ECON3010 Problem Set 4 and Solutions (2018 ) PDF

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Problem Set 4 on Uncertainty, Competitive Equilibrium with taxation, Market Demand, and Consumer surplus with detailed Solutions. ...

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Econ 3010 Intermediate Microeconomic Theory

Alejandro Molnar Spring 2018

Problem Set 4 Due in class on: Friday Mar 20 Instructions: Please turn in your problem set using a cover sheet with your name on it. The cover sheet could be the printout of the problem set or a blank page. Points on this problem set: 160

Q1. Uncertainty and insurance. 28 points Bob has an initial wealth of $900 and a 20% chance of getting in a (minor) car accident. If he gets in an accident, he will lose $500 in repairs to his car, leaving him with $400; if he doesnt, he loses √ nothing. He maximizes expected utility, and his Bernoulli utility function is u(w) = w. (A) What is Bob’s expected wealth? Solution: Bob’s expected wealth is given by: (0.2)$400 + (0.8)$900 = $80 + $720 = $800 (B) What is Bob’s expected utility? √ √ Solution: His expected utility is given by: (0.2)u(400)+(0.8)u(900) = (0.2)( 400)+(0.8)( 900) = (0.2)20 + (0.8)30 = 28 (C) What is Bob’s certainty equivalent wealth level, i.e., the certain level of wealth that would give him the same expected utility as his uncertain situation? (Hint: find this by equating the expected utility from an unknown but certain level of wealth wc to the expected utility you found in (b).) Solution: His certainty equivalent wealth is the certain wealth wCE that gives him the same expected utility as the uncertain situation he starts out with, i.e., the certain wealth wCE that gives √ him an expected utility of 28. Solving u(wCE ) = 28 for wC E gives us wCE = 28 or wCE = (28)2 = 784. If we asked Bob how much he’d be willing to pay for full insurance, the answer would be $900 − $784 = $116, which is $16 dollars more than his expected loss of $100. √ (D) How do your answers to the above questions change if his utility function is u(w) = 10 w + 20 √ instead of u(w) = w? Solution: His expected loss and expected wealth are unchanged at $100 and $800, respectively. His expected utility is now (0.2)u(400) + (0.8)u(900) = 40 + 240 + 20 = 300 instead of 28. To get his certainty equivalent wealth, solve u(wCE ) = 300 for wCE to get wCE = $784, which is the same as before. The maximum amount he would pay for full insurance is therefore also the same as before ($900 − $784 = $116). The point here is that nothing observable changes: with the data available it √ would be impossible to√distinguish between an individual maximizing u(w) = w and an individual maximizing u(w) = 10 w + 20.

Q2. Selling an asset. 24 points Yang owns a machine (asset) that provides a flow of services over time. Suppose the value of the asset changes over time t (given in years) and the value is given by

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V (t) = −3000 + 500t − 10t2 (A) When is the asset value at a maximum? Find this time and call it t1 . What is the value of the asset at time t1 ? Solution: The problem is to find the maximum of V (t). By the first-order condition (derivative w.r.t t), V ′ (t) = 500 − 20t = 0. Solving for t gives t1 = 500/20 = 25. Now plug this back into the value function to get V (t1 = 25) = −3000 + 500 · (25) − 10 · (25)2 = 3250. (B) Suppose the interest rate is 10%. When should the machine be sold? Find this time t2 . What is the asset value at this time? Solution: The rate-of-return of the asset at time t is V ′ (t)/V (t) = (500 − 20t)/(−3000 + 500t − 10t2 ) The asset should be sold when the rate of return equals to the interest rate. If the asset’s rate of return is higher, he should keep it. If the interest rate is higher, he should sell it earlier and make a deposit to earn interest. Solve the following equation for t: V ′ (t)/V (t) = (500 − 20t)/(−3000 + 500t − 10t2 ) = 0.1 We √get t2 − 70t + 800 = 0. Solving this equation with the quadratic formula, we find that t = √ √ 70± 702 −4·800 = 35 ± 12 1700 = 35 ± 5 17. Thus, 2 t ≈ 14.3845, 55.6155, and t = 14.3845 maximizes the value. When t2 =14.3845, the asset value at t2 is V (t2 = 14.3845) = −3000 + 500 · (14.3845) − 10 · (14.3845)2 = 2123.11 (C) What is the payoff at time t1 if he sells at time t2 in part B) and invests the money at 10% compounding yearly for the remaining years? Compare the payoff to that in part (A). Solution: We found earlier that t1 =25 and t2 =14.3845. The payoff at year 25 from selling at year 14.3845 and then investing the 2123.11 at 10% interest per year for the remaining 10.6155 years is: 2123.11 · (1 + 0.1)10.6155 ≈ 5839.51. Compared to A), 5500.90 > 3250 = V (t1 = 25).

Q3. Consumer surplus. 36 points Emily consumes peaches and other things. We’ll call peaches x and everything else that she spends her money on y. Her utility function for peaches and spending money on other things is given by

u(x, y) = 100x −

x2 2

+y

(A) What kind of utility function does Emily have? Solution: Emily has a quasilinear utility function, since it is of the form u(x, y) = f (x) + y. (B) What are her demand curve and inverse demand curves for peaches? 2

Solution: To get the demand curve, we want to solve the problem of maximizing utility subject to the budget constraint. max 100x − x

x2 +y 2

subject to m = px x + y ⇒ y = m − px x

One way to do this is to substitute y = m − px x and take the derivative with respect to x; we get that the first order condition for peaches is 100 − x − px = 0. This implies that the demand curve for peaches and the inverse demand curve for peaches are, respectively: x = 100 − px

px = 100 − x

(C) If the price of peaches is $50, how many peaches will she consume? What if the price of peaches is $80? Solution: Plugging in the prices, we get the consumption at each price is: x(px = 50) = 100 − 50 = 50

x(px = 80) = 100 − 80 = 20

(D) Suppose Emily has $4000 in total to spend each month. What is her total utility from peaches and money to spend on other things if the price of peaches is $50? What if the price of peaches is $80? Solution: Given the budget constraint we wrote above, y = m − px x, and the quantities we found in part (C), we can solve for the remaining income spent on y for each price and then calculate the utilities as follows: If px = 50 ⇒ y = 4000 − 50 · 50 = 1500 So utility is: u(50, 1500) = 100 · 50 −

502 + 1500 = 5250 2

If px = 80 ⇒ y = 4000 − 80 · 20 = 2400 So utility is: u(20, 2400) = 100 · 20 −

202 + 2400 = 4200 2

(E) By how much does utility decrease when the price changes from $50 to $80? Solution: Taking the difference, we see that the utility decreases by 5250 − 4200 = 1050 when the price increases from $50 to $80. (F) What is the change in her consumer surplus when the price changes from $50 to $80? Solution: We know that the inverse demand curve for x is p = 100 − x so we can draw the demand curve. The consumer surplus when the price is $50 is the triangle bounded by the line p = 50 and the y-intercept of the demand curve (the blue area + the red area). At $80 the consumer surplus is the triangle bounded by p = 80 and the y-intercept (the red area). We can see that the triangle’s height is the difference between the y-intercept and the market price (pintercept − pmarket price ), and the base is the quantity demanded at the market price (Qmarket price ) (p −p )∗Q so the area is intercept market 2price market price .

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The y-intercept is 100. At the market price of 50, the quantity demanded is 50, so the CS is = 200. The change in consumer = 1250. Similarly, the CS when p = 80 is (100−80)∗20 2 surplus is 1250 − 200 = 1050. (100−50)∗50 2

Q4. Market demand. 30 points In Fargo, North Dakota, there are two kinds of consumers: Buick owners and Dodge owners. Every Buick owner has a demand function for gasoline given by: DB (p) = 20 − 5p for p ≤ 4, and DB (p) = 0 for p > 4. Every Dodge owner has a demand function for gasoline given by: DD (p) = 15 − 3p for p ≤ 5 and DD (p) = 0 for p > 5. (Quantities are measured in gallons per week and prices are measured in dollars.) Suppose that Fargo has 100 Buick owners and 50 Dodge owners, for a total of 150 consumers. (A) If the price of gasoline is $3, what is the total amount demanded by each individual Buick owner? And by each individual Dodge owner? Solution: Plugging in the price, the total amount demanded by each individual Buick and Dodge owner is: DB (p = 3 < 4) = 20 − 5 · 3 = 5

DD (p = 3 < 5) = 15 − 3 · 3 = 6

(B) If the price of gasoline is $3, what is the total amount demanded by all Buick owners? And by all Dodge owners? Solution: Multiplying the individual demands we found in part (A) by the number of Buick and Dodge owners, we get that the total amount demanded by all Buick and Dodge owners is: DB = 5 ∗ 100 = 500

DD = 6 ∗ 50 = 300 4

(C) What is the total amount demanded by all consumers in Fargo at a price of $3? Solution: Adding the total amount demanded by all Buick and Dodge owners together gives us the total amount of gas demanded for all consumers in Fargo: D = DB + DD = 500 + 300 = 800

(D) Draw a graph with the demand curve representing the total demand by Buick owners and a demand curve representing total demand by Dodge owners. Then draw the market demand curve for the whole town of Fargo. Be sure to label the axes, relevant quantities, and each curve clearly. Solution:

$/gallon

5 4

Total demand by Dodge owners Total demand by Buick owners

Market demand for whole town

750

2000

2750

gallons/week

(E) At what prices does the market demand curve have kinks? Solution: As you can see from the graph, the market demand curve has kinks at p = 4 and p = 5, due to the changes in demand for Buick and Dodge owners at those prices. (F) When the price of gasoline is $1 per gallon, how much does weekly demand fall when price rises by 10 cents? Solution: When the price of gasoline is $1 per gallon, weekly demand for gasoline is D = 1500 + 600 = 2100. When the price of gasoline is $1.10 per gallon, weekly demand for gasoline is D = 1450 + 585 = 2035. Thus, when the price rises by 10 cents from this initial price, the weekly demand falls by 65 gallons. (G) When the price of gasoline is $4.50 per gallon, how much does weekly demand fall when price rises by 10 cents? Solution: When the price of gasoline is $4.50 per gallon, weekly demand for gasoline is D = 0 + 75 = 75. When the price of gasoline is $4.60 per gallon, weekly demand for gasoline is D = 0 + 60 = 60. Thus, when the price rises by 10 cents from this initial price, the weekly demand falls by 15 gallons.

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(H) When the price of gasoline is $10 per gallon, how much does weekly demand fall when price rises by 10 cents? Solution: When the price of gasoline is $10 per gallon, weekly demand for gasoline is zero, so when the price rises by 10 cents the demand will remain at zero.

Q5. Competitive equilibrium with taxation. 32 points The demand curve for cigarettes is given by D(pD ) = 100 − 20pD and the supply curve is given by S(pS ) = 30pS (A) What is the equilibrium price? What is the equilibrium quantity? Solution: In an equilibrium without the tax the market clears so that the demand (buyer) price pD equals the supply (seller) price pS , and the demand quantity D(pD ) equals the supply quantity S(pS ). D(pD ) = 100 − 20pD = 30pS = S(pS ) and pD = pS The solutions are pD = pS = $2

and

D(pD ) = S(pS ) = 60

The equilibrium price is $2 and the equilibrium quantity is 60. (B) Draw the consumer surplus and producer surplus on a graph, and calculate their numerical value. (Hint: These surpluses are represented by triangles, you can calculate them by recalling the function for the area of a triangle). Solution:

P & p D = 5 − D / 20

%$)+#( +(&"+)

p=

'+!"!(!+# (%+( +(&"+)

p S = S / 30

& 

Q



6

Consumer and Producer Surplus are: CS = (5 − 2) × 60/2 = 90

P S = (2 − 0).60/2 = 60

The sum of consumer surplus and producer surplus is 90 + 60 = 150 Assume that a tax of $1 per cigarette is imposed on consumers. Consider the following cases with the tax. (C) Write an equation between the price paid by buyers pD and the price received by sellers pS . Solution: The tax is $1 per cigarette, pD = pS + 1. (D) What is the new equilibrium price paid by buyers and what is the equilibrium price received by sellers? What is the new equilibrium quantity? Solution: In an equilibrium with a unit tax of $1, the demand (buyer) price pD is $1 higher than the supply (seller) price pS , and the demand quantity D(pD ) equals to the supply quantity S(pS ). D(pD ) = 100 − 20pD = S(pS ) = 30pS , pD = pS + 1 The solution to these equations is pD = $2.6 pS = $1.6 D(pD ) = S(pS ) = 48 The equilibrium price paid by buyers is $2.6, the equilibrium price received by sellers is $1.6, and the equilibrium quantity is 48. (E) Calculate the consumer surplus, producer surplus, tax revenue and deadweight loss; and draw them on a graph. Solution: Consumer and Producer Surplus is: CS = (5 − 2.6) × 48/2 = 57.6

P S = (1.6 − 0) × 48/2 = 38.4 Tax revenue is: T R = (2.6 − 1.6) × 48 = 48 Deadweight loss is: DW L = (2.6 − 1.6) × (60 − 48)/2 = 6

The sum of consumer surplus, producer surplus, tax revenue and deadweight loss is 57.6+38.4+48+6 = 150 (same as in part B)

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P & p D = 5 − D / 20 %$)+#( +(&"+)

-! * %))

p= . ,$+

(!!$"'+!"!(!+#

(%+( +(&"+)

p S = S / 30

p=

&



Q

 

(F) For each unit of cigarette consumed, a second hand smoker John is harmed. We describe the damage done to John by saying that smoking hurts him by an amount of $1 per cigarette. How much is John hurt without the tax? How much is he hurt with the tax? Solution: Compare the negative externality in two cases: Without the tax, John is hurt by $1 × 60 = $60. With the tax, John is hurt by $1 × 48 = $48. An imposed tax on cigarettes makes the second hand smoker somewhat better off by reducing the externality.

Q6. Equilibrium supply and taxation. [0 points] Note: this problem is optional, provided to you for further practice, and will not be graded. A solution will be posted along with the solutions to this problem set. Suppose the demand curve for flannel shirts is given by Qd (pd ) = 85 − 5pd and the supply of flannel shirts is inelastically supplied at a quantity of 5. (A) What is the equilibrium price and quantity of flannel shirts? Solution: In equilibrium, we must have quantity supplied equal to the quantity demanded. By setting these two equations equal to each other, we can solve for the equilibrium price and quantity. Note that in this case, since the supply is inelastic, the quantity demanded must also equal 5 in equilibrium. Plugging this back in, the price must be 16. 5 = 85 − 5p ⇒ p = 16

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Now suppose that the demand curve for flannel shirts is the same as before, but that the supply curve of flannel shirts is given by Qs (ps ) = 2.5ps − 20 (B) What is the new equilibrium price and quantity of flannel shirts? Solution: In equilibrium, we must have quantity supplied equal to the quantity demanded. By setting these two equations equal to each other, we can solve for the equilibrium price and quantity. 2.5p − 20 = 85 − 5p

⇒ p = ps = pd = 14 ⇒ q = qs = qd = 15

(C) What is the elasticity of supply with respect to price at the equilibrium? Hint: recall the general (x) x formula for elasticities is εx = ∂f∂x . f (x) Solution: Using the equation for elasticity, we can solve for the elasticity of supply with respect to price as: εs,p =

ps ∂qs ps · = 2.5 · 2.5ps − 20 ∂ps qs

Plugging in the equilibrium values we found above, we get: εs,p = 2.5

14 7 = ≈ 2.33 2.5 · 14 − 20 3

Finally, suppose that a tax of $1.20 per flannel shirt is imposed on consumers. (D) Write the equation that describes the relationship between the price paid by consumers, pd , and the price received by sellers, ps , as a function of the tax. Solution: pd = ps + t = ps + 1.2 (E) What is the new equilibrium quantity of flannel shirts? What is the new equilibrium price paid by consumers and what is the equilibrium price received by sellers? Solution: Again we set quantity supplied equal to quantity demanded, but now we use the price relationship from part (D). This gives us: 2.5ps − 20 = 85 − 5pd

2.5ps − 20 = 85 − 5(ps + 1.2) ⇒ ps = 13.2

⇒ pd = ps + 1.2 = 14.4 ⇒ qs = qd = 13

(F) What is the tax revenue on flannel shirts collected by the government? Solution: The tax revenue is 13 × 1.2 = 15.6 9...