Hello, I'm struggling with this question recently, may I get some assistance and guide, please. Thank You.

4. A professor chooses how much time to spend on three activities: re- search, teaching, and administrative duties. The amount of time spent conducting research is written x, the amount of time spent teaching y, and the amount of time spent attending to administrative duties z. The professor has a total amount of time t > 1, and has preferences over these three activities represented by a utility function u : R; - R, with u(x, y, z) = Inx + Iny + alnz, where a> 0 is a fixed parameter. The time spent conducting research must be at least 1 or else the professor will be fired; y and z cannot be negative. (a) Write down the professor's constrained optimization problem. [4] (b) Prove that the professor's objective function is concave and that [8] the professor's constraint set is convex. (c) Why will a non-degenerate constraint qualification (NDCQ) hold? [2] (d) Write down the Lagrangian [ for this problem. Compute the first- [4] order conditions. Write down the associated complementary slack- ness, feasibility, and non-negative multiplier conditions. (e) Briefly explain why the solution to the equations in your answer to [2] part (d) will generate the solution to the problem in part (a). (f) Find the optimal amount of time spent on each activity (x*, y* , z* ) [8] when t > a + 2. What happens when t s a + 2? (g) Derive the marginal utility of time as a function of t and plot it in a [8] graph. Is this function continuous at t = a + 2? Is it differentiable? (h) Show that the multiplier associated with the constraint that at least [4] time 1 must be spent conducting research is given by max o, a + 1 1- 1- 1 } . Interpret this multiplier both when t > a + 2 and when t s a + 2