Economic growth and economic development 166

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 > 0 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, 1 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 = 0. 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