Problem Problem 3.21(b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more of x? Explain.

Transcription

1 EconS 301 Problem 3.1 Suppose a consumer s preferences for two goods can be represented by the Cobb Douglas utility function U = Ax α y β where A, α, and β are positive constants. The marginal utilities are MU x = αax α 1 y β and MU y = βax α y β 1. Answer the following questions. Problem 3.1(a) Is the assumption that more is better satisfied for both goods? Yes, the more is better assumption is satisfied for both goods since both marginal utilities are always positive. Problem 3.1(b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more of x? Explain. Since we do not know the value of α, only that it is positive, we need to specify three possible cases: When α1, the marginal utility of x diminishes as x increases. When α1, the marginal utility of x remains constant as x increases. When α1, the marginal utility of x increases as x increases. 1

2 Problem 3.1(c) What is MRS x,y? MRS 1 MU x Ax y y xy, 1 MUy Ax y x Problem 3.1(d) Is the MRS diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? As the consumer substitutes for x for y, the MRS x,y will diminish because x is in the denominator and drives down the entire fraction. Problem 4. The utility function that Ann receives by consuming food F and clothing C is given by U(F,C) = FC + F The marginal utilities of food and clothing are MU F = C + 1 and MU C = F Food costs $1 a unit, and clothing costs $ a unit. Ann s income is $. Answer the following questions. Problem 4.(a) Ann is currently spending all of her income. She is buying 8 units of food. How many units of clothing is she consuming? If Ann is spending all of her income then F C 8 C C 14 C 7

3 Problem 4.(b) Graph her budget line. Place the number of units of clothing on the vertical axis and the number of units of food on the horizontal axis. Plot her current consumption basket. Remember: Use the Budget Line and set the quantity of good y equal to zero to find the horizontal intercept, and then the quantity of good x equal to zero to find the vertical intercept. Problem 4.(b, cont.) Clothing Food Problem 4.(c) Draw the indifference curve associated with a utility level of 36 and the indifference curve associated with a utility of 7. Are the indifference curves bowed in toward the origin? Problem 4.(c, cont.) Clothing U=7 U=36 Yes, the indifference curves are convex, i.e., bowed in toward the origin. Also, note that they intersect the F axis; a common property in quasilinear utility functionsastheoneinthisexercise Food 3

4 Problem 4.(d) Using a graph (and no algebra), find the utilitymaximizing choice of food and clothing. That is, find the tangency between the budget line and the indifference curves. Problem 4.(d, cont.) Clothing U= U=36 Optimum at F=1, C= Food Problem 4.(e) Using algebra, find the utility maximizing choice of food and clothing. Problem 4.(e, cont.) The tangency condition requires that: MU MU Plugging in the known information yields: C 1 1 F C F F C P F P C Substituting this result into the budget line results in: (C ) C 4C 0 C 5 4

5 Problem 4.(e, cont.) Finally, plugging this result, C=5 units, back into the tangency condition C+=F, yields F=(5)+=1. Summarizing, at the optimum the consumer chooses: C=5, i.e., 5 units of clothing, and F=1, i.e., 1 units of food. Problem 5.19 Lou s preferences over pizza (x) and other goods (y) are given by U(x,y)=xy, with associated marginal utilities MU x =y and MU y =x. His income is $. Problem 5.19(a) Calculate his optimal basket when P x =4 and P y =1. Using the tangency condition, y/x=4, and the budget constraint, 4x+y=, 4x 4x 8x x 15 4(15) y y 60 Lou s initial optimum is the basket (x, y) = (15, 60) with a utility of 900. Problem 5.19(b) Calculate the income and substitution effects of a decrease in the price of food from $4 to $3. First we need to find the decomposition basket. This would satisfy the new tangency condition, y/x=3, i.e., y=3x. In addition, it would give him as much utility as before the price change, i.e. xy=900. y 3x x(3x) 900 x 300 x y 900 y

6 Problem 5.19(b, cont.) This gives (x,y)=( 3,30 3, or approximately (17.3,51.9). Now we need the final basket, which satisfies the same tangency condition as the decomposition basket and also the new budget constraint, 3x+y=: y 3 3(0) y x y 3x y 60 3x 3x 6x x 0 Together, these conditions imply that (x, y) = (0, 60). The substitution effect is therefore =.3, and the income effect is =.7. Problem 5.19(c) Calculate the compensating variation of the price change. The compensating variation is the amount of income Lou would be willing to give up after the price change to maintain the level of utility he had before the price change. This equals the difference between the consumer s actual income, $, and the income needed to buy the decomposition basket at the new prices. This latter income equals: 3* *51.9 = 3.8. Therefore, the compensating variation is CV= 3.8 = $16.0. Problem 5.19(d) Calculate the equivalent variation of the price change. The equivalent variation is the amount of income that Lou would need to be given before the price change in order to leave him as well off as he would be after the price change. After the price change his utility level is 0(60)=1,00. Therefore the additional income should be such that it allows Lou to purchase a bundle (x, y) satisfying the initial tangency condition with this utility. This implies (x,y)=( 3,40 3, or approximately (17.3, 69.). How much income would Lou need to purchase this bundle under the original prices? He would need 4(17.3) = That is, he would need to increase his income by EV=138.4 =$18.40 dollars in order to be as well off as if the price of pizza were to decrease instead. Problem 5.0 Carina buys two goods, food F and clothing C, with the utility function U=FC+F Her marginal utility of food is MU f = C+1 and her marginal utility of clothing is MU c =F. She has an income of $0. The price of clothing is $4. 6

7 Problem 5.0(a) Derive the equation representing Carina s demand for food, and draw this demand curve for prices of food ranging between 1 and 6. MU F = C + 1 MU C = F Tangency: MU F /MU C = P F / P C, or (C + 1)/ F = P F /4, implying 4C + 4 = P F F. (Eq1) Budget Line: P F F + P C C = I, which implies P F F + 4C = 0. (Eq) Substituting (Eq1) into (Eq): 4C C = 0. Thus C =, independent of P F. From the budget line, we see that P F F + 4() = 0, which simplifies to P F F = 1. Solving for F we obtain the demand for F, Problem 5.0(a, cont.) The figure plots F = 1/P F. Note that, since the price is in the vertical axis, we first need to solve for the price P F, which yields an inverse demand function P F P F = 1/F Demand for food 6 F 1 F = 1/P F Problem 5.0(b) Calculate the income and substitution effects on Carina s consumption of food when the price rises from 1 to 4. Draw a graph illustrating these effects. [Your graph does not need to be to scale, but it should be consistent with the data.] Problem 5.0(b, cont.) Initial Basket: From the demand for food in (a), F = 1/1 = 1, and C =. Also, the initial level of utility is U = FC + F = 1() + 1 = 36. Final Basket: From the demand for food in (a), we know that F = 1/4 = 3, and C =. And the final level of utility is, U = 3() + 3 = 9. Decomposition Basket: Must be on initial indifference curve, with U = FC + F = 36 (Eq 3) 7

8 Problem 5.0(b, cont.) Tangency condition satisfied with final price: MU F /MU C = P F / P C. (C + 1)/ F = 4/4 => C + 1 = F. (Eq 4) Eq 3, FC + F = 36, can be written as F(C + 1) = 36. Plugging Eq 4 into the rewritten Eq 3 we have (C+1)(C+1)=(C + 1) = 36, and thus, C = 5. Problem 5.0(b, cont.) Income effect on F: F final basket F decomposition basket = 3 6 = 3. Substitution effect on F: F decomposition basket F initial basket = 6 1 = 6. Also, by Eq 4, C + 1 = F, or 5+1=F, entailing F = 6. Summarizing, the decomposition basket is F = 6, C = 5. Problem 5.0(b, cont.) Problem 5.0(c) Determine the numerical size of the compensating variation (in monetary terms) associated with the increase in the price of the good from $1 to $4. P F F + P C C = 4(6) + 4(5) = 44. So she would need an additional income of 4 (plus her actual income of 0). The compensating variation associated with the increase in the price of food is 4. 8

9 Problem 5.8 Consider Noah s preference for leisure (L) and other goods (Y), U ( L, Y ) L Y. The associated marginal utilities are: 1 1 MU L andmuy L Y Suppose that P Y =$1. Is Noah s supply of labor backward bending? Problem 5.8 (cont.) If Noah s wage rate is w, then the income he earns from working is (4 L)w. Since P Y = 1, the number of units of other goods he purchases is Y = (4 L)w. Also, the tangency condition gives us /. Combining the two conditions gives us 4 or, solving for L, 4/ 1. Problem 5.8 (cont.) Given that the optimal amount of leisure (L) he consumes is 4/ 1. Leisure decreases with an increase in the wage rate, w, and this is true no matter what the wage rate is; as depicted in the next slide. 1 Problem 5.8 (cont.) Since the amount of labor that Noah supplies equals 4 L= 4 4/ 1)=4/ 1. we see that his supply of labor always increases with an increase in the wage rate, w Hence, his labor supply curve is always positively sloped that is, it is not backward bending. Let s plot it

10 Amount of labor that Noah supplies, , as a function of w. Problem 6.19 A firm produces quantity Q of breakfast cereal using labor L and material M with the production function 1/ Q 50( ML) M L The marginal product functions for this production function are: M MPL 5 L MPM 5 L M 1 1 Problem 6.19(a) Are the returns to scale increasing, constant, or decreasing for this production function? Problem 6.19(b) Is the marginal product of labor ever diminishing for this production function? If so, when? Is it ever negative, and if so, when? To determine the nature of returns to scale, increase all inputs by some factor λ and determine if output goes up by a factor more than, less than, or equal to λ. Q 50( M L) M L 1/ 50 ( ML) 50( ML) 50( ML) M L M L M L Q Thus, by increasing all inputs by a factor λ, output goes up by a factor of λ. Since output goes up by the same factor as the inputs, this production function exhibits constant returns to scale.

11 Exercise A firm produces quantity Q of breakfast cereal using labor L 1/ and material M with the production function Q50( ML) M L. The marginal product functions for this production function are: M MPL 5 L MPM 5 L M 1 1 b) Is the marginal product of labor ever diminishing for this production function? If so, when? Is it ever negative, and if so, when? Answer Don t forget, M MP L 5 L 1 Suppose M>0. Holding M constant, increasing L will decrease the MP L. The marginal product of labor is decreasing for all levels of labor. The MP L, however, will never be negative since both components of the equation will always be greater or equal to zero. In fact, for this production function, MP L 1. 11