Part 2 (60 points): Consider the endogenous growth model with human capital accumulation we studied in Chapter 8. The representative household allocates one unit of time in each period between "working" and "schooling," and let u = (0, 1) denote the fraction of time for working. Assume this is a constant parameter. The representative household is endowed with H unit of human capital in time t, and it evolves according to the equation: H+1 b(1 u)H. (1) The representative firm uses only labour (in efficiency unit) as an input for production, and the production technology is given by: Y = zuH. (2) 1. Suppose the household does not save and consumes all the "income" from working. If w is the real wage rate per efficiency unit of labour supply in time t, what is the household's budget constraint? 2. Write down the representative firm's profit maximization problem. 3. We argued in class that w = z must hold for the labour market to be in equilibrium. Explain briefly why. 4. Using the labour market equilibrium condition, write down the equation for the equilibrium evolution of the human capital in this economy. 5. Under what condition, in terms of model parameters, can the human capital in this econ- omy continue to grow indefinitely? Draw a diagram to describe this situation. 6. Suppose, without any human activity involved, a new "innovation" suddenly increases the productivity of the representative firm. (a) Through what model parameter(s) would this innovation affect the economy? (b) If we assume the innovation will not affect the other model parameters that we as- sumed constant, how does this innovation affect the economy? Does the innovation affect the growth rate of the economy? (c) If this innovation is to affect the growth rate of the economy, what could be a possible "story," and what assumption do you need to modify for that story to make sense? 8 / 10