Question

Consider a preference-maximising consumer who consumes two commodities. The preferences of the consumer are given by the utility function U(x1, x2) = x1.

(a) Comment on the monotonicity and convexity of the consumer’s preferences.

Answer 1

Since utility function is just a function of good 1, utility is thus independent of good 2. Hence U can be written as:

$U(x_1, x_2) = x_1 = U(x_1)$

Since increasing x₁ increases utility for the consumer, hence preferences are monotonic in good 1.

It is neutral in good 2.

The graph for utility function can be drawn as follows -

ープ)( אי

Preferences are convex if means are better than the extremes i.e. if we take two combinations of goods X and Y i.e.  (x₁, y₁) and (x₂, y₂), on the same indifference curve, then any weighted average of the two i.e. Z = $\lambda$X + (1-$\lambda$)Y will be at least as good as, or, strictly preferred to, each of the extreme bundles for any $\lambda \in [0,1]$ .

In our case, if we take X and Y on the indifference curve, we see that any convex combination Z = $\lambda$X + (1-$\lambda$)Y for $\lambda \in [0,1]$ also lies on the same indifference curve.

i.e. $X \sim Y \sim Z$ .

Hence preferences are convex (not strictly convex though).

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