Intermediate microeconomics

Professor Yongmin Chen

Topic 3. Consumer Choice

Budget Line

Your ability to consume is limited by the income you have. This is called the budget constraint. Suppose there are only two goods, X, Y, and your income is M.

Your budget constraint: p_XX + p_YY £ M

Budget line: p_XX + p_YY = M. Slope of the budget line.

Example. Suppose a consumer's budget line passes the point of Y = 30 on the vertical axis and X = 10 on the horizonal axis. The price of Good X is 3. What is the consumer's income? What is the price of Good Y? What is the equation for the budget line? What is the slope of the budget line?

How will changes in income or price change the budget line? When will changes in both income and price leave the budget line unchanged?

Maximizing Utility by the Consumer

Now we are ready to find out how the consumer maximizes utility from consumption subject to budget constraint. Two cases: interior solution and corner solution.

At an interior solution, the consumer chooses the consumption bundle at which her indifference curve is tangent to her budget line. (See a graphical illustration in class.) At the tangent point, the slope of the consumer's indifference curve (DY/DX) equals the slope of her budget line (‑P_X/P_Y). Therefore at the tangent point DY/DX = ‑ P_X/P_Y. But we know MRS_XY = ‑DY/DX. So at the tangent point MRS_XY = P_X/P_Y. Thus given the budget constraint, the consumer will maximize her utility by choosing the bundle (X,Y) such that

P_X×X + P_Y×Y = M, and MRS_XY = P_X/P_Y.

Example: If U(X,Y) = XY, P_X = 1, P_Y = 1, and M = 10, what consumption bundle will maximize utility? You solve the following equations (Why): Y/X = 1, and X + Y = 10 to get the optimal (X,Y). What will be the new optimal consumption bundle if P_X increases to 2?

In general, we shall be interested in situations where the optimum occurs at interior. But we also need to be aware of the possibility of a corner solution. At a corner solution, it is optimal for the consumer to consume zero of some goods. (See a graphical illustration in class.)

Example. Why do companies pay overtime? Suppose initially a worker works 8 hours a day at $10/hour. When the company wants the worker to work more hours, it typically pays a higher rate for the overtime period, say at $20/hour. Why doesn't the company raise the overall wage rate to induce more work hours? Think a model where the two goods you choose from are consumption and leisure, and your income is your wage income.

Example. A consumer is willing to trade two books for one CD. He currently is purchasing as many books as Cds. The price of Cds is three times of that of books. Is the consumer maximizing his utility? Should he buy more books and fewer Cds, or buy fewer books and more Cds?