Consider the continuous time Solow growth model. Assume that population and tech- nology grow at constant rates over time L . E = n and g = g, production is Cobb-Doulgas Y (t) = K 3)\" (A (t) L 6))\". Capital depreciatas at rate 6 . K=I(t)6K(t), and the savings rate is constant I (t) = sY (t) (a) Let y (t) = L denote output per eiciency unit of labor and k ()5) denote AmL) _ capital per efciency unit of labor. Derive the equation for k. (1)) Plot the equation for Ii: _(not the components of the equation seperately but the entire equation, that is Is: is measured on the vertical axis and Is: on the horizontal axis). Explain how this compares to the phase diagram for the Solow model in discrete time and the graph of actual investment vs. replacement investment for the continuous time model. (c) Under what circumstances do is and y converge to steady state values using your diagram? Explain. Solve for Hand y'\". (d) Consider a one-time increase in the population due to an inux of immigrants. What are the short-run and long-run effects on this economy? Explain using your diagram, and plot the evolution of the growth rates of capital and output per effective unit of labor as well as the level of output per worker. I (e) Consider an income tax so that instead of receiving wL + rK = Y consumers receive (1 1') wL + (1 T) rK = (1 7') Y. Trace the consequences of this tax for outcome per worker in the short and long run. Assume that the economy starts with no tax in steady state