Question

5. A consumer maximizes her satisfaction by finding the highest indifference curve that could be reached, given her budget constraint. Consider a consumer with utility function u(X, Y). When there is a small movement down in indifference curve, the additional consumption of good X, will generate marginal utility MUX. This results in a total increase in utility of MUX ∆X. At the same time, the reduced consumption of good Y, ∆Y, will lower utility per unit by MUY, resulting in a total loss of MUY ∆Y. Because all points on an indifference curve generate the same level of utility, moving down an indifference curve means the utility level does not change. So the total gain in utility associated with the increase in X must balance the loss due to reduction of consumption in Y. Show how you can get MUX/PX = MUY/PY, that is, marginal utility of good X, divided by the price of X, equals marginal utility of good Y, divided by the price of Y, which means the marginal utility per dollar of expenditure on good X is the same as that on good Y.

Answer 1

Indifference curve represents various combinations of two goods x and y which give a consumer equal satisfaction or utility. Budget line shows the various combinations which a consumer can buy with his given income.

Any rational consumer aims to maximize his/her utility subject to the budget constraint or budget line he faces. This optimization problem of maximizing the utility given a budget constraint becomes:

Maximize U subject to Px.X + Py.Y = M, where X is good 1 and y is good 2, Px and Py are the prices of good X and Y respectively, m is income of the consumer. Setting up lagrange and solving for first order condition yields and   All steps are clearly explained in the attached page.