Problems With Solutions, Intermediate Microeconomics .

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Problems with solutions, Intermediate microeconomics, part 1Niklas Jakobsson, nja@nova.noKatarina.Katz@kau.seProblem 1. DemandBengt’s utility function is U(x1, x2) x1 ln x2x1 - stampsx2 - beerBengts budget p1 x1 p2 x2 mp1 – price of stampsp2 – price of beerm – Bengt’s budgeta) What is Bengt’s demand for beer and stamps?b) Is it true that Bengt would spend every krona in additional income on stamps?c) What happens to demand when Bengt’s income changes (i.e. find the income elasticity)?d) What happens to demand when p1 and p2 increase (i.e. find the price elasticities)?Problem 2. DemandJan has fallen on hard times. His income per week is 400 kr, spending 200 kr on food and 200 kron all other goods. However, he is also receiving a social allowance in the form of 10 foodstamps per week. The coupons can be exchanged for 10 kr worth of food, and he only has to pay5 kr for such coupons. Show the budget line with and without the food stamps. If Jan hashomothetic preferences, how much more food will he buy when he receives the food stamps?Problem 3. DemandFind the demand functions for the individuals below, the budget constraint is pxx pyy m2 3Bill: U(x ,x ) x y122/5 3/5Buster: U(x, y) x y2Ben: U(x, y) (x 1) (y 2)Barbara: U(x ,x ) 3x 2y12Beth: U(x, y) min{x , y}3

Problem 4. DemandBirgitta spends 150 SEK per month on coffee and buns at the cafeteria. A cup of coffee costs 15 SEKand a bun costs 10 SEK.a) Write the equation for Birgitta’s cafeteria budget constraint and draw it in a diagram.b) Assume that Birgitta never drinks coffee without eating one bun, and never eats bunswithout drinking coffee. How much of each will she consume? Draw some of her indifferencecurves.c) What do we call goods that are always consumed in the same proportion?Problem 1 Slutsky equationTomas is trying to figure out how to supplement the study allowances of 500 kr a week. He isconsidering a part-time job at a gas station. The wage is 50 kr per hour. His utility function is U(C, L) C*L where C is his consumption measured in SEK and L his leisure measured in hours. The amount ofleisure time that he has left after allowing for necessary activities is 50 hours a week.a. What is the monetary value of Tomas' endowment?b. Draw Tomas' budget set (horizontal axis: leisure and vertical axis: consumption).c. Set up the maximisation problem and decide optimal consumption and leisure.d. Let Y study allowance and T total amount of leisure time. Express his demand for consumptionas a function of study allowance and wage.e. Express his supply function for labour as a function of study allowance and wage.f. How many hours would Tomas work if he did not receive any study allowance?Problem 2 Slutsky equationAssume that the function U(x, y) x0.3y0.5 is the utility function of a person who consumes two goodsin quantities x and y, respectively.The price of x ispx 5 and the price of y is py 8This person s income is m 160.a) Find the optimal consumption choice of this person.b) Verify that at the optimum that you found the marginal rate of substitution equals the price ratio.Explain in terms of economic theory why this should be the case!c) Assume that the price of x falls to px 4.

i. Draw the old and the new budget constraints in a diagram. (Indicate at what values theyintersect the axes).ii. Calculate the person s demand for x and y at the new price.iii. Calculate the compensated income, m .iv. Decompose the change in demand for good x into a substitution and an income effect.Problem 1. Consumer’s surplusMattias has quasilinear preferences and his demand function for books is B 15 – 0.5p.a) Write the inverse demand functionb) Mattias is currently consuming 10 books at a price of 10 kr. How much money would he be willingto pay to have this amount, rather than no books at all? What is his level of consumer's surplus?Problem 2. Consumer’s surplus0.1Suppose Birgitta has the utility function U x10.9x2. She has an income of 100 and P 1 and P 121. Calculate compensating and equivalent variation when the price of x1 increases to 2. Also, tryto estimate the change in consumer's surplus measured by the area below the demand function.Problem 3. Consumer’s surplusExplain the concept of “consumer surplus” in words and illustrate by a diagram.Problem 4. Consumer’s surplusThe inverse demand curve (the demand curve but with p instead of q on the left hand side) isgiven by p(q) 100-10q. The consumer consumes five units of the good (q).a) How much money would you have to pay to compensate her for reducing herconsumption to zero? (The consumer is not paying anything for the goods.)b) Suppose now that the consumer is buying the goods at a price of 50 per unit. If you nowrequire her to reduce her purchases to zero, how much does she need to getcompensated? Hint: The number you will find is the net consumer’s surplus.Problem 5. Consumer’s surplusNew housing is planned in Karlstad but the location where it is to be built is used as a popularrecreation area for people in neighbouring parts of the city. In order to decide whether to build ornot, the city authorities want to make a survey to measure the decrease in welfare due the loss of

this recreation area. They are told by an economist that two measures are possible, compensatingvariation (CV) and equivalent variation (EV).a) How should they formulate the question if they want to measure the compensatingvariation?b) How should they phrase it if they want to measure the equivalent variation?Problem 1. Market demandLinus has a demand function q 10 - 2pa. What is the price elasticity of demand when the price is 3?b. At what price is the elasticity of demand equal to -1?c. Suppose that his demand function takes the general form q a - bp. Write down an algebraicexpression for his elasticity of demand at an arbitrary price p.Problem 2. Market demandThe demand function is q(p) (p 1)-2a. What is the price elasticity of demand?b. At what price is the price elasticity of demand equal to minus one?c. Write an expression for total revenue as a function of the price.d. Answer a-c when the demand function takes the more general form q(p) (p a)b where a 0 andb -1.Problem 3. Market demandFind the price elasticity of demand for the following demand functions.a) D(p) 30-6pb) D(p) 60-pc) D(p) a-bpd) D(p) 40p-2e) D(p) Ap-bf)D(p) (p 3)-2Problem 1. EquilibriumSuppose we have the following demand and supply equationsD(p) 200 - pS(p) 150 p

a. What is the equilibrium price and quantity?b. The government decides to restrict the industry to selling only 160 units by imposing a maximumprice and rationing the good. What maximum price should the government impose?c. The government doesn't want the firms in the industry to receive more than the minimum pricethat it would take to have them supply 160 units of the good. Therefore, they issue 160 rationcoupons. If the ration coupons were freely bought and sold on the open market, what would be theequilibrium price of these coupons?d. Calculate the dead-weight loss from restricting the supply of the goods. Will the dead-weightloss increase or decrease if the government would not allow the coupons to be sold on the openmarket?Problem 2. EquilibriumThe demand curve is qD 100 - 5p and the supply curve is qS 5p.a. A quantity tax of 2 kr per unit is placed on the good. Calculate the dead-weight loss of the tax.b. A value (ad valorem) tax of 20 % is placed on the good. Calculate the dead-weight loss of the tax.Problem 3. EquilibriumAssume that both demand and supply for a good are linear functions of its price:D(p) a bp, a 0, b 0S(p) c dp, c 0, d 0a) Draw curves that fit this description in a diagram.b) Assume that a tax t per unit has to be paid by the consumer. Show the effects on demand, supply,equilibrium price, quantity consumed and consumer and producer welfare in your diagram.c) Assume instead that an equally large tax has to be paid by the producer. What are the effects nowon demand, supply, equilibrium price, quantity consumed and consumer and producer welfare. (Usea diagram to illustrate.)Problem 1. Intertemporal choiceSuppose that a consumer has an endowment of 200.000 kr each period (period 1 and 2). He canborrow money at an interest rate of 200%, and he can lend money at a rate of 0%.a. Illustrate his budget set.b. The consumer is offered an investment that will change his endowment to m 300.000 and m 1150.000. Would the consumer be better or worse off, or can't you tell?2

c. If he is offered m 150.000 and m 300.000, is he better or worse off?12Problem 2. Intertemporal choiceMainy Landin has an income of 200.000 kr this year and she expects an income of 110.000 kr nextyear. She can borrow and lend money at an interest rate of 10%. Consumption goods cost 1 kr andthere is no inflation.a. What is the present value of Mainy's endowment?b. What is the future value of Mainy's endowment?c. Suppose that Mainy has the utility function U c1c2. Write down Mainy's marginal rate ofsubstitution.d. Set this slope equal to the slope of the budget line and solve for the consumption in period 1 and2. Will she borrow or save in the first period.e. d, but the interest rate is 20%. Will Mainy be better or worse off?Problem 1. UncertaintyJonas Thern maximises expected utility:U(π1, π2,c1,c2) π1c1 π2c2Jonas's friend Stefan Schwarz has offered to bet him 10.000 kr on the outcome of the toss of a coin. Ifthe coin comes up head, Jonas must pay Stefan 10.000 kr, and if the coin comes up tails, Stefan mustpay Jonas 10.000 kr. If Jonas doesn't accept the bet, he will have 100.000 kr with certainty. Let Event1 be "coin comes up heads".a. What is Jonas’s utility if he accepts the bet and if he decides not to bet? Does Jonas take the bet?b. Answer the question in a, if the bet is 100.000 kr.c. Answer the question if Jonas must pay Stefan 100.000 kr if he coin comes up head, but if the coincomes up tails Stefan must pay Jonas 500.000 kr.d. Klas Ingesson would also like to gamble with Jonas. He is very intelligent and realises the nature ofJonas' preferences. He offers him a bet that Jonas will take. Klas says: "If you loose you will give me10.000 kr. If you win, I will give you .?Problem 2. UncertaintyGabriel likes to gamble and his preferences are represented by the expected utility functionU π1c12 π2c22Gabriel has not worked out very well, he only have 1.000 kr. Thomas shuffled a deck of cards andoffered to bet Gabriel 200 kr that he would not cut a spade from the deck.

a. Show that Gabriel refuses the bet.b. Would Gabriel accept the bet if they would bet 1.000 kr instead of 200 kr?c. Sketch one of Gabriel's indifference curves (let Event 2 be the event that a card drawn from a fairdeck of cards is a spade)d. On the same graph, sketch the indifference curve when the gamble is that he would not cut ablack card from the deck.Problem 3. UncertaintyConsider an individual with an income of 100. She has the option of participating in a lottery whereshe can win 30 with a probability of 0.5, and loose 30 with a probability of 0.5. Would she participateif she is risk averse? What id she is a risk lover? ExplainAnswers to the problemsProblem 1. Demanda) Given prices, p1 and p2, find the quantities x1 and x2 which maximise Bengt’s utility!Necessary condition:MRS MU 1MU 2MU1 1 p1p 2MU2 1/x2MRS x2Therefore the optimum occurs whenx2 p1p2Money left to buy x1 for:m p 2 x2 m p 2x1 p1 m p1p2m p1m 1p1p1if m p1b) Yes, if m p1 he won’t buy any more beer when m increases.c)-d)

p1 dx1 p1m p1m p12 2 2 dp1 x1p1 m p1p1 m p1p1mm p1 p2 dx2 p2p p 12 2 1dp2 x2p2 p1p2 m1 dx1 m 1 mp1m dm x1 p1 m p1 p1 (m p1 )p1 m(m p1 )(m p1 ) m2 dx2 mm 0 0dm x2x2Problem 2. Demandy 40010 food stamps per week, price 5 kr. Can be exchanged for 10 kr worth of food.Old budget line: max 400 other and 400 food.New budget line: max 400 other and 450 food, kink at 100 food and 350 other.Homothetic preferences: The income expansion path is a straight line through origin.Since Jan spent half his income on food, he will continue doing so. His income increases by 50 kr, thushe spends 25 kr more on food.Problem 3. DemandBill: u 2xy 3 x u 2x 2 y 2 y2xy3 Px 3x2y2 Py(simplify)2y Px3x Py(solve for y)3 Px xy 2Py(insert this into the budget restriction)

3 Px x3Px x5m Py 2 P Px x 2 Px x 2 Px xy2mx 5P(solve for x)(insert this into the expression for y)x3 Px 2m 6m 3my 2 P 5P 10P 5Py xyyWe can also solve this by transforming the utility function to a Cobb-Douglas:v(x2y3)1/5 x2/5 y3/5am 2mx P 5PxxBen: u3 2 x 1 y 2 x u22 2 x 1 y 2 y2(x 1) (y 2)3 Px (x 1)2 3(y 2)2 Py(simplify)2(y 2) Px3(x 1) Py(solve for y)Px2y P 3(x 1) - 4yPx 3(x 1) 4y P2 -2y(insert this into the budget restriction)Px 3(x 1) 4m Px x Py (P2 - 2) (simplify)ym Px x Px3(x 1)2 - Py 233 Px x 2 Px x 2 Px - 2 Py(solve for x)33Px x 2 Px x m - 2 Px 2Py532 Px x m - 2 Px 2Py5(divide both sides with 2 Px)2 m 3 2 Px2 Pyx 5P -25P 25Pxxx2 m 3 4 Pyx 5P -5 5PxxPxy Py2 m 3 4 Py3(5 P - 5 5 P 1)xx-22(insert this into the expression for y)

Px 3 2 m Px 3 3 Px 3 4 Py Px 3y P 25P - P 25 P 2 5P P 21-2yxyyxy(simplify)m 6 Px 9 12 Px 3y P 10 - P 10 10 P 2 - 2yyym 3 6 Px 8y P 5 10 P - 10yym 3 3 Px 4y P 5 5P -5yyProblem 4. DemandC denotes the number of cups of coffee and B the number of buns:a) 15C 10B 150. The budget line intersects the “coffee-axis” at C 10 and the “bun-axis” at B 15.b) 6 of each. Her indifference curves are L-shaped with the corners on the 45-degree line.c) Perfect complements.Problem 1 Slutsky equationa. Value of endowment: 500 50*50 3.000b. Connect the points (0,0), (50, 0). (50, 500) and (0, 3000)c. d. and e. L LeisureY non-labour incomeH Labour suppliedBudget constraints: C Y wH and L T - HC Y w(T-L) C wL Y wTMaximise U(C,L) CLs.t. C wL Y wTMRS: MUC L and MUL C Set MRS equal to the slope of the budget line:C w L 1C wLInsert this into the budget restriction:C C Y wTwhich solves part d.

or wL wL Y wT which gives usand solves e.With T 50, w 50 and Y 500 we get the answer to b. as L 5 25 30 and labour supply 50–30 20while C 500 20*50 1500f. Labour supplied if Y 0:L 50/2 25 and labour supply H 50-25 25Problem 2 Slutsky equationa. Lagrange function: L(x, y, λ) x0.3y0.5 - λ (5x 8y – 160)First order conditions for maximum: 0.3 x-0.7y0.5 - 5 λ 00.5 x0.3y-0.5 - 8 λ 05x 8y – 160 0Piecewise division of the first two gives x 0.96 y, insertion into the constraint gives x 12, y 12.5(You could also have taken the demand functions with a Cobb-Douglas utility function as known.)MRS MUx/MUy 0.3 x-0.7y0.5/0.5 x0.3y-0.5 0.6y/x 0.6*12.5/12 0.625px/ py 5/8 0.625b) The MRS shows the maximum amount of y that she could trade for one unit of x, without losingutility. If it is lower than the relative price of x, she would be better off consuming less of x and moreof y. If it is higher than the relative price of x, she would be better off consuming more of x and lessof y.For example, if MUx MUy 1 and the price of x is twice that of y, she could give up one unit of yand get two units of x, gain two units of utility and give up one unit. If the price of y is twice the priceof x, she could give up one unit of y (and one unit of utility) and get two units of x (and two units ofutility).c)i. The old budget constraint intersects the axes at (0, 20) and (32, 0), the new at (0, 20) and(40, 0)ii. x 15, y 12.5iii. m’ 148iv. substitution effect: x(4, 148) – x(5, 160) 13.9 – 12 1.9income effect: x(4, 160) – x(4, 148) 15 – 13.9 1.1

Problem 1. Consumer’s surplusa) P 30 - 2Bb) Willingness to pay: 10*10 Consumer’s surplus:(30-10) * 10 200230-202 10 100Problem 2. Consumer’s surplusU ( x, y) x 0,1 y 0,9m0 100, p0 q0 1 where p is the price of x, q the price yCaclulate CV and EV when p increases to 2i) Initially U is a Cobb-Douglas function so we know thatam 0,1 100 10p1bm 0,9 100y 90q1x U 10 0,1 90 0,9ii) If p 2 then x 0,1 *100*1/2 5y is independent of p, and therefore unchanged.U 50,1 900,9iii) CV: Let m’ 100 CV and assume that p 2x 0,05m’y 0,9m’The definition of CV 0,1m' 0,9m',) U (10,90) 10 0,190 0,9210,10,1(m' ) 0,1 0,9 0,9 (m' ) 0,9 10 0,190 0,90,1210 0,1 2 0,1 100 0,9 0,9 0,9(m' ) 0,1 0,9 0,10,1 0,9 0,9U(0,1 10 m' 0,1 CV 7,18 2 0,1 100 0,9 2 0,1 100 0,1 0,9 2 0,1 100 107,177iv) EV: Find the income m’ 100 –EV such that the maximal utility when p 2 and m 100 is the sameas when p 1 and m m’If p 2 and m 100, U 50,1900,9 according to ii)

If p 1, m m’ then x 0,1m’ och y 0,9m’U (0,1) 0,1 (m' ) 0,1 (0,9) 0,9 (m' ) 0,9 m' (0,1) 0,1 (0,9)0,9 0,1 (9) 0,9 m'should be equal to50,1900,9After simplification, this implies that m’ 100*2-0,1 93,30EV 6,7Problem 3. Consumer’s surplusThe consumer surplus is a (monetary) measure of the utility a consumer derives from her/hisconsumption of the good and the price paid for it and it corresponds to the area under the demandcurve, but above the price (assuming that the income effect is so small that it can be disregarded).pp* and q* are the price and the quantityCCSp*demanded at this price.q*If you see the demand curve as that of an individual, the reason for CS is that the reservation price isdecreasing in q and the person would have been willing to pay more if the seller could have chargedthe highest possible price for each unit. If you see it as the market, aggregate demand differentindividuals have different reservation prices and - since perfect price discrimination is usually notpossible – the market price for all buyers will be the reservation price of the marginal buyer. (Eitheranswer is OK.)Problem 4. Consumer’s surplusa) 5*50 (100-50)*5/2 375. (Hint: It is easier to see this if you plot the demand curve in afigure.)b) 375-(5*50) 125Problem 5. Consumer’s surplusa) To ask for the compensating variation: What compensation (for example, in the form of a taxdecrease) would make you agree to have the new housing construction?b) To ask for the equivalent variation: How much would you be willing to give up (say, in theform of a tax increase) if this could stop the construction plans?

Problem 1. Market demandDemand function D(p) q 10 -2pa) What is the price elasticity of demand when p 3dq 2dpq(3) 10 – 2*3 4ε -2 * ¾ -3/2b) At what price and quantity is ε -1?ε -1 dqp 1dp 10 2 pp 2 110 2 p2 p 10 2 p10 2 p 5 p2p 2 .5p c) If the demand function is D(p) a-bp, what is the elasticity of demand? dq pp bdp qa bpProblem 2. Market demandq(p) (p 1)-2a) The price elasticity of demand dq pp2p 2( p 1) 3 2 p( p 1) 3 ( p 1) 2 2dp qp 1( p 1)b) If p -1 2p 1 2 p p 1 p 1p 1c) Total revenue R pq p(p 1)-2d) Answers to a)-c) when q(p) (p a)b, a o and b -1 bpp a -1 when p -a/(b 1)R(p) p(a p)bProblem 3. Market demanda) dD(p)/dp -6 and p/q p/(30-6p), so ϵ -6p/(30-6p)

b) ϵ -p/(60-p)c)ϵ -bp/(a-bp)d) ϵ -2e) ϵ -bf)ϵ -2p/(p 3)Problem 1. Equilibriuma. D(p) S(p)200 - p 150 pp 25D(p) 200 - 25 175b. When is S 160:160 150 pp 10c. Willingness to pay at q 160:160 200 - pp 40Maximum price - minimum price 40 - 10 30 price of the coupons.d. Dead-weight loss of restricting the supply:DW 30 (175-160) 225.2Problem 2. EquilibriumqD 100 - 5p qS 5pqSpS 5No tax:100 - 5p 5pp 10q 50Tax:qSpS 5 2 q 5p - 10100 - 5p 5p - 10p 11

q 45The revenue per unit is (according to the old supply curve):45p 5 9N.B. the price increase is not equal to the tax.Problem 3. Equilibriuma) With p on the vertical axis and q on the horizontal: The demand curve intersects the p-axis abovethe q-axis and slopes downwards to the right, t

Problem 1. Intertemporal choice Suppose that a consumer has an endowment of 200.000 kr each period (period 1 and 2). He can borrow money at an interest rate of 200%, and he can lend money at a rate of 0%. a. Illustrate his budget set. b. The consumer is offered an investment that will change his endowment to m 1 300.000 and m 2 150.000.

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