Introduction to Modern Economic Growth result when the TFP estimate is constructed using the end date factor shares, i.e., as xˆ (t, t + T ) = g (t, t + T ) − αK (t + T ) gK (t, t + T ) − αL (t + T ) gL (t, t + T ) Explain the importance of differences in factor proportions (capital-labor ratio) between the beginning and end dates in these results Exercise 3.2 Consider the economy with labor market imperfections as in the second part of Exercise 2.10 from the previous chapter, where workers were paid a fraction β > of output Show that in this economy the fundamental growth accounting equation leads to biased estimates of TFP Exercise 3.3 For the Cobb-Douglas production function from Example 3.1 Y (t) = A (t) K (t)a L (t)1−α , derive an exact analogue of (3.8) and show how the rate of convergence, i.e., the coefficient in front of (log y (t) − log y ∗ (t)), changes as a function of log y (t) Exercise 3.4 Consider once again the production function in Example 3.1 Suppose that two countries, and 2, have exactly the same technology and the same parameters α, n, δ and g, thus the same y ∗ (t) Suppose that we start with y1 (0) = 2y2 (0) at time t = Using the parameter values in Example 3.1 calculate how long it would take for the income gap between the two countries to decline to 10% Exercise 3.5 Consider a collection of Solow economies, each with different levels of δ, s and n Show that an equivalent of the conditional convergence regression equation (3.10) can be derived from an analogue of (3.8) in this case Exercise 3.6 Prove Proposition 3.2 Exercise 3.7 In the augmented Solow model (cfr Proposition 3.2) determine the impact of increase in sk , sh and n on h∗ and k∗ Exercise 3.8 Suppose the world is given by the augmented Solow growth model with the production function (3.13) Derive the equivalent of the fundamental growth accounting equation in this case and explain how one might use available data in order to estimate TFP growth using this equation Exercise 3.9 Consider the basic Solow model with no population growth and no technological progress, and a production function of the form F (K, H), where 152