Exercise 2. The basic Solow model in continuous time This exercise asks you to analyse the Solow model in continuous time as given by the six equations (21)-(26). The restrictions on the parameters a, B, s, n and o are the same as in the model in discrete time, except we do not have to assume /> -1. It is assumed that n+ d>0. We use (for any point t in time) the definitions k = K/L and y = Y/L etc. 1. Show that from k= K/L it follows k/k=K/K-L/L. (Hint: You can do the 'log-dif-trick', that is, first take logs on both sides of k = K/L and then differentiate with respect to time). Then show that the Solow model in continuous time implies the Solow equation: k = SBk" - (n+ 0)k. (41) Compare with (32) for the model in discrete time and give an intuitive explanation like the one given for (32) in this chapter.2. Illustrate the above Solow equation in a Solow diagram with k along the horizontal axis and the curve sBk" as well as the ray (n + o)k along the vertical axis. Demonstrate (from the Solow diagram) that from any initial value, ko > 0, the capital intensity, k, will converge to a specific steady state level, k*, in the long run (as t - 09). Nk College3. Compute the steady state values, k", y" and c*, for the capital-labour ratio, income per worker, and consumption per worker, respectively (use k = 0). Explain and compare to the parallel expressions found in this chapter for the model in discrete time. Also compute the steady state values, , and w*, for the real rental rates for capital and labour and compare again to the model in discrete time. How are * and w affected by an increase in s? Explain. What are the growth rates of y and Y in steady state?4. Show that (it follows from the model that) the growth rate in k at any time is: = sBk"-1 - (n+o). (42 k Illustrate this in a modified Solow diagram. Show that the growth rate, y/y, at any time t is a times the growth rate, k/k, of k. Assume that initially Ko