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2. (28 points) Consider a worker who achieves a happiness level u(x, () if consuming x units of consumable goods and working f hours. a) (4 points) What do you expect the signs of u', and u', to be? Provide an intuitive explanation for both. b) (4 points) Suppose each consumable item costs p and the worker re- ceives w per hour worked. The worker cannot spend more than they make, and hours worked must be between 20 and 40. Write the con- strained utility optimisation problem. c) (2 point) Suppose workers successfully unionise and the wage w in- creases as a result, but nothing else changes. What is the effect on the worker's optimised utility? Justify your answer. Now suppose u(x, () = vx + (40 -() and p = 1. d) (18 points) When will the worker optimally choose to work part-time (i.e. 20 hours)? Full-time (i.e. 40 hours)? Neither (i.e. between 20 and 40 hours)? Justify your answer using the Kuhn-Tucker method