2. Embodied technological progress. (This follows Solow, 1960, and Sato, 1966.) One View of technological progress is that the productivity of capital goods built at t depends on the state of technology at t and is unaffected by subsequent technological progress. This is known as embodied technological progress (technological progress must be \"embodied\" in new capital before it can raise output). This problem asks you to investigate its effects. (a) As a preliminary, let us modify the basic Solow model to make technological progress capital-augmenting rather than labor-augmenting. So that a balanced growth path exists, assume that the production function is CobbDouglas: Y(t) : [A(t)K(t)]\"L(t)1'". Assume that A grows at rate ,u: Aft) = MIG). Show that the economy converges to a balanced growth path, and find the growth rates of Y and K on the balanced growth path. (Hint: Show that we can write Y/(Ach) as a function of K/ (A'pL), where (p = cit/(1 (I). Then analyze the dynamics of K/ (A'PL). ) (b) Now consider embodied technological progress. Specifically, let the production function be Y(t) = ](t)\"Mt - 03, where ](t) is the effective capital stock. The dynamics of ](t) are given byj(t) = SA (DY (t) 510). The presence of the A(t} term in this expression means that the productivity of investment at t depends on the technology at t. Show that the economy converges to a balanced growth path. What are the growth rates of Y and I on the balanced growth path? (Hint: Let ](t) = ](t)/A(t). Then use the same approach as in (a), focusing on \"(24%) instead of K/(A'V'L).) (c) What is the elasticity of output on the balanced growth path with respect to s? (d) In the vicinity of the balanced growth path, how rapidly does the economy converge to the balanced growth path