Consider a representative agent economy with two types of goods: those produced at home (ch) and those produced in the market (cm). Households have preferences over consumption of each good and leisure according to the following utility representation: (0 (c) + (1 - 0) ()) t=0 Households own the technology to produce the home good for their own consumption It combines household capital (i.e.: appliances) and home labor to produce the home good: c = A (k) (n) -a Competitive firms own the technology to produce the market good. It combines rented capital and labor from the households according to the function c+i+= A (km)" (n?")-a Note the resource constraints specify that only the market good can be used for investment in each type of physical capital. Each capital stock evolves according to a standard law of motion with a common depreciation rate 6. 1. Carefully define a Competitive Equilibrium. Be sure to specify all optimization problems and as few aggregate state variables as is sufficient. 2. Characterize the solution and solve for the steady state. 3 3. Does the first welfare theorem hold? Prove your claim. 4 4. Consider an one-time permanent increase in A" at some time T. How will output in the market and home sectors differ in the new steady state? 5. Market hours and home hours have near zero correlation (-0.12) in the data. Can this model reproduce that in response to changed in A" alone? Explain what would be required of the utility function.