Problem Set #5 Solutions ECON 2106H - J. Turner 1) i) Income Consumption Curve (ICC): Optimal combinations of X and Y (where Y usually represents “all other goods”), keeping prices fixed, and letting income vary. ii) Engel Curve: Relationship between the quantity of a good consumed and income. a) Slope of budget constraint = −PPxy Examination of the graph shows the slope of the budget constraint to be -1, so... −=−11Px which can easily be solved for Px=1. b) 5 ICC Other 4 Goods 3 2 1 2/3 4/3 2 8/3 X Note that all arcs should be tangent to the budget constraints. C) Engle Curve Income 5 4 3 2 1 4/3 8/3 X 2/3 2 10/32) i) Price Consumption Curve: For a given (i.e. fixed) income and price of y (generally assumed to be all other goods), the PCC is the set of optimal bundles of x and y for different prices of good x. ii) Demand Curve: The amount that an individual or a market is willing to buy at different prices for the good. a) Other Goods 400 PCC 50 100 150 200 225 400 800 X b) P 4 2 1 .5 50 150 200 225 Q Note: Not all demand curves are straight lines, this example just happened to work out like this. 3) Y ICC I2 I1 I0 Soap S* Income Engle CurveSoap S* 4) a) In part (a) , we showed that if bacon costs $1.50 per pound, and hamburger costs $1.00 per pound, that no hamburger was purchased. This is one point on a demand curve for hamburger (i.e. P=$1.50 Q=0). In fact, for any price less than $.75/pound, QD=0, using the same analysis as we did in problem 3a). At a price of $.75/pound, Elvis is indifferent between any point on his budget constraint, and thus the demand curve is flat at $.75 from QH=0 to Qh=4M. (Note: This analysis assumes the price of bacon is constant). PH .75 4 QH Additional comments on the above demand curve derivation: B2 mil IC 2M B.C. 4 mil H Optimal Bundle: 3M/PH If PH =.75 then this line is both the If PH=.75, then we can see above budget constraint and Elvis’ highest that Elvis consumes only hamburger indifference curve. QH= 3 mil/PH. This is the part of Elvis’ Demand curve with Q higher than 4 mil. The same analysis as above can give us Elvis’ Demand Curve for Bacon (assumes price of hamburger fixed at $1).PB 2 1.5 mil QB 4 b) In part b) we assume Elvis’ preferences change so that now the slope of his indifference curve is -1 (vs. Previously -2 .... if you are confused see the answer sheet to problem set #3...problem #3, parts a & b) Now his demand curves are: Demand for Hamburger: (Note: Again we assume the price of bacon fixed at $1.50) PH $1.50 QH 2 mil Demand for Bacon: (Note: Again we assume price of hamburger is fixed at $1.) PB $1 QB 3 Mil 5) a) i) Perfect Compliments (e.g. left and right shoes)ii) Some good like salt. It doesn’t matter what your income is, you will always consume about the same amount of salt. Income Income Engle Curve for X Engle Curve for Z slope = 2 QX Z* QZ c) Yes 6) a) Income Consumption Curve G Len’s ICs ICC M Len’s B.C. (slope of budget constraint= -4)b) Engle Curve Income Engle Curve for G G Income Engle Curve for M slope = PM=2 Mixed Bags Engle Curve for M: Since Len spends all of his income on M, his Engle Curve is: PMQM=Income 2QM=Income 7. The crucial assumption that is being made when indifference curves are “infinitely close together” is that individuals have preferences over goods that are defined even to fractions of goods. This means that I can evaluate, for example, my value of 1.34 M&M’s, and compare that to my value of 1.35 M&M’s. Clearly, this assumption makes more sense for some goods (like coke) than for others (like compact disks). Nonetheless, it is an assumption that is always implicit in the definition of indifference curves. So what is the rationale for doing this? The simplest answer is mathematical and graphical simplicity. Perfect divisibility is what allows us to draw smooth demand curves, Indifference curves, and virtually all other curves we draw. Other than allowing us to solve for fractional values, however, the assumption of divisibility of goods does not change the outcome of the analysis. It is for this reason, that some answers you may get in economics will be fractional.8. a) P 100 40 b) Demand curve is a crooked line as shown below. This is how you do it. Step 1. Convert equations for individual demand curves to: Q=100-P and Q=20-.5P Step 2. The horizontal sum of these two lines is Q=120-1.5P, (which is the same as P=80 - (2/3)Q after doing a little algebra) Step 3. This is not the market demand yet since for prices greater than 40 Len’s quantity demanded of coffee is zero and not a negative amount. When the price is greater than 40, the market demand curve is the same as that part of Jason’s demand curve since Jason is the only individual willing to purchase any of X when the price is above 40. Step 4. Therefore, the demand curve is a kinked line like below, where: P= 100-Q if Q<=60 (i.e., P>=40) 80-(2/3)Q if Q>=60 (i.e., P<=40) Jason’s Demand Curve Len’s Demand Curve 20 100 Q Market Demand Curve 20 60 100 120 Q c) P 100 40 d) Best way to do this is to horizontally sum the demand curves that look like Jason’s and then horizontally sum the demand curves that look like Len’s. Step 1. Again, convert equations for individual demand curves to: Q=100-P for Jason and Q=20-.5P for Len. Step 2. The horizontal sum of 10 of Jason’s demand curves gives us Q=10(100-P)=1000-10P,(which is the same as P=100 - .1Q after doing a little algebra) Step 3. The horizontal sum of 20