Financial management, mathematics, Lagrange function, Kuhn-Tucker conditions, consumer's optimization
Take the fictional nation of Panem, for example. The civilian population is subject to rationing of basic consumer goods. The rationing method is based on the use of redeemable coupons issued by the government.
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Let's consider the case of a two-good world where both goods, x and y, are rationed. The consumer's utility function is U (x, y). The consumer has a fixed monetary budget of B and faces monetary prices Px and Py. In addition, the consumer has a coupon allocation, denoted C, which he can use to buy both x and y at a coupon price of cx and cy.
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[...] Mathematics for Economics and Finance The consumer seeks to maximise the numerical value of her utility function U by adjusting quantities of commodities x and y. She has to take into account prices px, py, as well as coupon values cx and cy. The consumer is also constrained in how much she can buy in x and y according to its income and coupon allocation C. The consumer's optimization problem has a unique solution if there exist some couple such that: and In order to achieve that, function U needs to be concave and differetiable. [...]
[...] The coupon C is not going to affect the composition of the optimal allocation - C is part of the budget constraint, so quantities of x and y are bound to change commensurately. Indeed, if the government decides to increase C by then both quantities x and y will go up by thanks to a positive income effect. If the government wants to affect consumer expenditure in a meaningful way, they would do well to adjust coupon prices cx and cy, in order to influence the optimality condition, and thus the marginal rate of substitution through relative prices. [...]
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