3, output (y) is given by y = (I/2)x. After k crosses the level 3, an additional unit of k only yields one-seventh additional units of y. This happens until k reaches 10. After that, each additional unit of k produces only one-tenth additional units of y. (To draw this graph, you may want to measure the y axis in larger units than the k axis; otherwise, the graph is going to look way too flat) On a graph, plot this production function. What are the capital-output ratios at k = 2, 6, and 12? Note that the answers you get in the case k = 6 and 12 are different from what happens at the margin (when you increase capital by one unit). Think about why this is happening.
(c) Now let us suppose that Ping Pong saves 20% of its output and that the capital stock is perfectly durable and does not depreciate from year to year. If you are told what k(t) is, describe precisely how you would calculate k(t + I). In your formula, note two things: (i) convert all percentages to fractions (e.g., 3% = 0.03) before inserting them into the formula and (ii) retnember that the capital-output ratio depends on what the going value k(t) is, so that you may want to use a symbol like 0 for the capital-output ratio, to be replaced by the appropriate number once you know the value of k(t) (as in the next question).
(d) Now, using a calculator if you need to and starting from the point k(t) = 3 at time t, calculate the value of k(t + l). Likewise do so if k(t) = 10. From these answers, can you guess in what range the long-run value of k in Ping Pong most lie?
(e) Calculate the long-run value of k in Ping Pong. (Hint: You can do this by playing with different values or, more quickly, by setting up an equation that tells you how to find this value.)