1. (24 points) Consider consumer happiness given by the function V = u(x1, x2, . . . , xN) where xi is the quantity of good i consumed.

a) (1 point) How many first order partial derivatives does u have?

b) (2 points) Choose one of the partial derivatives above and explain what it represents intuitively in the context of consumer happiness.

c) (5 points) If u is homogeneous of degree 1, are its partial derivatives homogeneous? If so, of what degree? Explain. For the rest of the problem, consider the simplified version with two goods, V = u(x, y).

d) (1 point) Suppose the goods cost p and q per unit, respectively. Write the function for total consumer expenditure given x units of the first good and y units of the second good are purchased and consumed. Now suppose the consumer seeks to minimise expenditure while achieving a predetermined utility level V = c.

e) (2 points) Using implicit differentiation, find y 0 (x) given V = c.

f) (3 points) Write the first order condition of the minimisation problem in terms of x, p, and q only.

g) (2 points) Describe what the marginal rate of substitution means in this context and what the first order condition implies for its value.

h) (3 points) Suppose u is homogenous of degree 1 and (x , y ) minimises expenditure while delivering V = c units of utility. What is 1 the amount of good 1 and good 2 minimising total expenditure that delivers V = 2c units of utility? Explain. Let u take the Cobb-Douglas form u(x, y) = x y 1 where 0 < < 1.

i) (5 points) Find the quantities (x, y) minimising expenditure while achieving V = c in terms of p, q, c, and .