3.15. Consider the following variant of our model with physical and human capital:

where a and a are the fractions of the stocks of physical and human capital used in the education sector. This model assumes that human capital is produced in its own sector with its own production function. Bodies (L) are useful only as something to be educated, not as an input into the production of final goods. Similarly, knowledge (A) is useful only as something that can be conveyed to students, not as a direct input into goods production.

(a) Define k = K/AL and h = H/AL. Derive equations for k and h.

(b) Find an equation describing the set of combinations of h and k such that k = 0. Sketch in (h, k) space. Do the same for h = 0.

(c) Does this economy have a balanced growth path? If so, is it unique? Is it stable? What are the growth rates of output per person, physical capital per person, and human capital per person on the balanced growth path?

(d) Suppose the economy is initially on a balanced growth path, and that there is a permanent increase in s. How does this change affect the path of output per person over time?

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