14.01SC Principles of Microeconomics Scholar, Midterm 1 Solution Set

14.01 Fall 2010: Midterm 1 Solution Set 1. True/False/Uncertain Questions Write whether each statement is True, False or Uncertain. You should fully explain your answer, including diagrams where appropriate. Points will be given based on your explanation. (a) A consumer finds two goods to be perfectly substitutable. Claim: The optimal bundle for this consumer will always be a corner solution. False. Consider for example the stility function uy. The optimal bundle will be a corner solution if the prices of the two goods are different; if the prices are the same, then any allocation is optimal. (b) Innovations in the production of batteries lead to a rightward shift in the market supply for hybrid cars, while demand stays the same. Since this leads to a decrease in the equilibrium price and an increase in the equilibrium quantity, demand is more inelastic at the new equilibrium. Uncertain. This is true in the case of linear demand, since elasticity of demand is increasing in price along a linear demand curve. However, this is false in the case of a constant elasticity demand (c) A consumer has selected an optimal bundle of two goods that includes some of each good. The price of one good increases. Claim: her utility is lower after the price increase compared to before True. If the consumer was indifferent to one of the goods, she would have selocted a bundle of only the other good. Clearly, she obtains utility from both goods; thus when the price of either increases and her budget construint shifts in, she will be on a lower indifference curve. (d) When market demand and supply shift in opposite directions, we can unambiguously say how the equilibrium price and quantity change. False. When demand and supply shift in the opposite direction we can unambiguously say how the equilibrium price changes (equilibrium price increuses if supply shifts to the left and demand shifts to the right and vice versa). However, the effect on equilibrium quantity is ambiguans; it could either increase or decrease in both cases. 2. Market demand for frozen yogurt Market surveys show that there are two types of consumers for frozen yogurt. The first type like frozen yogurt and have an inverse domand curve of P5Q. The second type are crazy about frozen yogurt and have an inverse demand curve of P20-Q. In the town of Smallville there are only 2 consumers: one of them likes frozen yogurt and the other is crazy about frozen yogurt. (a) Using the individual demand curves above, derive the market demand for frozen yogurt in Smallville. Plot the market demand curves. The demand of the first group in and the demand of the second group is Q-20-P Summing the two demand curves, we get a market demand curve of QP +0= 39-3P PA 20-PP>5 The muerse market demand curve is then P 20-QQ≤15 10-9913 (b) Suppose that the market supply for frozen yogurt in Smallville is given by Q2+ P. Find the equilibrium price and quantity. How much does each consumer buy at the equilibrium price? (Hint: Check the equilibrium price and quantity you get on a graph) The market demand curve is piecewise linear. However, since at p5Q715Q it follows that supply would intersect demand to the left of the kink. Hence, Q5 Q or 2+P=20-Pand hence P9. This means that Q11. Only the cruzy consumer boys at the equilibrium price. 3. Consumer preferences and optimal allocations Mary is starting a jewelry collection. She wants to own matched sets of three bracelets and one necklace that can be worn together, and she doesn't want to own any bracelets or necklaces that are not in a matched set of this size. (a) Draw Mary's indifference curves and write her utility function, Put bracelets on the y axis and necklaces on the x axis. Assume she receives utility of 3 utils from each matched jewelry set she owas. U(b,n)= 3 min(n, 1 3 or (b,n)=3min(306) (b) Currently, Mary has 32 dollars to spend. The price of necklaces is p, 2 and the price of bracelets is p2. What is the optimal allocation of necklaces and bracelets for Mary? How much utility does she achieve from this allocation? We can find the optimal allocution by finding the intersection of the lines h=3n and 2n+24=32. Substituting in to the budget constraint, this yields the following. 20+G = 32 3 We have 4,6 12, 12. (c) Due to a shortage of gold, the price of necklaces increases to po 10. What is the new allocation of necklaces and bracelets at this income level, and what utility does Mary obtain? What is the proportional decrease in her utility? Re-solve the problem with a new budget constraint. This yields 10n6n32 We have n2, 66, 6. (d) Luckily, Mary's parents value their daughter's utility, and are willing to give her enough income to ensure that she has the same utility she did prior to the price change. How much extra money do they have to give her? Mary's parents have to give her sufficient income to buy two more matched sets at the new prices. This requires the purchase of treo necklaces and sir bracelets, at a cost of 2+10+0+2 32. They have to double her income, increasing it by 892, in order to maintain her at the same utility level. (e) Mary has a sister Lily who doesn't like wearing matched sets of jewelry and has different preferences. Her utility function is ". If she started with the same jewelry budget as Mary of 32 dollars and. then faced the same price shock, what would be the decrease in her utility when the price of necklaces increases from $2 to $10? First, solve for Lily's original optimal allocation. We set the ratio of marginal stilities equal to the ratio of prices. 2nh 2 2 b 2 Substituting into the budget construint, this yields 2(2n)+2n32 3