2. OLG model with labor-augmenting technological progress Consider the OLG model in which the production function takes the form F(K, L) = K (AL)-a so that A+ is a measure of labor efficiency. There is growth in At according to A++1 (1+g) At. Similarly, population growth follows Lt+1 = (1+n) Lt. (5) (a) Argue that, to obtain a steady-state representation of the economy, we should use capital per effective worker k = K/(ALt) rather than (Kt/Lt) as in class. Express y=Y/(A+Lt) as a function of kt. (b) Using the new definitions for yt and kt, write the firm maximization problem, and use the first order conditions to derive expressions for the rental rater and wage rate w. Assume households have logarithmic preferences of the form U(cit, C2t+1) = log ct + Blog C2,1+1 (13) so that the savings rate s(r++1) = /(1+B). (No need to prove this fact; we derived it in class, and you can use it freely.) (c) Derive the law of motion of capital per effective worker k++1. Start by using the accumulation equation for aggregate capital Kt+1 = Ls(rt+1)Wt. (d) Interpret the law of motion of capital. What is the main difference compared to the formula derived in class? (e) Finally, rearrange the law of motion of k into an expression for aggregate capital K+ as a function of aggregate output Y = K(AL)-a. Interpret.