Consider the following Romer model: Yt = A t Lyt At+1 = AtLat Lyt + Lat = Lt Lyt Lat = x, where Yt is output at time t, At is knowledge, Lyt is the number of workers producing output, Lat is the number of workers producing knowledge, Lt is the total number of workers in the economy, and , , and x are parameters. Suppose Lt = L, (0, 1], and x > 1. (a) Derive the expression for the growth rate of knowledge. (b) Derive the expression for the growth rate of output per worker, yt = Yt L . (c) Let A0 denote the initial value for knowledge. Derive the expression for the level of output per worker at any time t as a function of A0 and parameters. (d) Draw a diagram with output per worker on the vertical axis (using a ratio scale) and time on the horizontal axis. Suppose there is a one-time permanent reduction in the workforce because of out-migration. Show graphically what will happen to output per worker over time.

2. Consider the following Romer model: Y: 2 1ng AAt+1 wAtLat Ly: + Lat : Lt where Y; is output at time t, A; is knowledge, Ly; is the number of workers producing output, Lat is the number of workers producing knowledge, L, is the total number of workers in the economy, and a, (D, and a? are parameters. Suppose L, : L, a e (0, 1], and 5: > 1. (a) Derive the expression for the growth rate of knowledge. (b) Derive the expression for the growth rate of output per worker, y, : ZLL. (c) Let A0 denote the initial value for knowledge. Derive the expression for the level of output per worker at any time t as a function of A0 and parameters. (d) Draw a diagram with output per worker on the vertical axis (using a ratio scale) and time on the horizontal axis. Suppose there is a one-time permanent reduction in the workforce because of outmigration. Show graphically what will happen to output per worker over time