Q1) Show how to arrive to Equation (8.16) of the textbook, and then find the optimal amount to spend on education E and the number of children m. Provide a brief interpretation about how these equation help understand the demographic transition.

Q2) The importance of growth rates versus productivity levels. Consider two economies, A and B. Both economies are described by the model in Section 8.3, having a population growth function similar in form to that in Figure 8.6. You know that the population growth function’s peak is at L*/L = 0.02, or 2% per year. Both economies start with a productivity level of B = 1. In economy A, productivity grows at 0.5% per year for 1000 years. In economy B, productivity is stagnant at B = 1 for 800 years. Then, for the next 200 years productivity grows at the rate 2.5% per year.

(a) In the year 800, how much larger is productivity in economy A than in economy B?

(b) In the year 1000, how much larger is productivity in economy A than in economy B?

(c) Given what you know about the population growth function, will economy A ever take-off to sustained growth in income per capita? Will economy B be able to transition to sustained growth?