Question

Suppose u(††x) = ?_{l=1}^{L} f_l(x_l) is strictly quasiconcave with f'_l(x_l) > 0 for all l = 1, • • • , L. The consumer faces fixed prices p ? 0 and has wealth w > 0. Assume x(p, w) ? 0. (a) Show that if one good displays increasing marginal utility at x(p, w), all other goods must display diminishing marginal utility there. (b) Show that if one good displays increasing marginal utility, then one good is normal and all other goods are inferior. (c) Show that if all goods display diminishing marginal utility at x(p, w), then all goods are normal.

          Suppose
u(††x) = ?_{l=1}^{L} f_l(x_l)
is strictly quasiconcave with f'_l(x_l) > 0 for all l = 1, • • • , L. The consumer faces fixed prices p ? 0
and has wealth w > 0. Assume x(p, w) ? 0.
(a) Show that if one good displays increasing marginal utility at x(p, w), all other goods must
display diminishing marginal utility there.
(b) Show that if one good displays increasing marginal utility, then one good is normal and all
other goods are inferior.
(c) Show that if all goods display diminishing marginal utility at x(p, w), then all goods are
normal.

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Suppose
u(††x) = ?l=1^L fl(xl)
is strictly quasiconcave with f'l(xl) > 0 for all l = 1, • • • , L. The consumer faces fixed prices p ? 0
and has wealth w > 0. Assume x(p, w) ? 0.
(a) Show that if one good displays increasing marginal utility at x(p, w), all other goods must
display diminishing marginal utility there.
(b) Show that if one good displays increasing marginal utility, then one good is normal and all
other goods are inferior.
(c) Show that if all goods display diminishing marginal utility at x(p, w), then all goods are
normal.

Added by Noelia B.

Principles of Economics

Gregory Mankiw 8th Edition

Instant Answer

Solved by Expert Akash M

10/18/2023

Step 1

Step 1: **Understanding Quasiconcavity and Marginal Utility** - A function is strictly quasiconcave if for any two points \( x \) and \( y \), and for any \( \lambda \in (0, 1) \), we have: \[ u(\lambda x + (1-\lambda)y) > \min(u(x), u(y)) \] - Marginal Show more…

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Suppose u(x) = ̲̲̲̲̲̲̲̲∑(l=1 to L) fl(xl) is strictly quasiconcave with f'l(xl) > 0 for all l = 1, ... , L. The consumer faces fixed prices p ≫ 0 and has wealth w > 0. Assume x(p, w) ≫ 0. (a) Show that if one good displays increasing marginal utility at x(p, w), all other goods must display diminishing marginal utility there. (b) Show that if one good displays increasing marginal utility, then one good is normal and all other goods are inferior. (c) Show that if all goods display diminishing marginal utility at x(p,w), then all goods are normal.

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