Problem 1 (3 marks)

Let f(q) be a general (nonhomothetic) utility function and define the corresponding cost function C(u,p) by

C(u,p) = min{p · q : f(q) ≥ u}

Suppose the consumer is engaging in cost minimizing behaviour during periods 0 and 1 so that we have:

p0  · q0     =   C(u0 ,p0 )   (1)

p1  · q1     =   C(u1 ,p1 ).   (2)

Assume that the observed quantity vectors for periods 0 and 1 are given by the following expressions using Shephard’s Lemma:

q0     =   ∇p C(u0 ,p0 )                                                  (3)

q1     =   ∇p C(u1 ,p1 ).                                                    (4)

Hicks (1941-42) defined the following (theoretical) measures of welfare change:

V (p1 ,q0 ,q1 )   ≡ C(u1 ,p1 ) − C(u0 ,p1 )   (the compensating variation).             (6)

1.  (1 mark) Obtain an observable first-order approximation to the equivalent variation. Hint:

C(u1 ,p0 ) ≈ C(u1 ,p1 ) + ∇p C(u1 ,p1 ) · (p0 − p1 ).