Suppose we have a representative firm that produces an output (Y) by using two inputs: capital (K) and effective human capital. Effective human capital is the number of workers (N) multiplied by the productivity of each worker (e). So the production function looks like: Y = AF(K, eN). Henceforth, workers with higher human capital will be paid higher according to their effective human capital. Each worker is paid as wage rate (w) multiplied by their effective human capital, which gives the wage payment (weN). The firm lives for two periods and maximizes lifetime profit.

1. Please set up the firm's intertemporal objective function (value function) and their law of motion for capital. Let's use equity financing to simplify the structure.
2. Please derive the optimal condition for investment demand.
3. Please derive the optimal condition for labor demand (Nt), not (eN), and draw the labor demand curve.
4. All else equal, what happens to the labor demand curve if workers are more educated with higher human capital (e goes up)? Please explain why using MPN.
5. Now the government (not modeled in the economy) has decided to impose a corporate income tax. For each unit value of dividend the firm earns in the first period, it has to pay tax to the government. So the retained dividend for period one becomes (1- )D1. Please modify your firm's problem, and derive the new optimality conditions for capital (Kt) and (Nt).
6. What happens if the government raises the corporate tax rate ()?