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CHAPTER 2&4: BUDGET CONSTRAINT & UTILITY

CHAPTER 5: CHOICE

1 If two goods are perfect substitutes, what is the demand function for good 2?

2 Suppose that indifference curves are described by straight lines with a slope of −b Givenarbitrary prices and money income p1, p2, and m, what will the consumer’s optimal choiceslook like?

3 Suppose that a consumer always consumes 2 spoons of sugar with each cup of coffee Ifthe price of sugar is p1 per spoonful and the price of coffee is p2 per cup and the consumerhas m dollars to spend on coffee and sugar, how much will he or she want to purchase?4 Suppose that you have highly non convex preferences for ice cream and olives, like thosegiven in the text, and that you face prices p1, p2 and have m dollars to spend List the choicesfor the optimal consumption bundles.

5 If a consumer has a utility function , what fraction of herincome will she spend on good 2?

6 For what kind of preferences will the consumer be just as well-off facing a quantity tax asan income tax?

5.4 We know that you’ll either consume all ice cream or all olives Thus the two choices forthe optimal consumption bundles will be x1 = m/p1, x2 = 0, or x1 = 0, x2 = m/p2.

5.5 This is a Cobb-Douglas utility function, so she will spend 4/(1 + 4) = 4/5 of her incomeon good 2.

5.6 For kinked preferences, such as perfect complements, where the change in price doesn’tinduce any change in demand.

CHAPTER 15: MARKET DEMAND

1 If the market demand curve is D(p) = 100 - 0.5p, what is the inverse demand curve?

2 An addict’s demand function for a drug may be very inelastic, but the market demandfunction might be quite elastic How can this be?

3 If D(p) = 12 - 2p, what price will maximize revenue?

4 Suppose that the demand curve for a good is given by D(p) = 100/p What price willmaximize revenue?

5 True or false? In a two good model if one good is an inferior good the other good must be aluxury good.

15.1 The inverse demand curve is P(q) = 200 - 2q.

15.2 The decision about whether to consume the drug at all could well be price sensitive, sothe adjustment of market demand on the extensive margin would contribute to the elasticityof the market demand.

15.3 Revenue is R(p) = 12p - 2p^2, which is maximized at p = 3.

15.4 Revenue is pD(p) = 100, regardless of the price, so all prices maximize revenue.

15.5 True The weighted average of the income elasticities must be 1, so if one good has anegative income elasticity, the other good must have an elasticity greater than 1 to get theaverage to be 1.

CHAPTER 19: TECHNOLOGY

4 The technical rate of substitution between factors x2 and x1 is −4 If you desire to producethe same amount of output but cut your use of x1 by 3 units, how many more units of x2 willyou need?

5 True or false? If the law of diminishing marginal product did not hold, the world’s foodsupply could be grown in a flowerpot.

6 In a production process is it possible to have decreasing marginal product in an input andyet increasing returns to scale?

6 Yes the marginal product of an input refers to the change in production that occurs whenall other inputs are held constant

CHAPTER 20: PROFIT MAXIMIZATION

1 In the short run, if the price of the fixed factor is increased, what will happen to profits?2 If a firm had everywhere increasing returns to scale, what would happen to its profits ifprices remained fixed and if it doubled its scale of operation?

3 If a firm had decreasing returns to scale at all levels of output and it divided up into twoequal-size smaller firms, what would happen to its overall profits?

6 If p/MP1 > w1, then should the firm increase or decrease the amount of factor 1 in order toincrease profits?

7 Suppose a firm is maximizing profits in the short run with variable factor x1 and fixedfactor x2 If the price of x2 goes down, what happens to the firm’s use of x1? What happensto the firm’s level of profits?

8 A profit-maximizing competitive firm that is making positive profits in long-runequilibrium (may/may not) have a technology with constant returns to scale.

20.1 Profits will decrease.

20.2 Profit would increase, since output would go up more than the cost of the inputs.

20.3 If the firm really had decreasing returns to scale, dividing the scale of all inputs by 2would produce more than half as much output Thus the subdivided firm would make moreprofits than the big firm This is one argument why having everywhere decreasing returns toscale is implausible.

20.6 Increase.

20.7 The use of x1 does not change, and profits will increase.20.8 May not.

CHAPTER 21: COST MINIMIZATION

1 Prove that a profit-maximizing firm will always minimize costs.

2 If a firm is producing where MP1/w1 > MP2/w2, what can it do to reduce costs butmaintain the same output?

3 Suppose that a cost-minimizing firm uses two inputs that are perfect substitutes If the twoinputs are priced the same, what do the conditional factor demands look like for the inputs?

21.1 Since profit is equal to total revenue minus total costs, if a firm is not minimizing coststhen there exists a way for the firm to increase profits; however, this contradicts the fact thatthe firm is a profit maximizer.

21.2 Increase the use of factor 1 and decrease the use of factor 2.

21.3 Since the inputs are identically priced perfect substitutes, the firm will be indifferentbetween which of the inputs it uses Thus the firm will use any amount of the two inputs suchthat x1 + x2 = y

CHAPTER 22: COST CURVES

1 Which of the following are true? (1) Average fixed costs never increase with output; (2)average total costs are always greater than or equal to average variable costs; (3) average costcan never rise while marginal costs are declining.

2 A firm produces identical outputs at two different plants If the marginal cost at the firstplant exceeds the marginal cost at the second plant, how can the firm reduce costs andmaintain the same level of output?

3 True or false? In the long run a firm always operates at the minimum level of average costsfor the optimally sized plant to produce a given amount of output.

22.1 True, true, false.

22.2 By simultaneously producing more output at the second plant and reducing productionat the first plant, the firm can reduce costs.

22.3 False.

CHAPTER 23: FIRM SUPPLY

1 A firm has a cost function given by c(y) = 10y^2 + 1000 What is its supply curve?

2 A firm has a cost function given by c(y) = 10y^2 + 1000 At what output is average costminimized?

3 If the supply curve is given by S(p) = 100 + 20p, what is the formula for the inverse supplycurve?

4 A firm has a supply function given by S(p) = 4p Its fixed costs are 100 If the pricechanges from 10 to 20, what is the change in its profits?

5 If the long-run cost function is c(y) = y2 +1, what is the long-run supply curve of the firm?6 Classify each of the following as either technological or market constraints: the price ofinputs, the number of other firms in the market, the quantity of output produced, and theability to produce more given the current input levels.

7 What is the major assumption that characterizes a purely competitive market?

8 In a purely competitive market a firm’s marginal revenue is always equal to what? Aprofit-maximizing firm in such a market will operate at what level of output?

9 If average variable costs exceed the market price, what level of output should the firmproduce? What if there are no fixed costs?

10 Is it ever better for a perfectly competitive firm to produce output even though it is losingmoney? If so, when?

11 In a perfectly competitive market what is the relationship between the market price andthe cost of production for all firms in the industry?

23.1 The inverse supply curve is p = 20y, so the supply curve is y = p/2023.2 Set AC = MC to find 10y + 1000/y = 20y Solve to get y∗ = 10.23.3 Solve for p to get Ps(y) = (y - 100)/20.

23.4 At 10 the supply is 40 and at 20 the supply is 80 The producer’s surplus is composed ofa rectangle of area 10 × 40 plus a triangle of area 12 × 10 × 40, whichgives a total change inproducer’s surplus of 600 This is the same as the change in profits, since the fixed costsdon’t change.

23.5 The supply curve is given by y = p/2 for all p ≥ 2, and y = 0 for al p ≤ 2 At p = 2 thefirm is indifferent between supplying 1 unit of output or not supplying it.

23.6 Mostly technical (in more advanced models this could be market) market, could beeither market or technical, technical.

23.7 That all firms in the industry take the market price as given.

23.8 The market price A profit-maximizing firm will set its output such that the marginalcost of producing the last unit of output is equal to its marginal revenue, which in the case ofpure competition is equal to the market price.

23.9 The firm should produce zero output (with or without fixed costs).

23.10 In the short run, if the market price is greater than the average variable cost, a firmshould produce some output even though it is losing money This is true because the firmwould have lost more had it not produced since it must still pay fixed costs However, in thelong run there are no fixed costs, and therefore any firm that is losing money can producezero output and lose a maximum of zero dollars.

23.11 The market price must be equal to the marginal cost of production for all firms in theindustry

CHAPTER 25: MONOPOLY

1 The market demand curve for heroin is said to be highly inelastic Heroin supply is alsosaid to be monopolized by the Mafia, which we assume to be interested in maximizingprofits Are these two statements consistent?

2 The monopolist faces a demand curve given by D(p) = 100-2p Its cost function is c(y) =2y What is its optimal level of output and price?

3 The monopolist faces a demand curve given by D(p) = 10p^-3 Its cost function is c(y) =2y What is its optimal level of output and price?

4 If D(p) = 100/p and c(y) = y^2, what is the optimal level of output of the monopolist? (Becareful.)

6 What is the answer to the above question if the demand curve facing the monopolist hasconstant elasticity?

25.3 The demand curve has a constant elasticity of -3 Using the formula p[1 + 1/ǫ] = MC,we substitute to get p[1 - 1/3] = 2 Solving, we get p = 3 Substitute back into the demandfunction to get the quantity produced: D(3) = 10 × 3-3.

25.4 The demand curve has a constant elasticity of -1 Thus marginal revenue is zero for alllevels of output Hence it can never be equal to marginal cost.

25.6 In this case p = kMC, where k = 1/(1 - 1/3) = 3/2 Thus the price rises by $9.

CHAPTER 27: FACTOR MARKETS

1 We saw that a monopolist never produced where the demand for output was inelastic Willa monopsonist produce where a factor is inelastically supplied?

2 In our example of the minimum wage, what would happen if the labor market wasdominated by a monopsonist and the government set a wage that was above the competitivewage?

27.1 Sure A monopsonist can produce at any level of supply elasticity.

27.2 Since the supply of labor would exceed the demand for labor at such a wage, we wouldpresumably see unemployment

5 Draw a set of reaction curves that result in an unstable equilibrium.6 Do oligopolies produce an efficient level of output?

28.1 In equilibrium each firm will produce (a-c)/3b, so the total industry output is 2(a - c)/3b.28.2 Nothing Since all firms have the same marginal cost, it doesn’t matter which of themproduces the output.

28.5 Make f2(y1) steeper than f1(y2).

28.6 In general, no Only in the case of the Bertrand solution does price equal the marginalcost

32.2 No For this would mean that at the allegedly Pareto efficient allocation there is someway to make everyone better off, contradicting the assumption of Pareto efficiency.

32.3 If we know the contract curve, then any trading should end up somewhere on the curve;however, we don’t know where.

32.4 Yes, but not without making someone else worse off.

32.5 The value of excess demand in the remaining two markets must sum to zero.

CHAPTER 33: PRODUCTION

1 The competitive price of coconuts is $6 per pound and the price of fish is $3 per pound Ifsociety were to give up 1 pound of coconuts, how many more pounds of fish could beproduced?

3 In what sense is a competitive equilibrium a good or bad thing for a given economy?4 If Robinson’s marginal rate of substitution between coconuts and fish is -2 and themarginal rate of transformation between the two goods is -1, what should he do if he wants toincrease his utility?

5 Suppose that Robinson and Friday both want 60 pounds of fish and 60 pounds of coconutsper day Using the production rates given in the chapter, how many hours must Robinson andFriday work per day if they don’t help each other? Suppose they decide to work together inthe most efficient manner possible Now how many hours each day do they have to work?What is the economic explanation for the reduction in hours?

ANSWER:

33.1 Giving up 1 coconut frees up $6 worth of resources that could be used to produce 2pounds (equals $6 worth) of fish.

33.3 Given a few assumptions, an economy that is in competitive equilibrium is Paretoefficient It is generally recognized that this is a good thing for a society since it implies thatthere are no opportunities to make any individual in the economy better off without hurtingsomeone else However, it may be that the society would prefer a different distribution ofwelfare; that is, it may be that society prefers making one group better off at the expense ofanother group.

33.4 He should produce more fish His marginal rate of substitution indicates that he iswilling to give up two coconuts for an additional fish The marginal rate of transformationimplies that he only has to give up one coconut to get an additional fish Therefore, by givingup a single coconut (even though he would have been willing to give up two) he can have anadditional fish.

33.5 Both would have to work 9 hours per day If they both work for 6 hours per day(Robinson producing coconuts, and Friday catching fish) and give half of their totalproduction to the other, they can produce the same output The reduction in the hours of workfrom 9 to 6 hours per day is due to rearranging production based on each individual’scomparative advantage.