Short Questions
1. A firm faces the following production function: y = F(z1, z2) = 100z11/2z22
The price of z1, w1, is $10, and the price of z2, w2, is $5.
a)When z1 is fixed at 100 calculate and graph the following functions: TP(z2), AP(z2), MP(z2), VC(y), AVC(y), SMC(y)
b)When z2 is fixed at 10 calculate and graph the following functions: TP(z1), AP(z1), MP(z1), VC(y), AVC(y), SMC(y)2. Suppose a firm's short run total costs are given by: TC(y) = 3y2+y+500
a) What are the firm's variable costs? Fixed costs?
b) Calculate and graph the firm's short run average costs (SAC(y)) and short run marginal costs (SMC(y))
c) Calculate the point where SAC=SMC and mark it on your graph. What point on the SAC curve does this correspond to?3. Hair Apparent is one of many identical firms in the highly competitive baldness treatment industry. Its cost function is given by TC = H2 + 4 where H is the number of heads treated (number of units of output). Derive an expression for and graph Hair Apparent's average cost curve, average variable cost curve, and marginal cost curve.
Answers:
1.a) TP(z2)=1000z22, AP(z2)=1000z2, MP(z2)=2000z2, VC(y)=5(y/1000)1/2, AVC(y)=5(1/1000y)1/2, SMC(y)=2.5(y/1000)1/2
b) TP(z1)=10000z11/2, AP(z1)=10000z1(-1/2), MP(z1)=5000z1(-1/2), VC(y)=y2/107, AVC(y)=y/107, SMC(y)=2y/107
2. a) VC(y)=3y2+y, FC(y)=500
b) SAC(y)=3y+1+500/y, SMC(y)=6y+1 . On the diagram SAC(y) is in RED (curved line) and SMC(y) is in GREEN (straight line from the origin)
c) y=12.9, SAC(12.9)= SMC(12.9)=78.4 The point represents the MIN SAC(y).
Short Questions
1. Natural Farms Ltd. produces yogurt using milk and bacteria culture in fixed proportions. It takes one pint of milk and one ounce of bacteria culture to make a pint of yogurt. The price of milk is $0.50 per pint, and bacteria culture costs $0.10 an ounce. Draw one or two isoquants for Natural Farms Ltd. How much does it cost to produce one pint of yogurt? Two pints? Three pints? What is Natural Farms' cost function?
2. State whether the following production functions exhibit decreasing returns to scale, constant returns to scale, or increasing returns to scale:
a). f(x1, x2) = 5x11/4x21/4
b). F(K,L) = (K+L)2
c). f(x1,x2) = 2(x1 + x2)
Give an explanation for your answer.3. For the production functions in the previous question calculate the marginal product of x1 (or MPK for part b) and the technical rate of substitution. Graph one or two isoquants for each production function (To check that you have the right answer remember: what is the relationship between the technical rate of substitution and the slope of the isoquant?)
Answers: 1. $0.60;$1.2;$1.8; LTC(y)=0.6y.
2. a)DRTS; b)IRTS; c) CRTS
Long Questions
4. Consider the Cobb-Douglas production function: f(x1, x2) = A x1ax2b.
For the following sets of parameter values, does the function exhibit constant, increasing, or decreasing returns to scale?
a). A=1, a=1/3, b=2/3
b). A=2, a=1/3, b=2/3
c). A=1, a=1/4, b=1/4
d). A=a=b=15. Blooming Blossoms produces flowers using two inputs, seeds (measured in ounces - z1) and fertilizer (measured in ounces - z2). The production function for flowers is given by: y = 1000z11/4z21/4
a). Suppose initially that Blooming Blossoms only has only 16 ounces of Precious Petunia seeds. Calculate Blooming Blossoms' Total Product and Short Run Marginal Product functions.
b). Does the law of diminishing marginal product hold for Blooming Blossoms' total product function?
c). T/F/U. If the law of diminishing marginal product did not hold, Blooming Blossoms could grow the entire world's petunia supply from 16 ounces of seeds.
d). Now suppose that Blooming Blossom can buy as many Precious Petunia seeds as it needs. Graph one of Blooming Blossom's isoquants.
e). Calculate the solution to Blooming Blossom's cost minimization problem.6. Dan Donaldson runs Draper Dan, a company that makes curtains. Curtain manufacturing uses cloth and labor in fixed proportions: it takes exactly five meters of cloth and 3 hours of labor to make 1 curtain. Derive Draper Dan's cost function (a) in terms of input prices and output and (b) when the price of cloth, w1, is $3/metre and the wage rate w2 is $10 per hour.
7. Herongate Horses produces saddles (y) using two inputs, leather (z1) and labor (z2). Herongate Horses's production function is given by: F(z1, z2) = (1/4)z1z2
a). Does Herongate Horses's production function exhibit constant, increasing, or decreasing returns to scale? Explain your answer fully.
b). Graph Herongate Horses's isoquant for y=4. Calculate the slope of the isoquant as a function of z1 and z2.
c). Calculate Herongate Horses's conditional input demands for z1 and z2 as a function of y, when w1=5 and w2=20.
d). Using your answer to (c), find Herongate Horses's total cost function. Graph the total cost function, TC(y). Find and graph the long run average cost function, LAC(y).
e). Suppose now that z2 is fixed at 2. Find Herongate Horses's total product function, TP(z1).
f). Continuing on from (e), find Herongate Horses's variable cost function, VC(y). Find and graph the average variable cost function, AVC(y).
g). Compare your answers from part (d) and part (f). Are the shape of the functions the same or different? Can you explain why?8. A firm's production function is given by Q = (K+L)2 where Q is output, and K and L are capital and labor inputs.
a). Does the production function exhibit constant, increasing, or decreasing returns to scale?
b). Draw isoquants for Q=4 and Q=9. What is the firm's marginal rate of technical substitution?
c). Suppose the price of capital, r, is $1 and the price of labor, w, is $2. How much capital and labor will the firm use to produce Q=4? Q=9?
d). What is the firm's cost function when r=$1 and w=$2?9. Floppy Corp produces software using two inputs, large (5 1/4 inch) disks, L, and small (3 1/2 inch) disks, S. Its production function is given by: Q=(L/2) + S where Q is output.
a). Does the production function exhibit constant, increasing, or decreasing returns to scale?
b). Draw isoquants for Q=2 and Q=3. What is the firm's marginal rate of technical substitution?
c). Suppose small disks cost $3 each and large disks cost $1 each. How many of each type of disk will Floppy Corp use to produce Q=2? Q=3?
d). What is the firm's cost function when small disks cost $3 and large disks cost $1?Answers:
4. a) CRTS; b) CRTS; c) DRTS; d) IRTS;
5. a) TP(z2)=2000z21/4; MP(z2)=500z2(-3/4); b) yes; c) F; d)smooth and convex; e) In general, if y=Az1az2b then the conditional input demand functions are given by: z1= (y/A) (1/(a+b))(aw1/bw2)(b/(a+b)) and z2=(y/A) (1/(a+b))(bw1/aw2)(a/(a+b)) . In this case a=b=1/4 and A=1000.
6.a) TC(y)=(5w1+3w2)y; b) TC(y)=45y where w1 is the price of cloth and w2 is the price of labor.
7. a) IRTS; b) slope=z2/z1; c) z1=4y1/2 and z2=y1/2 ; d) TC(y)=40y1/2 LAC(y)=40/y1/2 ; e) TP(z1)=z1/2 ; f) VC(y)=10y, AVC(y)=10 ; g) different because LR allows for more flexibility (substitution in production).
8. a) IRTS; b) MRTS=1, isoquants are straight lines; c) K*=Q1/2 ,L*=0, if Q=4 then K*=2 and L*=0, if Q=9 then K*=3 and L*=0; d) TC(Q) = rQ1/2 .
9. a) CRTS; B)MRTS=1/2 (L on the horizontal axis); c) wL/wS=1/3 ; S*=0 and L*=2Q. If Q=2 then L*=4 and S*=0 and if Q=3 then L*=6 and S*=0; d) TC(Q)=2wLQ.
III The Theory of Perfect Competition
Short Questions
1. An industry has 50 identical, perfectly competitive firm. Each firm has a short run cost function given by: C=192+12q2 . What are the firm and industry short run supply functions?
Answers:q=p/24; Q=50p/24.
Long Questions
1. B.D. Shovel Ltd. is one of many identical competitive firms producing spades. Its cost function is given by C(Q)=Q2+4, where Q is the number of spades produced.
a). Give an equation for and graph the spade industry long run supply curve.
b). Suppose the demand for spades is given by QD=D(p)=5000-500p. Graph the demand curve. Find the equilibrium price and quantity of spades. (Hint: at the equilibrium QD=QS).
c). Bowing to pressure from the mechanical digger lobby, the government decides to impose a $1 per unit tax on spades. What is the effect of the tax on the price paid by consumers and the equilibrium quantity?2. The market demand curve for swim goggles is given by: QD = 225 - 25P where QD is the quantity of swim goggles demanded and P is the price of swim goggles. The supply curve for swim goggles is QS = - 150 + 50P where QS is the quantity of swim goggles supplied.
a). Find the equilibrium price of swim goggles and quantity sold.
b). Calculate the consumers' surplus and producers' surplus at the price and quantity found in part (a).
c). Explain in words and using diagrams what is meant by the terms "consumers' surplus" and "producers' surplus".3. The demand for cigarettes is given by: QD = 140,000-25,000P.
where QD is the quantity of cigarettes demanded (in packs) and P is the price of a pack of cigarettes. The supply of cigarettes is given by: QS = 20,000+75,000P where QS is the quantity of cigarettes supplied (in packs).
Suppose that a tax of $0.40 per pack was imposed on cigarettes.a). How much would consumers' surplus be reduced? How much would producers' surplus be reduced?
b). What is the deadweight loss associated with the tax? Explain using words and diagrams what is meant by the term "deadweight loss".4. Tim Long is is one of many identical dairy farmers in Subsidyland. His cost function for milk production is given by: C=Q2/200 where C represents costs (in dollars) and Q represents daily milk production in litres. The government of Subsidyland decides that, in order to guarantee farmers a reasonable and stable income, it will intervene in the milk industry. It decides to issue a limited number of milk production quotas. Only farmers with quotas will be allowed to sell milk. Farmers with quotas can sell up to 40 litres of milk a day, at a guaranteed price of $0.25 a litre. Quotas, once issued, cannot be bought and sold.
a). How much (per day) would Tim Long be prepared to pay for a milk quota?
b). If Tim is issued a milk quota by the government, what will be his economic profits?5. Hair Apparent is one of many identical firms in the highly competitive baldness treatment industry.
Its cost function is given by: C = (1/4)y2+1a). Give an equation for and graph the firm's short-run supply curve.
b). Give an equation for and graph the industry's long-run supply curve.
c). If the demand for baldness treatment is given by Q = 108 - 12p how many firms will there be in the baldness treatment industry in the long-run competitive equilibrium?
d). How much profits will a typical firm make in the long-run equilibrium? How much producer's surplus?6. Violent Toys Inc is a competitive firm which produces Teenage Mutant Ninja Tortoise dolls according to the cost function: C=Q2/20 where Q is the number of dolls produced. The dolls sell for $10 each.
After losing a court case, Violent Toys is ordered to pay a licence fee to Turtles Ltd.a). If the licence fee is $500, regardless of how many dolls Violent Toys produces, what happens to Violent Toy's cost function? To the marginal cost function? To the level of output? Explain you answer with math and with a graph.
b). If the licence fee is $5 per doll, what happens to Violent Toy's cost function? To the marginal cost function? To the level of output? Explain you answer with math and with a graph.Answers:
1. a) LRS is given by p=4 and is a horizontal line. b) The demand is adownward sloping straight line. The equilibrium price and quantity are p*=4 and Q*=3000. c) The tax will increase the total cost by Q ($1 times the quantity produced Q). The tax will increase the price by $1 and the quantity will decrease by 500.
2. a) p=5, Q=100; b) CS=200, PS=100.
3. a) CS will be reduced by 31.875 and PS will be reduced by 10.625; b) DWL=1500.
4. Tim will be willing to pay at most 3.125. This is his economic profit.
5. a)p=y/2, b) LRS is given by p=1; c) At p=1 the quantity demanded (and supplied) in the market is 96. Each firm produces 2 units, therefore n=48. d) profit =0, Producer Surplus=0.
6. a) The total cost will increase by 500, MC will not change, therefore the output will not change. b) The total cost will increase by 5Q, MC will increase by 5 and the output will decrease by 50.
IV: Monopoly
Short Questions
1. Calculate the marginal revenue curves associated with the following inverse demand or demand functions
a). p = 130 - y
b). y = 100 - p
c). p = 1/yAnswers: a) 130-2y b) 100-2y c) 0.
Long Questions
1. The (inverse) market demand for snowshovels in Spuzzum is given by p = 50 - .5y
a). Snowgone Inc. is the only supplier of showshovels in Spuzzum.
Its total cost function is given by TC = 10yCalculate:
i). the profit maximizing level of output
ii). the profit maximizing price
iii). the consumers surplus
iv). the monopoly profits
v). the burden of monopoly (deadweight loss)b). Snowgone Inc. loses a legal battle and as a result has to pay licensing fee of $700 per year to MasterShovels Ltd. Its total costs therefore increase to TC = 10y + 700
With this new cost function, once again calculate
i). the profit maximizing level of output
ii). the profit maximizing price
iii). the consumers surplus
iv). the monopoly profits
v). the burden of monopoly (deadweight loss)Are your answers the same as in part (a) or different? Explain why.
c). If all potential showshovel suppliers have cost functions as shown in part (a), is Snowgone Inc. a natural monopoly? Explain (Hint: Calculate the residual demand function. Would another firm, facing this demand function, be able to make positive profits?).
d). If all potential showshovel suppliers have cost functions as shown in part (b), is Snowgone Inc. a natural monopoly? Explain.
Answers: a) y=40, p=30, CS=400, Profit=800, DWL=400; b) y=40, p=30, CS=400, Profit=100, DWL=400; c) NO the residual demand is y=60-2p.The profit for the potential entrant is 200. d) YES Because of the fixed cost the market cannot support the second firm. The potential entrant makes negative profits if enters, therefore will stay out.
V: Price Discrimination and Monopoly Practices
Short Questions
1. Grocery stores often sell more expensive and/or poorer quality food in poor areas.
a). Explain this using the theory of price discrimination. Illustrate your answer with a diagram.
b). Give at least two reasons why the price elasticity of demand might be different for rich and poor people.Answers:1.
- lower income individuals may be less likely to have a car therefore find it harder to travel, shop for bargains
- lower income individuals may have cash flow problems, therefore may be unable to stock up on, e.g., baked beans at 59 cents per can.
- lower income individuals may not have access to deep freezes, may be less informed about alternatives available (don't subscribe to a newspaper)
- there may be less competition in poor areas so each firm may face a relatively inelastic demand compared to stores in more competitive rich areas (compare quality of food in restaurants in poor area v. food quality in restaurants in wealthier area).Long Questions
1. A theater has a monopoly on the rights to show movies in the town of Moosejaw. The monopolist knows that the price elasticities of demand for movies by adults and by children are 2 and 4 respectively. Suppose the monopolist can charge different prices for adults and children. Explain, using the monopolist's profit-maximizing conditions, why the price of an adult ticket is higher than the price of a child's ticket.
2. A monopolist produces a commodity, Q, from a single plant but sells it in two separate markets. The total cost of production is TC = 100+Q2 The inverse demand functions for the two markets are
p1=240 - q1
p2 = 200 - 0.5q2
where q1 + q2=Q.
a). Calculate the profit maximizing price and quantity when the monopolist is able to price discriminate.
b). Calculate the profit maximizing price and quantity when the monopolist is not able to price discriminate.
c). Calculate the consumers' surplus and monopolist profits associated with the outcomes in part (a) and part (b).3. A monopolist has total production costs given by:
TC(Q) = 0.5Q2 The demand function in the "home" market is: P = 20 - 0.5 Qa). If the monopolist sells all its output in the home market, what is the simple monopoly, equilibrium, price?
b). Now suppose the monopolist has a choice between two markets, the home market, and a foreign market in which the monopolist can sell any amount of Q at a price of $12 per unit.
i). Will the monopolist sell in the foreign market?
ii). If the monopolist does sell in the foreign market, what would happen to the price the monopolist charges in the home market?
4.a). Find the simple monopoly equilibrium price under the following conditions. The "home" consumers' inverse demand function is $P = 12 - 0.25Q where Q represents the monopolist's production and sales, per period. The monopolist produces Q at a cost represented by the total cost function, TC= F+0.125Q2
b). At the equilibrium in part (a), suppose a second, "foreign" market were available to the monopolist. In this second market, the monopolist could sell any amount of Q for $6 per unit, and the monopolist is assured that no sales of Q from the second market to the "home" market in part (a) would occur.
i). Explain carefully whether or not the monopolist would sell Q at $6 in the second market.
ii). Would the "home" consumers be better, worse off, or unaffected if the monopolist made sales in the second market? Please explain your answer. [Hint: Remember MR1=MR2. What is MR1 as calculated in part (a)? Part (b) if the monopolist sells in the second market? What will happen to prices?]Answers:
1. Use the formula that expresses MR as a function of the price elasticity of demand. MR = P(1 - 1/e) where e is the elasticity
2. a) q1=40, q2=40, p1=200, p2=180, CS1=800, CS2=400, Profit=8700; b) q1=53, q2=27, p=187, CS1=1423, CS2=178, Profit=8434;
3. a) p=15; b) i) Yes, the monopoly will sell 4 units in the foreign market and 8 units in the domestic market; ii) The price in the domestic market will increase to 16.
4. a) p=8; b) i) Yes, the monopoly will sell 12 units in the foreign market and 12 units in the domestic market; ii) The price in the domestic market will increase to 9 and the consumers will be worse off.