Question 5 Solving Consumer's Choice Problem under CES Utility, Numerical (sympy solution not required) The consumer with income Y has a preference represented by CES utility function U(q1,q2) = (904 + q24). Given prices for the two goods, denoted by P₁, P2, solve the consumer's optimal choice problem following the steps below. (a) Write down the consumer's maximization problem, i.e be clear about (1) the choice variables, (2) the objective function and (3) the constraint. (b) Write down the Lagrangian for this optimization problem. (c) Derive the three first-order conditions.

3. Let there be two goods x and y with prices px and py respectively, and let M be the consumer's income. Let the consumer's preferences be represented by the utility function min(2x, y). Find the consumer's expenditure function. Note: you cannot use the Lagrange method as the utility function is not differentiable - but you have used the min U = utility function before. (4)

A consumer has the following utility function (shown in image) where ? is the number of spa days and ? is the number of city breaks consumed. Suppose that the price of a spa day is £200 and the price of a city break is £300. (i) Set up the economic problem and find the numbers of spa days and city breaks that minimise expenditure if 12,800 units of utility are to be obtained.

image attached