Question

. Which of the following utility functions satisfy (i) strict convexity; (ii) monotonicity:

(a) U(x, y) = min[x, y];

(b) U(x, y) = x^a*y^(1−a);

(c) U(x, y) = x + y;

(d) Lexicographic preference.

(e) U(x, y) = y

Answer 1

Monotonic preferences are ones in which having more is always better. preferences are monotone if more of any good x and y makes the agent strictly better off.

More of one good but no less of the other is the condition of srtict monotonicity

convexity is when averages are preffered to extremes. if the agent is indifferent between x and y then she prefers the average tx + (1 − t)y to either x or y.

a)U(x, y) = min[x, y]The indifference curves of perfect compliments ie the utility function mentioned above are L–shaped,, with the kink along the line x=y. so indifference curve is not strictly increasing . this is because such functions dont follow strict monotonicity but follows monotonicity ie with 2 left and 3 right shoes and with 2 left and 2 right shoes the utility of the person remains the same . 1 extra right show doesn't increase the utility of consumer.

convexity : this preference is convex but not strictly convex.

b) U(x, y) = x^a*y^(1−a)

such type of functions are called cobb douglas functions.

in this in the MRS x is in the denominator , so we get a downward sloping indifference curve and strict convex prefernces . as x increases there is a reduction in MRS.

strict monotonicity exists in cobb douglas function ie more of good is better.

c) U(x, y) = x + y

this is the case of perfect subsititutes. the indifference curves are straight lines .As a result , the preferences are weakly convex. with perfect substitutes, strict convexity does not hold.

(indifference curves are not strictly convex). MRS is constant; it is not diminishing.

monotonicity in subsitute goods exist ie these goods are monotone (more the better)

d) Lexicographic preference

in lexicographic prefernce the consumer is conserned about only one good no matter how much the other good is .

to show strong monotonicty assume x>=y and x not equal to y

so this implies x1>y1 and x2>=y2 or that x1=y1 and x2>y2

in either case x>y.

lexicographic function has a strictly convex prefernce