Problem Set 1

Problem Set 1 - Problem Set #1 Name: _____________________...

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Unformatted text preview: Problem Set #1 Name: _____________________ Econ326 Intermediate Micro Student ID: _____________________ Instructor: Ginger Jin Section No. _____________________ Due Date: February 14, 2011 TA Name: _____________________ =============================================================== 1. Sally consumes two goods, X and Y. Her utility function is given by the expression The current market price for X is $10, while the market price for Y is $5. Sally's current income is $600. a. Sketch two indifference curves for Sally in her consumption of X and Y so that one curve represents U=240000 and the other curve represents U=480000. Mark two points on each curve, write down the precise numbers of X and Y for each point. (Note: you can choose whatever unit of X and Y that you think are most appropriate in the following graph.) b. Calculate the marginal rate of substitution for each of the four points you have marked on the above graph. MRS = MUx /MUy = Y2/2XY= Y/2X 1 Problem Set #1 Name: _____________________ Econ326 Intermediate Micro Student ID: _____________________ Instructor: Ginger Jin Section No. _____________________ Due Date: February 14, 2011 TA Name: _____________________ =============================================================== !" #" $%&" '(" )*+,,-.-" ,((" -(" /0+-,111" '((" ,(" ,,+.-'0/" -1" 1(" ,(" '/" c. Write the expression for Sally's budget constraint. Graph the budget constraint and determine its slope. Sally’s Budget constraint is given by 10X + 5Y = 600 and its slope is -2.This can be seen by rewriting the budget constraint as Y= 120-2X ( Form : y= m*x+c, where ‘m’ is slope of the line) d. Determine the X,Y combination which maximizes Sally's utility, given her budget constraint. Show her optimum point and its corresponding indifference curve on the graph. Compute the utility level at this optimum point. Note that this utility function belongs to the category Cobb – Douglas Utility functions. For a utility function of the general form U = A X! Y" , we can write the optimal demands (X*,Y*) as follows: X* = [!M] / [(!+ ")Px ] Y* = ["M] / [(!+ ")Py ] Where M : Income, Px : Price of good X; Py : Price of good Y For our problem, ! = 1 and " =2; M= 600;Px=10;Py=5. Therefore, X* = 20 ; Y*= 80. The utility corresponding to the optimum point is U = 3*20*1600 = 96000 U=3*20*6400=384000 2 Problem Set #1 Name: _____________________ Econ326 Intermediate Micro Student ID: _____________________ Instructor: Ginger Jin Section No. _____________________ Due Date: February 14, 2011 TA Name: _____________________ =============================================================== e. Suppose Sally’s utility function changes to U=2lnX + 4lnY. Recalculate the optimal point that Sally will choose (given the same budget constraint). Is this answer different from the solution in part d? Why? If we recalculate the (X*,Y*) using 1. MRS = Px /Py ie. Y/2X = 2 2. Budget constraint : 10x + 5Y = 600 We obtain (X*,Y*) = (20,80) This is the same as in part d because this utility function is just a MONOTONIC TRANSFORMATION of the previous utility function and therefore will result in the same optimal consumption bundle . 3 Problem Set #1 Name: _____________________ Econ326 Intermediate Micro Student ID: _____________________ Instructor: Ginger Jin Section No. _____________________ Due Date: February 14, 2011 TA Name: _____________________ =============================================================== 2. Antonio buys five new college textbooks during his first year at school at a cost of $80 each. Used books cost only $50 each. When the bookstore announces that there will be a 10 percent increase in the price of new books and a 5 percent increase in the price of used books, Antonio’s father offers him $40 extra. a. How much did Antonio spend on textbooks before the price hike? With new books on the vertical axis and used books on the horizontal axis, draw his budget line and his consumption bundle of textbooks before the price change. 4 Problem Set #1 Name: _____________________ Econ326 Intermediate Micro Student ID: _____________________ Instructor: Ginger Jin Section No. _____________________ Due Date: February 14, 2011 TA Name: _____________________ =============================================================== Let N1 be the number of new books and N2 be the number of used books Antonio uses before the price hike. Let his total budget before the price hike be M1 Before the price hike, his budget constraint could be written as : 80 * N1 + 50*N2 = M1 Using N1 = 5, we can write this as 80 * 5 + 50*N2 = M1 After the price hike, his budget becomes M2 = M1+40, so that he can continue to consume the same combination of books as earlier. Thus his budget constraint becomes 88* 5 + 52.5*N2 = M2 Using M2= M1+40 =80 * 5 + 50*N2 We get 88* 5 + 52.5* N2= 80*5 + 50N2 +40 Solving which gives us that N2 = 0 Thus M1= 400. i.e Anonio spent $400 before the price hike. Therefore his consumption bundle before the price change was (X,Y) = (0,5). 5 Problem Set #1 Name: _____________________ Econ326 Intermediate Micro Student ID: _____________________ Instructor: Ginger Jin Section No. _____________________ Due Date: February 14, 2011 TA Name: _____________________ =============================================================== b. Let X=# of used books, Y=# of new books. Suppose Antonio’s utility is U=100X+165Y. Draw the indifference curve that Antonio is on before the price change. Calculate the marginal rate of substitution at this point. Has Antonio made an optimal choice of new versus used textbooks before the price change? Explain why. MRS = 100/165 = .606 Before the price change, Antonio consumes (0,5) so his utility U = 100*0+5*165 = 825. Yes, Antonio has made an optimal choice, because new books and old books are perfect substitutes and MRS =.606 < .625 = (Px/Py). Therefore, Indifference curves are flatter than the budget line and Antonio is better off consuming only good Y i.e new books. c. What happens to Antonio’s budget line after the price change (including both price changes and the extra money from his father)? Illustrate the new budget line in the above graph. Slope of the Budget Line is given by Px/Py The slope of the budget line changes from - .625 (i.e 50/ 80) to -.59 (i.e 52.5/88) d. How many new and used textbooks will Antonio buy after the price hike? Is Antonio worse or better off after the price change? Explain. Since new books and old books are perfect substitutes and MRS =.606 > .5966 = (P’x/P’y). Therefore, Indifference curves are steeper than the budget line and Antonio is better off consuming only good X i.e used books. Substituting N1 ‘=0 in budget constraint, we get 0+ 52.5 *N2’ = 440 Ie. N2 ‘ = 8.3810 # 8 Antonio’s utility is now U’ = 100*8 + 0 = 800. He is worse off than before as a result of the price change. 6 QUESTION 3 Elmer’s utility function is U (x, y ) = min{x, y 2 }. a. If Elmer consumes 4 units of x and 3 units of y, his utility is ___4___. If Elmer consumes 4 units of x and 2 units of y, his utility is ___4___. If Elmer consumers 5 units of x and 2 units of y, his utility is ___4___. Answer: U (4, 3) = min{4, 9} = 4 U (4, 2) = min{4, 4} = 4 U (5, 2) = min{5, 4} = 4 Hint: Since the above 3 points gives the same utility level, connecting the 3 points will give you an idea of the shape of the indifference curve, which is right angled. And preference is this form is called perfect complementary. On the graph below, draw the indifference curves that go through (4,2), (1,1) and (16,5) respectively. By now, you may notice that Elmer’s indifference curves have kinks. If you connect the kinks in one curve, what is the equation for this curve? (16,5) (1,1) (4,2) Hint: The steps to draw the indifference curves is to first identify which indifference curves do these points lie. And secondly, draw these indifference curves. Answer: Step1 : U (4, 2) = min{4, 4} = 4, thus point (4, 2) is on indifference curve that yields the utility level of 4. 1 U (1, 1) = min{1, 1} = 1, thus point (1, 1) is on indifference curve that yields the utility level of 1. U (16, 5) = min{16, 25} = 16, thus point (16, 5) is on indifference curve that yields the utility level of 16. Step2 :To graph indifference curve U (x, y ) = 4, the key is to find the kink point, where we find by equating the two component of the min function, x = y 2 = 4 ⇒ (x, y ) = (4, 2) To graph indifference curve U (x, y ) = 1, again we find the kink point by equating the two component of the min function, x = y 2 = 1 ⇒ (x, y ) = (1, 1) To graph indifference curve U (x, y ) = 16, again we find the kink point by equating the two component of the min function, x = y 2 = 16 ⇒ (x, y ) = (16, 4) The kinks lies on the locus x = y 2 or y = √ x. b. On the same graph, draw Elmer’s budget line where the price of x is 1, the price of y is 3, and his income is 4. Write down the budget constraint equation. Answer: The budget constraint equation is x + 3y = 4 c. What bundle does Elmer choose in this situation? Write down your steps below. Hint: The optimal bundle with perfect complementary goods is the intersection between the locus and the budget line. Answer : Solving the system of the following two equations x + 3y = 4 x = y2 Plug the second equation into the first one, we get y 2 + 3y = 4 y 2 + 3y − 4 = 0 (y + 4)(y − 1) = 0 y=1 2 and x = 1. Thus the optimal bundle Elmer chooses is (x, y ) = (1, 1). d. Suppose that Elmer’s income increases to 8 while both prices remain unchanged. What bundle does Elmer choose now? Does doubling the income lead to doubling consumption in x and y? Explain. Answer : Again solving the system of the following two equations with the new income level, x + 3y = 8 x = y2 Plug the second equation into the first one, we get y 2 + 3y = 8 y 2 + 3y − 8 = 0 y ￿ 1. 7 x ￿ 2. 9 The new optimal bundle Elmer chooses is (x, y ) = (1.7, 2.9). Thus the consumption is not doubling the original bundle after the income doubled. It is because we have nonlinear locus, and the direction of indifference curve increasing does not follow a straight line. Mathematically, when utility is not homogeneous of degree one in two goods, we lose the property of doubling the income doubles the optimal consumption in two goods. 3 QUESTION 4. Gentle Charlie consumes only apples and bananas. Let the number of apples be xA , and the number of bananas be xB . His utility function is U (xA , xB ) = xA ∗ xB . The price of apples is $2, the price of bananas is $8, and Charlie’s income is $160 a day. The price of bananas suddenly falls to $2. a. Before the price change, how many apples and bananas do Charlie consume? With these consumptions, how much utility can he achieve? Write down your steps below. Answer: Let’s focus on the the market before the price change. Denote the optimal consumption bundle before the price change as (x0 , x0 ), where the superscript 0 stands for initial AB optimal choices. Denote original price pair as (pA , pB ). We have two conditions to solve for ( x 0 , x0 ) , AB M RSAB = pA pB which can be written as M RSAB = M UA xB pA 1 = = = M UB xA pB 4 and hence we have relationship x0 = 4x0 . A B Also we have the condition that the optimal bundle (x0 , x0 ) is on the budget constraint, or AB pA x 0 + pB x 0 = I A B 2x0 + 8x0 = 160 A B Plug x0 = 4x0 into the budget constraint, we have A B 8x0 + 8x0 = 160 B B x0 = 10 B and x0 = 4x0 = 40. Thus we have solve for the optimal bundle is that A B (x0 , x0 ) = (40, 10) AB which corresponds to point A in the graph. And the utility level achieved by this bundle is U = 40 ∗ 10 = 400 On the graph, Charlie’s original budget line is in blue and A is his original chosen consumption. 4 b. If, after the price change, Charlie’s income had changed so that he stays on the same indifference curve as before, how much would his new income be and how many apples and bananas will he consumer with this income and the new prices? Write down your steps below. Hint: The key to this answer is that when Charlie “stays on the same indifference curve as before ”, he is still making his optimal consumption choice, but under the new price ratio, so we have condition M RSAB = pA /p￿ . And graphically, we have the budget line with the B slope representing the new price ratio tangent with the original indifference curve. So the steps part(b) takes is to first solve for the optimal consumption bundle, we denote as (x1 , x1 ) that AB makes Charlie stays on the same indifference curve under the new price ratio. Secondly, we can solve for the amount of money Charlie needs to be able to afford bundle (x1 , x1 ), which is his AB new income, we denote as I ￿ . Answer : Denote the new price of banana as p￿ . (x1 , x1 ) satisfies the condition that it B AB makes Charlie stays on “the same indifference curve as before”, which the one we solved in part(a), xA xB = 400, thus (x1 , x1 ) must satisfy equation AB x1 x1 = 400 AB Also (x1 , x1 ) is an optimal choice for Charlie, we have M RSAB = AB M RSAB = M UA xB pA = = ￿ =1 M UB xA pB 5 pA pB , or Thus we have the second equation x1 /x1 = 1 or x1 = x1 . Plug it into x1 x1 = 400, we A B A B AB have x1 x1 = 400 AA So we solve for x1 = x1 = 20, which is the consumption bundle charlie chose under the A B new price if he is to keep himself on the same indifference curve as before. It is indicated in the graph as B. The income that Charlie needs to spend on (x1 , x1 ) is AB pA x1 + p￿ x1 = 2 ∗ 20 + 2 ∗ 20 = 80 A BB In the graph, the budget line corresponding to this income is in light blue line. The bundle that Charlie would choose at this income is labelled at B. c. After the price change (and income remains at $160 a day), how many apples and bananas will Charlie actually buy? Write your steps below. Answer: Denote the consumption bundle after the price change as (x2 , x2 ), and the new AB price pair as (pA , p￿ ). We have two conditions to solve for (x2 , x2 ), B AB M RSAB = pA pB which can be written as M RSAB = M UA xB pA = = =1 M UB xA pB and hence we have relationship x2 = x2 . A B Also we have the condition that the optimal bundle (x2 , x2 ) is on the budget constraint, or AB pA x 2 + pB x 2 = I A B 2x0 + 2x2 = 160 A B Plug x2 = x2 into the budget constraint, we have A B 2x2 + 2x2 = 160 B B x2 = 40 B and x2 = x2 = 40. Thus we have solve for the optimal bundle is that A B (x2 , x2 ) = (40, 40) AB d. With points A, B, C on the graph, from which point to which point is the substitution 6 effect of the price fall of bananas? From which point to which point is the income effect? How would the consumption of apples and bananas change because of the substitution effect? How would the consumption of apples and bananas change because of the income effect? Answer : Substitution effect is the change in consumption bundle due to price change and keeps the consumer at the original indifference curve. And hence, from A to B is the substitution effect of the price fall of bananas. From B point to C is the income effect. Because of the substitution effect, consumption of apples drops from 40 to 20 and bananas increases from 10 to 20. Because of income effect, the consumption of apples increases from 20 to 40, and bananas increases from 20 to 40. e. Is banana a Giffen good? Is apple an inferior good? Explain. Answer : A Giffen good is one that the consumption of it change in the same direction as its price changes. Consumption of Banana increases when its price decreases, thus Banana is not Giffen good. An inferior good is one that the consumption of it decreases when the income increases, and consumption of it increases when income decreases (keeping prices unchanged). Consumption of apple increases when income increases, so apple is not an inferior good. 7 ...
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This note was uploaded on 09/07/2011 for the course ECON 326 taught by Professor Hulten during the Spring '08 term at Maryland.

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