Economics 2150A/B Lecture Notes - Budget Constraint, International Polar Year, Indifference Curve

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15 Nov 2012

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Consumer enjoys consuming x and y but only if for every 3 units of x, they consume 7 units of y. Should switch the constants to make the totals equal. (7x, 3y) (7(3),3(7)) (21,21) U2 does not represent the same, by power of and negative. U3 does represent the same, because there are 2 negative components. Budget constraint (bc) is the set of affordable bundles given prices and income. Budget line set of all bundles the consumer can purchase, while spending all his income. Graphing budget line, i = pxx + pyy. I = 12 px = 2 py = 1. Change in i, won"t change the slope, it shifts the budget line instead. Change in px or py, will change the slope and 1 or both the intercepts. Special case: px, py change by the same percentage, then the slope doesn"t change. Leads to a shift in the budget line only. Constraint: budget constraint, which included income and prices.

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Given prices for two goods PX = 2 and PY = 3 and a budget of $120 a consumer is trying to maximize the utility function U(X, Y ) = 2X0.3Y 0.7 . Use the Lagrange method to solve this problem, but donât worry about the second order conditions (if you wish, you can actually check that the MRS is decreasing and this utility function is well behaved). a) Write and solve the utility maximization problem subject to the budget constraint. What is the resulting value of the utility function? (U=49.0481) b) Write and solve the dual problem. Do you get the same optimal solution (the same optimal combination of X and Y)? What is the resulting expenditure? Does that match the budget of your original (primal) problem? (X=18, Y=28 should be the same as for a or very close due to rounding) c) Assume now general values for prices and income. Solve for the optimal amounts of X and Y as functions of PX, PY , and I. What conclusion can you draw regarding how you spend your budget when dealing with Cobb Douglas functions with exponents that add up to 1. You can actually try to solve the general case where we replace the exponents 0.3 and 0.7 with some Î± and Î², where Î± + Î² = 1. (Hint: you should find that you spend percent of your income on X and Î² percent of your income on Y. See if you can get there by solving!)

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1. Economists use indifference curves to characterize the preferences of a rational consumer. There are some plausible assumptions that lead to a set of âwell-behavedâ (i.e., downward-sloping and strictly convex) indifference curves. Which one of the following is NOT among the assumptions?

a. All goods must be economic âgoods.â

b. Preferences are complete.

c. Preferences are transitive.

d. All goods are normal goods

2. Given the prices of good X and Y, the (negative of the) slope of the budget constraint (i.e., - PX/PY) measures

a. the amount of good Y the consumer is willing to forgo in order to consume one more unit of good X.

b. the ratio of the consumer's income to the price of the X good.

c. the quantity of the Y good that must be forgone in order to purchase one more unit of the X good.

d. the quantity of the X good that must be forgone in order to purchase one more unit of the Y good.

3. Which of the following statements is incorrect for an inferior good?

a. An inferior good may have an upward-sloping demand curve.

b. The income elasticity of demand for an inferior good is negative.

c. An inferior good is an economic âbad.â

d. All of the above.

True or False. If False explain why it's false.

1. If all prices are doubled an money income is the left the same, the budget set does not change because the relative prices do not change.

2. A decrease in income pivots the budget line around the bundle initially consumed.

3. If preferences are well-behaved, then for any commodity bundle (x1,x2), the set of commodity bundles that are worse than (x1,x2) is a convex set.

4. The marginal rate of substitution measures the distance betwee one indifference curve and the next one.

5. If someone has the utility function U= 2min(x, y), then x and y are perfect complements for that person.

6. At the boundary optimum, a consumers indifference curve must be tangent to her budget line.

7. If two goods are substitutes, then an increase in the price of one of them will increase the demand for the other.

8. An Engel curve is a demand curve with the vertical and horizontal axes reversed.

9. An inferior good is less durable than a normal good.

10. When other variables are held fixed, the demand for a Giffen good rises when income increase.

Economics 2150A/B Lecture Notes - Budget Constraint, International Polar Year, Indifference Curve

ECON 2150A/B Full Course Notes

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