ECO 3145: Spring/Summer 2022 Quiz 3
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ECO 3145: Spring/Summer 2022
I) For each question, choose the correct answer. 1 × 5 = 5 marks
Let f and g be functions deﬁned on Rn and c a real number. Consider the following two problems,
Problem 1: min f (x) and Problem 2: min f (x) subject to g(x) = c.
1. Any solution of problem 1 is also a solution of problem 2. True or false?
2. If Problem 1 does not have a solution, then Problem 2 does not have a solution. True or false?
3. Problem 2 is equivalent to max - f (x) subject to - g(x) = -c. True or false?
4. In Problem 2, quasi-convexity of f is a suﬃcient condition for a point satisfying the ﬁrst-order conditions to be a global minimum. True or false?
5. Consider the function f (x1 , x2 ) = ln(x1 ) - x2 . f is
a) quasiconcave and quasiconvex b) quasiconvex
c) quasiconcave d) no correct answer
II) Consider the problem of a consumer who must choose between two types of goods, good 1 (x1 ) and good 2 (x2 ) costing respectively p1 and p2 per unit. He is endowed with an income m and has a quasi-concave utility function u deﬁned by u(x1 , x2 ) =
2 ln(x1 )＋5 ln(x2 ).
1. Write down the problem of the consumer. 1 mark
2. Determine the optimal choice of good 1 and good 2, x1(x) = x1 (p1 , p2 , m) and x2(x) = x2 (p1 , p2 , m). 2 marks
3. By how much will the consumer’s utility change if his income increases by 1 unit. 1 mark
4. Find the optimal amounts of good 1 and good 2 the consumer will choose if p1 = 2, p2 = 5 and m = 25. 1 mark
III) Consider the optimization problem of the objective function f (x, y) = x2＋y2 - 3xy＋1 subject to -x＋3y - 4 = 0.
1. Write down the Lagrangian function and the ﬁrst-order conditions. 1 mark
2. Determine the stationary points. 2 marks
3. Does the stationary point represent a local maximum or a local minimum? Justify your answer. 2 marks