2. (50 points) Consider the following Hansen-Prescott economy developed in class. At each date there is one good produced with two technologies: Malthus and Solow ac- cording with the following constant-returns-to-scale production functions: Ys,t = AsY* Ns,t, 7> 1, Ym,t = AmNi LI-H, HE (0, 1), where (Ni, A;) are the labor input and productivity parameter in each sector and L is the land input (in fixed supply). There is an infinitely-lived representative household with preferences over consumption described by: Bt log (C). 1= 0 Households are endowed with one unit of productive time each period and L = 1 units of land. Households rent labor and land services to competitive firms. Normalize the population size to one. Assume (AmLI- > Aa. a. (5) Write down the problem of the representative household and the first order conditions. b. (5) Write down the problem of the firms in each sector and the first order condi- tions. c. (5) Define a competitive equilibrium. d. (10) Solve for the competitive equilibrium. e. (10) Solve for the time of adoption of the Solow technology. f. (5) Illustrate the time path of the economy starting at date 0