范里安微观作业(不知名A习题四次)

Problem set 1 Due date: April-18 1. Homer and Barney subsist solely on the consumption of two good: beer and donuts. The price of beer is $4 each and the price of donuts is $1 each. The marginal utility Homer received from the last beer he consumed is 8 and his marginal utility from the last donut is 2. Barneys marginal utility from the last beer is 1 and his marginal utility from the last donut is 3. (a) Which consumer is not maximizing his utility? Why? (b) How should he change his allocation? 2. Guy likes to eat hot dogs, but franks and buns must be purchased separately. Guy needs an equal number of franks and buns (1 frank + 1 bun = 1 hot dog; they are perfect complements) and has a budget of $10 for hot dogs. Hot dog buns always cost $0.25 each. (a) If franks cost $0.75 each, how many franks and buns will Guy buy? (b) If the price of franks increases to cost $1.00 each, how will Guy react? Explain your answers by drawing the indiff erence curves and budget lines. 3. For the Constant Elasticity of Substitution (CES) utility function (q 1 + q 2) 1 , derive the optimal quantity demanded of q1and q2as a function of their prices (p1and p2) and income (Y ). 4. Suppose a persons utility for goods x and y is given by u(x;y) = aln(x)+bln(y),a,b 0. (a) Compute the demand functions for goods x and y. (b) Compute the eff ect on each demand of a change in the price of the other good. (c) Are these two goods complements or substitutes? Explain your answer. (d) Compute the eff ect on each demand of a change in income. (e) Are these two goods normal or inferior? Explain your answer. 1 5. Suppose instead a persons utility for goods x and y is given by v(x;y) = xayb. Explain why the answers to a)-e) in the above question remain the same in this environment. 6. Your are giving the following partial information about a consumers purchases. He con- sumes only two goods. over what range of quantity of good 2 consumed in year 2 would you conclude? Year 1 Year 2 Quantity Price Quantity Price Good 1 100 100 120 100 Good 2 100 100 x 80 (a) That his behavior is inconsistent (contradict with weak axiom)? (b) That his consumption bundle in year 1 is revealed preferred to that in year 2? (c) That his consumption bundle in year 2 is revealed preferred to that in year 1? Teaching assistant: 邓子琛; tel Email: 2 Intermediate Microeconomics Problem Set #1 1. Homer and Barney subsist solely on the consumption of two good: beer and donuts. The price of beer is $4 each and the price of donuts is $1 each. The marginal utility Homer received from the last beer he consumed is 8 and his marginal utility from the last donut is 2. Barneys marginal utility from the last beer is 1 and his marginal utility from the last donut is 3. a) Which consumer is not maximizing his utility? Why? At the maximum, MRS = MRT or equivalently marginal utility per dollar spent is equalized across all goods. For Homer: MRS = MUD MUB = 2 8 = 1 4 = pD pB = MRT or MUD pD = 2 1 = 8 4 = MUB pB In other words, his marginal utility per dollar spent is equalized across goods. Therefore, Homer is maximizing his utility. For Barney, MRS = MUD MUB = 3 1 6= 1 4 = pD pB or MUD pD = 3 1 6= 1 4 = MUB pB In other words, his marginal utility per dollar spent is not equalized across goods. Therefore, Barney is not maximizing his utility. b) How should he change his allocation? Barney is consuming too much beer. He is getting signifi cantly less utility per dollar spent on his last beer purchase than on his last donut purchase. Therefore, he should spend more money on donuts until the point where the marginal utility per dollar spent on each good is equal. 2. Guy likes to eat hot dogs, but franks and buns must be purchased separately. Guy needs an equal number of franks and buns (1 frank + 1 bun = 1 hot dog; they are perfect complements) and has a budget of $10 for hot dogs. Hot dog buns always cost $0.25 each. a) If franks cost $0.75 each, how many franks and buns will Guy buy? Intuitively, each frank and bun pair costs $0.75+$0.25 = $1.00, so Guy will buy 10 of each. Mathematically, it is a system of two equations B = F 0.75F + 0.25B = 10 the solution to which is F = 10 and B = 10. b) If the price of franks increases to cost $1.00 each, how will Guy react? Explain your answers by drawing the indiff erence curves and budget lines. 1 Intuitively, each frank and bun pair now costs $1.00+$0.25 = $1.25, so Guy will only buy 8 of each. Mathemat- ically, it is a system of two equations B = F F + 0.25B = 10 the solution to which is F = 8 and B = 8. 3. For the Constant Elasticity of Substitution (CES) utility function (q 1+ q 2) 1 , derive the optimal quantity demanded of q1and q2as a function of their prices (p1and p2) and income (Y ) using the Langrangian method. (Hint: look at problems 32, 37,and 38 in the textbook.) The algebra is somewhat tedious, so bear with me. Note fi rst that the utility function can be raised to the power without changing the problem, as exponentiation is a monotone transformation. Utilizing this property, the Lagrangian for the problem can be written L = q 1+ q 2+ (Y p1q1 p2q2) Diff erentiating with respect to q1and q2 obtains the fi rst-order conditions for an interior maximum L q1 =q1 1 p1= 0 L q2 =q1 2 p2= 0 Combining these two conditions yields the following system of two equations in two unknowns, p1 p2 = q1 1 q1 2 (1) Y=p1q1+ p2q2(2) Rearranging equation (1) yields the following p1 p2 = q1 1 q1 2 ?p 1 p2 ?1 1 = q1 q2 q1=q2 ?p 1 p2 ?1 1 Substituting into equation (2), Y=p1q1+ p2q2 Y=p1q2 ?p 1 p2 ?1 1 + p2q2 Y=q2 p1 ?p 1 p2 ?1 1 + p2 ! q2= Y ? p1 ? p1 p2 ?1 1 + p2 ? 2 q1= Y ? p1 ? p1 p2 ?1 1 + p2 ? ?p 1 p2 ?1 1 = Y ? p1 ? p1 p2 ?1 1 + p2 ? ?p 1 p2 ?1 1 = Y ? p1 ? p2 p1 ?1 1 ? p1 p2 ?1 1 + p2 ? p2 p1 ?1 1 ? = Y ? p1+ p2 ? p2 p1 ?1 1 ? which gives the optimal quantity demanded of q1and q2as a function of the prices p1and p2and income Y . 4. Suppose a persons utility for goods x and y is given by u(x,y) = aln(x) + bln(y). a) Compute the demand functions for goods x and y. The Lagrangian can be written as follows, L = aln(x) + bln(y) + (I pxx pyy) Diff erentiating with respect to x and x obtains the fi rst-order conditions for an interior maximum L x = a x px= 0 L x = b y py= 0 which is equivalent to MRS = MUx MUy = px py = MRT. Combining this with the budget constraint (obtained by diff erentiating with respect to ), we get the following system of two equations in two unknowns a b y x = px py (3) I=pxx + pyy(4) By rewriting the fi rst equation, we can obtain x as a function of y and prices, a b y x = px py x=y a b py px Substituting into the second equation, I=pxx + pyy I=y a b py px px+ pyy y= I py ?a b + 1? = I py ?a+b b ? =I b a + b 1 py x=I a a + b 1 px which provides the standard result for a Cobb-Douglas utility function (yes, this is equivalent to a Cobb-Douglas utility function as we will show below): a a+b and b a+b dictate the share of income spent on each good. 3 b) Compute the eff ect on each demand of a change in the price of the other good. y=I b a + b 1 py dy dpx =0 x=I a a + b 1 px dx dpy =0 The price of the other good has no impact on demand for the good in question. c) Are these two goods complements or substitutes? Explain your answer. The goods are neither complements nor substitutes; they are completely unrelated. d) Compute the eff ect on each demand of a change in income. y=I b a + b 1 py dy dI = b a + b 1 py x=I a a + b 1 px dx dI = a a + b 1 px e) Are these two goods normal or inferior? Explain your answer. Since a a+b is positive and both prices must be positive, demand for both goods increases as income increases. This means that both goods are normal. 5. Suppose instead a persons utility for goods x and y is given by v(x,y) = xayb.Explain why the answers to a)-e) in the above question remain the same in this environment. (Hint: the function f(x) = exis a strictly increasing transformation.) Utility maximization problems are identical up to a monotonic (or strictly increasing) transformation of the utility function. All that needs to be shown is that there exists a strictly increasing function f(x) such that f(u(x,y) from question 4 equals v(x,y) from this question. The function f(x) = exmay be a good place to start. u(x,y)=aln(x) + bln(y) f(x)=ex f(u(x,y)=ealn(x)+bln(y) =eln(x a)eln(ya), utilizing properties of logs =xaya= v(x,y) So, since there is a monotonic function that transforms u(x,y) into v(x,y), the maximization problem is unchanged and all of the same results hold. 4 第二次作业 Due date: 4-25 April 17, 2012 1. 对下列效用函数推导对商品1的需求函数，反需求函数，恩格尔曲线；在图上大致画出 价格提供曲线，收入提供曲线；说明商品一是否正常品(normal good)、劣质品(inferior good)、一般商品(ordinary good, 根据教材中的定义ordinary good是指需求曲线具有负斜 率的商品)、吉芬商品，商品二与商品一是替代还是互补关系 (a) u(x1,x2) = 2x1+ x2 (b) u = minx1,2x2 (c) u = xa 1xb2 (d) u = lnx1+ x2 2. Mr. Cog works in a machine factory. He can work as many hours per day as he wishes at a wage rate of w. Let C be the number of dollars he has to spend on consumer goods and let R be the number of hours of leisure that he chooses. (a) Suppose that Mr. Cog earns $8 an hour and has 18 hours per day to devote to labor or leisure, and suppose that he has $16 of nonlabor income per day. Write an equation for his budget between consumption and leisure. (b) draw his budget line in the graph below. His initial endowment is the point where he doesnt work, but keeps all of his leisure. Mark this point on the graph below with the letter A (c) If Mr. Cog has the utility function U(R;C) = CR, how many hours of leisure per day will he choose? How many hours per day will he work? (d) Suppose that Mr. Cogs wage rate rose to $12 an hour. Use red ink to draw his new budget line. (He still has $16 a day in nonlabor income.) If Mr. Cog continued to work exactly as many hours as he did before the wage increase, how much more money would he have each day to spend on consumption? But with his new budget 1 line, how many hours he will chooses to work? By what amount his consumption actually increase? (e) Suppose that Mr. Cog still receives $8 an hour but that his nonlabor income rises to $48 per day. Use black ink to draw his budget line. How many hours does he choose to work? (f) Suppose that Mr. Cog has a wage of $w per hour and a nonlabor income of $m. As before, assume that he has 18 hours to divide between labor and leisure. Cogs budget line has the equation C +wR = m+18w. Using the same methods you used in the chapter on demand functions, fi nd the amount of leisure that Mr. Cog will demand and his supply of labor as a function of wages and of nonlabor income. 3. 通过Slutsky证明Giff en商品必然为劣等商品(inferior good). 2 第二次作业答案 第一题 第二题： 第三题：根据 slutsky 方程 其中恒为负，如为正，必然有 为负，从而 Giffen 商品必然为劣等商品。 no answer Microeconomics Homework 3 Due date: May 21 Pls. answer the questions and fill the blank in the following problems. 1. Nickleby has an income of $2,000 this year, and he expects an income of $1,100 next year. He can borrow and lend money at an interest rate of 10%. Consumption goods cost $1 per unit this year and there is no inflation. a) What is the present value of Nicklebys endowment? What is the future value of his endowment? With blueink, show the combinations of consumption this year and consumption next year that he can aford. Label Nickelbys endowment with the letter E. b) Suppose that Nickleby has the utility function .Write an expression for Nicklebys marginal rate of substitution between consumption this year and consumption next year. (Your answer will be a function of the variables ) c) What is the slope of Nicklebys budget line? Write an equation that states that the slope of Nicklebys indifference curve is equal to the slope of his budget line when the interest rate is 10%. Also write down Nicklebys budget equation. d) Solve these two equations. Nickleby will consume Units in period 1 and units in period 2. Label this point A on your diagram. e) Will he borrow or save in the first period? How much? f) On your graph use red ink to show what Nicklebys budget line would be if the interest rate rose to 20%. Knowing that Nickleby chose the point A at a 10% interest rate, even without knowing his utility function, you can determine that his new choice cannot be on certain parts of his new budget line. Draw a squiggly mark over the part of his new budget line where that choice can not be. (Hint: Close your eyes and think of WARP.) g) Solve for Nicklebys optimal choice when the interest rate is 20%. h) Will he borrow or save in the first period? How much? 2. Dr. No owns a bond, serial number 007, issued by the James Company. The bond pays $200 for each of the next three years, at which time the bond is retired and pays its face value of $2,000. a) How much is the James bond 007 worth to Dr. No at an interest rate of 10%? b) How valuable is James bond 007 at an interest rate of 5%? c) Ms. Yes offers Dr. No $2,200 for the James bond 007. Should Dr. No say yes or no to Ms. Yes if the interest rate is 10%? What if the interest rate is 5%? d) In order to destroy the world, Dr. No hires Professor Know to develop a nasty zap beam. In order to lure Professor Know from his university position, Dr. No will have to pay the professor $200 a year. The nasty zap beam will take three years to develop, at the end of which it can be built for $2,000. If the interest rate is 5%, how much money will Dr. No need today to finance this dastardly program? If the interest rate were 10%, would the world be in more or less danger from Dr. No? 3. The certainty equivalent of a lottery is the amount of money you would have to be given with certainty to be just as well-off with that lottery. Suppose that your von Neumann-Morgenstern utility function over lotteries that give you an amount x if Event 1 happens and y if Event 1 does not happen is , where is the probability that Event 1 happens and is the probability that Event 1 does not happen. a) If , calculate the utility of a lottery that gives you $10,000 if Event 1 happens and $100 if Event 1 does not happen. b) If you were sure to receive $4,900, what would your utility be? (Hint: If you receive $4,900 with certainty, then you receive $4,900 in both events.) c) Given this utility function and , write a general formula for the certainty equivalent of a lottery that gives you $x if Event 1 happens and $y if Event 1 does not happen. d) Calculate the certainty equivalent of receiving $10,000 if Event 1 happens and $100 if Event 1 does not happen. 4. Fenner Smith is contemplating dividing his portfolio between two assets, a risky asset that has an expected return of 30% and a standard deviation of 10%, and a safe asset that has an expected return of 10% and a standard deviation of 0%. a) If Mr. Smith invests x percent of his wealth in the risky asset, what will be his expected return? b) If Mr. Smith invests x percent of his wealth in the risky asset, what will be the standard deviation of his wealth? c) Solve the above two equations for the expected return on Mr. Smiths wealth as a function of the standard deviation he accepts. d) Plot this “budget line” on the graph e) If Mr. Smiths utility function is , then what are Mr. Smiths optimal value of and ? (Hint: You will need to solve two equations in two unknowns. One of the equations is the budget constraint.) f) Plot Mr. Smiths optimal choice and an indifference curve through it in the graph. g) What fraction of his wealth should Mr. Smith invest in the risky asset? 5. Lolita, an intelligent and charming Holstein cow, consumes only two goods, cow feed (made of ground corn and oats) and hay. Her preferences are represented by the utility function where x is her consumption of cow feed and y is her consumption of hay. Lolita has been instructed in the mysteries of budgets and optimization and always maximizes her utility subject to her budget constraint. Lolita has an income of $m that she is allowed to spend as she wishes on cow feed and hay. The price of hay is always $1, and the price of cow feed will be denoted by p, where 0 p 1. a) Write Lolitas inverse demand function for cow feed. (Hint: Lolitas utility function is quasilinear. When y is the numeraire and the price of x is p, the inverse demand function for someone with quasilinear utility f(x) + y is found by simply setting ) b) If the price of cow feed is p and her income is m, how much hay does Lolita choose? (Hint: The money that she doesnt spend on feed is used to buy hay.) c) Plug these numbers into her utility function to to find out the utility level that she enjoys at this price and this income. d) Suppose that Lolitas daily income is $3 and that the price of feed is $50. What bundle does she buy? What bundle would she buy if the price of cow feed rose to $1? e) How much money would Lolita be willing to pay to avoid having the price of cow feed rise to $1? This amount is known as the f) Suppose that the price of cow feed rose to $1. How much extra money would you have to pay Lolita to make her as well-off as she was at the old prices? This amount is known