In the late 90s it was observed that the relative price of equipment (capital) has declined at an average annual rate of more than 3 percent. There has also been a negative correlation (-0.46) between the relative price of new equipment and new equipment investment. This can be interpreted as evidence that there has been significant technological change in the production of new equipment. Technological advances have made equipment less expensive, triggering increases in the accumulation of equipment both in the short and long run. Concrete examples in support of this interpretation abound: new and more powerful computers, faster and more efficient means of telecommunication and transportation, robotization of assembly lines, and so on. In this problem we are going to extend the Solow Growth Model to allow for such investment specific technological progress. Start with the standard Solow model with population growth and assume for simplicity that the production function is Cobb-Douglas: Yt = Kt L1t , where the population growth rate is deltaLt/Lt= n. Similarly, just as in the basic model, assume that investment and consumption are constant fractions of output It = sYt and Ct = (1 s)Yt. However, assume that the relationship between investment and capital accumulation is modified to:

Kt+1 Kt = qtIt Kt

where the variable qt represents the level of technology in the production of capital equipment and grows at an exogenously given rate , i.e. deltaqt / qt= . Intuitively, when qt is high, the same investment expenditure translates into a greater increase in the capital stock. (Note: another way to interpret qt is as the inverse of the relative price between machinery and output: when qt is high, machinery is relatively cheaper). (a) Transform the model (the production function, the equations for consumption and investment, and the capital accumulation equation) in per-worker form.

c) Suppose that capital per worker kt grows at a constant rate (we do not know that yet, but we will make a guess). Divide the capital accumulation equation by kt and use this assumption to prove that qtk1t has to be constant over time.