Exercise 2.20. In this exercise, we work through an alternative conception of technology, which will be useful in the next chapter. Consider the basic Solow model in continuous time and suppose that A (t) = A, so that there is no technological progress of the usual kind. However, assume that the relationship between investment and capital accumulation modified to K (t + 1) = (1 − δ) K (t) + q (t) I (t), where [q (t)]∞ t=0 is an exogenously given time-varying process. Intuitively, when q (t) is high, the same investment expenditure translates into a greater increase in the capital stock. Therefore, we can think of q (t) as the inverse of the relative prices of machinery to output. When q (t) is high, machinery is relatively cheaper. Gordon (1990) documented that the relative prices of durable machinery has been declining relative to output throughout the postwar era. This is quite plausible, especially given our recent experience with the decline in the relative price of computer hardware and software. Thus we may want to suppose that q˙(t) > 0. This exercise asks you to work through a model with this feature based on Greenwood, Hercowitz and Krusell (1997). (1) Suppose that q˙(t) /q (t) = γK > 0. Show that for a general production function, F (K, L), there exists no steady-state equilibrium. (2) Now suppose that the production function is Cobb-Douglas, F (K, L) = L1−αKα, and characterize the unique steady-state equilibrium. (3) Show that this steady-state equilibrium does not satisfy the Kaldor fact of constant K/Y . Is this a problem? [Hint: how is “K” measured in practice? How is it measured in this model?].