R&D Based growth model (Quality Improvement): Consider an economy where agents are lived only for one period. The nal good of this economy is produced, by a perfectly competitive rm, using a single intermediate good and labor: Yt = (AtL) 1 (Xt) (20) where 2 (0; 1) and Yt is output of the nal good in period t; L is the labour force the rm employs, which is standardized to be unity, L = 1. At is the quality of the intermediate good at time t and Xt is the amount of intermediate input. The intermediate input is produced using the nal good by a monopolist entrepreneur and producer. Suppose the price of the intermediate good is Pt and producing a unit of the intermediate good costs one unit (i.e., the marginal cost of the intermediate good is just one). (a) Derive the demand schedule for the quantity of the intermediate good. (b) Derive the equilibrium price for the intermediate good. (c) What is the equilibrium level of intermediate good production? What is the equilibrium level of prot for the rm in the intermediate good sector? (d) Derive the probability of innovating the new version of the product. (e) Derive the long run growth rate of the economy. How does an increase in the productivity of the research sector (e.g., an increase in the quality of higher education) aects the growth rate of the economy