Exercise 3.10. Consider a constant returns to scale production function for country j, Yj = F (Kj , AjHj ), where Kj is physical capital, Hj denotes the efficiency units of labor and Aj is labor-augmenting technology. Prove that if Kj/Yj = Kj0/Yj0 in two different countries j and j0 , than the rental rates of capital in the two countries, Rj and Rj0 will also be equal. Exercise 3.11. Imagine you have a cross-section of countries, i = 1, ..., N, and for each country, at a single point in time, you observe labor Li, capital Ki, total output Yi, and the share of capital in national income, σK i . Assume that all countries have access to a production technology of the following form F (L, K, A) where A is technology. Assume that F exhibits constant returns to scale in L and K, and all markets are competitive. (1) Explain how you would estimate relative differences in technology/productivity across countries due to the term A without making any further assumptions. Write down the equations that are involved in estimating the contribution of A to crosscountry income differences explicitly. (2) Suppose that the exercise in part 1 leads to large differences in productivity due to the A term. How would you interpret this? Does it imply that countries have access to different production possibility sets? (3) Now suppose that the true production function is F (H, K, A) where H denotes efficiency units of labor. What other types of data would you need in order to estimate the contribution of technology/productivity across countries to output differences. (4) Show that if H is calculated as in Section 3.5, but there are significant quality-ofschooling differences and no differences in A, this strategy will lead to significant differences in the estimates of A.