A student has two sources of expenses: living and consumption. If the student lives in an apartment with s square metres and consumes c consumables, then they receive utility

u(s, c) = s^1/5c^4/5

The student chooses apartment size and consumption amount to get the highest utility. Total living expenses are1000 + rs. Each unit of consumption costs 1. The student spends exactly 1800.

Under what conditions will the student live in a 20m²apartment? A 120m²apartment? An apartment between 20m²and 120m²? Now suppose each unit of consumption costs 2 (instead of 1). Everything else stays the same. Without solving the problem again, use the envelope theorem to approximate change in utility.