Question

A consumer’s preferences are said to be strictly monotonic if “more is better than less”, or equivalently, for a C 1 utility function u defined on R n , ∂u(x)/∂xi > 0 for all i = {1, . . . , n}. Consider a budget set with prices pi > 0 for all i defined by Xn i=1 pixi ≤ m and x1 ≥ 0, . . . , xn ≥ 0, for some income level m > 0. Prove that the budget constraint (the first constraint in the above list) always binds when a consumer has monotonic preferences

Answer 1

Monotonicity implies that whenever bundle x has more of every good than bundle y then x would be prefered to y or more is prefered to less. It ha stwo forms weak and strong monotonicity where strict monotonicity is the latter one where if $x\geq y$ and $x\neq y$ then .

Monotonicity also implies that preferences are convex.i.e. averages are prefered to extremes and marginal rate of substitution is decreasing as one subsititues one good for the other.

let bundles x,y,z belong to X_n

such that $y\geq xand z\geq x$ and $y\neq z$ then $\alpha y+(1-\alpha )z> x$ where $\alpha \epsilon (0,1)$

utility is given as having a positive slope implies $u(x)\geq u(y)\Leftrightarrow x\geq y$

so utility of a more prefered bundle is always more than less prefered bundle.

hence, consumer has rational preferences after satisfying strict monotonicity and convexity.

However, consumer is not free to choose any amount of good as there is a price involved with every unit of a good and the amount paid for good i isp_ix_i

given income of the consumer is m , this level of income posses a limit on the amount that can be purchased implies p_ixi $\leq$ m .

if vector $x=(x{_{1}},.....x_{n})$ and price vector is $p=(p{_{1}},.....p_{n})$ then total amount spend on these bundle of goods will be $\sum p_{i}x_{i}$ . Now given m, the only bundles which can be consumed are as

$B(m,p)= x_{i}\epsilon x | \sum p_{i}x_{i}\leq m$

so while maximining utility function to find optimal bundles consumed, it is subjected to the budget constraint as given above.