Quiz, week 10

  1. A headhunter is trying to lure you away from your studies. He knows that you currently study 7 hours per day, and spend 3 hours outdoors per day, giving a total utility of u(h,o)=u(7,3)u(h, o) = u(7, 3). It costs the headhunter c(h,o)c(h, o) to deliver hh hours of work and oo outdoor hours in the competing offer. If uu and cc are continuous on X={(h,o)∈[0,24]²:h+o≤24}X = \{(h, o) \in [0, 24]² : h + o \le 24\}, is there a solution to the problem min(h,o)∈Xc(h,o)s.t. u(h,o)≥u(7,3)?\begin{aligned} &\min_{(h, o) \in X} c(h, o) \\ &\text{s.t. } u(h, o) \ge u(7, 3)? \end{aligned}

  2. The government chooses a tax policy t∈CB(ℝ+)t \in CB(\mathbb{R}_+) to maximise social welfare W:CB(ℝ+)→ℝW : CB(\mathbb{R}_+) → \mathbb{R}, where distances in the domain and co-domain are measured with d∞d_\infty and d2d_2. Suppose WW is continuous. Does the extreme value theorem imply that there is an optimal tax policy?

  3. Suppose there are 10,000m²10,000 m² of land and 1000L1000L of water. These can be allocated to wine and to blueberry production. If (lv,wv)(l_v, w_v) is allocated to wine and (lb,wb)(l_b, w_b) is allocated to blueberry production, then the total output is f(lv,wv)f(l_v, w_v) and g(lb,wb)g(l_b, w_b), where ff and gg are continuous. Is the feasible set of possible wine and blueberry outputs compact?