Exam1_Sample - Exam 1 Sample Bin Xie March 3 2015 Multiple...

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Unformatted text preview: Exam 1 Sample Bin Xie March 3, 2015 Multiple Choice Questions 1. The non-satiation rule that "More is better" results in indierence curves: (a) sloping down. (b) not intersecting. (c) reecting greater preferences the further they are from the origin. (d) All of the above. • Answer: D 2. Suppose sugar and articial sweeteners are perfect substitutes for Sam. Her indierence curves for those two goods would be: (a) downward-sloping and convex to the origin. (b) downward-sloping and straight lines. (c) L-shaped. (d) downward-sloping and concave to the origin. • Answer: B 3. √ If the utility function (U ) between food (F ) and clothing (C ) can be represented as U = 6) C · F , the marginal rate of substitution of clothing for food equals: (a) −C/F . (b) −F/C . (c) − C/F . (d) − F/C . • Answer: A (M RS = − dC M UF C =− =− . dF M UC F ) 4. If the consumer's income increases while the prices of both goods remain unchanged, what will happen to the budget line? 1 (a) The budget line rotates inward from the intercept on the horizontal axis. (b) The budget line rotates outward from the intercept on the vertical axis. (c) The budget line shifts inward without a change in slope. (d) The budget line shifts outward without a change in slope. • Answer: D 5. Jane's utility function is represented as: U = F C , F is quantity of food and C is quantity of clothing. If her budget constraint is represented as: 120 = 2F + C , her optimal bundle of consumption (F, C) should be: (a) (40,40) (b) (30,60) (c) (45,30) (d) (50,20) 0.5 • 0.5 Answer: B 6. After Joyce and Larry purchased their rst house, they made additional home improvements in response to increases in income. After a while, their income rose so much that they could aord a larger home. Once they realized they would be moving, they reduced the amount of home improvements. Their Engel curve for home improvements on their current home is: (a) negatively sloped. (b) at. (c) positively sloped. (d) backward bending. • Answer: D 7. To separate the income and substitute eects, the imaginary budget line should be: (a) tangent to the new indierence curve and parallel to the new budget line. (b) tangent to the new indierence curve and parallel to the old budget line. (c) tangent to the old indierence curve and parallel to the new budget line. (d) tangent to the old indierence curve and parallel to the old budget line. • Answer: C 8. A consumer has the quasi-linear utility function U (q , q ) = 8q + q . If there is an increase in the price of good 1, the income eect of the quantity change of good 1 is: (a) positive. (b) negative. 1 2 2 2 1 2 (c) zero. (d) indeterminate. • Answer: C (Derive the demand function of good 1 through utility maximization. dq In the demand function there is no Y , so 1 = 0 meaning zero income eect.) dY 9. If the income elasticity of potatoes is -0.7, then the income eect caused by a price decrease of potatoes (a) tends to increase the consumption of potatoes. (b) tends to decrease the consumption of potatoes. (c) is less than the substitution eect. (d) None of the above. • Answer: B (Negative income elasticity indicates a negative income eect. Lower price - Higher real income - Decreased demand in terms of income eect) 10. In the relevant price range a demand curve for a Gien good would be: (a) upward sloping. (b) downward sloping. (c) horizontal. (d) vertical. • Answer: A (Income eect dominates substitution eect for a Gien good.) 11. The compensation variation and equivalent variation will be closer to each other when (a) the income elasticity is greater. (b) the budget share is greater. (c) the price change is smaller. (d) the income elasticity is smaller. • Answer: D 12. Suppose a victim of an accident brings the injurer to court. You are hired to determine the amount of damages. You are specically asked to nd a measure of the amount of money needed to restore the victim to the position he was in prior to the accident. What welfare measure will provide the most accurate measure of this amount? (a) compensating variation (b) equivalent variation (c) consumer surplus (d) the loss of utility 3 • Answer: A (By denition.) 13. A tax cut that raises the after-tax wage rate will most likely result in more hours worked if: (a) tax rates were low already. (b) the relevant portion of the labor supply curve is upward sloping. (c) the relevant portion of the labor supply curve is downward sloping. (d) workers can be easily fooled. • Answer: B (A tax cut essentially means a higher after-tax wage.) 14. In response to an increase in the wage rate, the income eect will usually cause a person to (a) supply fewer hours of labor. (b) supply more hours of labor. (c) supply the same hours of labor. (d) have a horizontal labor supply curve. • Answer: A (The leisure is assumed to be a normal good all along here. This question is not very precise and in the exam I will specify whether leisure is a inferior/normal good in similar questions.) Short-Answer Questions 1. (Market Equilibrium and Tax) The California cigarette market consists of the following supply and demand curves: Q = 150 − 20p, Q = 40p where Q is the number of packs of cigarettes per year, and p is the price per pack. (a) Compute the market equilibrium price and quantity. (b) Calculate the price elasticities of the demand and supply curves at the equilibrium price/quantity. (c) California imposes a tax on cigarettes of $0.90 per pack. Suppliers pay this tax to the government. Compute the after-tax price and quantity. How much do suppliers receive net of tax (per pack)? (d) Demand for cigarettes is generally more elastic over longer periods of time as consumers have more time to kick the habit. What does this imply about the tax incidence in the long run as compared to the short run? D • S Solution: (a) Set the supply and demand equal: 150 − 20p = 40p Solving for p: p = 2.5, Q = 100 So the price is $2.50 per pack and 100 million packs of cigarettes sold. (b) The slope of the demand equation is: dQ /dp = −20 The elasticity of demand is therefore ε = −20(2.5/100) = −0.5 The slope of the supply equation is: dQ /dp = 40 The elasticity of supply is then: ε = 40(2.5/100) = 1.0 ∗ ∗ D D S S 4 (c) The supply with the tax becomes:Q = 40(p0.90) The new equilibrium is at where 40(p − 0.90) = 150 − 20p. The equilibrium price is: p∗ = 3.10 and the equilibrium quantity is: Q∗ = 150 − 20(3.10) = 88. The price sellers earn net of tax (per pack) is 3.10 − 0.90 = 2.20. (d) The more elastic the demand curve is, the less of the burden falls on consumers. So over a longer period of time, the burden on consumers ($0.60) will move towards the suppliers. Price will fall from $3.10 as consumers quit smoking. (No numerical solution required in this question.) 2. √ (Demand Functions) Consider a consumer with the Cobb-Douglas utility function U (q , q ) = q q , where q and q are the quantities of goods 1 and 2 consumed, respectively. This consumer derives a level of utility denoted by U . The prices of goods 1 and 2 are denoted p and p . (a) Find out the demand functions of good 1 q = q (p , p , Y ) and good 2 q = q (p , p , Y ). (b) Derive the consumer's expenditure function, E(p , p , U ). (You can either use indirect utility function or solve the expenditure minimization problem.) ¯ (c) Derive the compensated expenditure function of good 1 H (p , p , U ). (d) Initially, Y = 100, p = 1 and p = 1. Calculate the total eect, substitution eect and income eect in terms of quantity change if the p increases to 5. S 1 1 2 1 2 2 0 1 2 1 1 1 1 2 2 2 1 2 0 1 1 2 1 2 2 1 • Solution: (a) q = D = 0.5Y and q = D = 0.5Y . p p (b) The indirect utility function is U (p , p , Y ) = ( 0.5Y ) ( 0.5Y ) . After the transforp p ¯ mation (E = Y ), the expenditure function is E = 2U √p p . ∂E ¯ (c) The compensated demand function could be derived: q = H (p , p , U ) = ∂p = ¯ p using Shephard Lemma. U p (d) Total100 (using uncompensated demand function): ∆D = D(p , p , Y )−D(p , p , Y ) = eect 0.5 · 100 0.5 · − = −40. 5 1 Substitution eect (using compensated demand function): ∆H = H(p , p , U ) − √ 1 1 H(p , p , U ) = 50 − 50 = 10 5 − 50. Utility level is calculated using indi5 1 rect utility function: U = U (1, 1, 100) = 50. Income √ Total eect - Substitution eect ∆D − ∆H = −40 − (10√5 − 50) = eect: 10 − 10 5. 3. (Government Policy: Subsidies) Consider Jen, a consumer with preferences U (H, F ) = F H , where H is the quantity of housing and F is the quantity of food (per month). Suppose Jen has a stipend of $600/month which she uses to purchase food at a price of $1/unit and housing at a price of $10/unit. 1 1 2 2 1 2 1 0.5 2 0.5 1 2 1 2 1 1 1 2 1 2 1 ∗ 1 2 1 ∗ 1 1 2 0 0 1/3 0 2/3 5 2 2 0 (a) Compute Jen's utility-maximizing(optimal) bundle of goods. (b) Suppose that Jen's employer subsidizes housing by paying 60% of her total housing costs, thereby eectively lowering the price Jen pays for housing to $4/unit. Compute Jen's new optimal consumption bundle. (c) How much does Jen's employer pay in total for this subsidy? How much utility does Jen enjoy with this subsidy (compute her utility at the optimal bundle). (d) Suppose that her employer simply gave Jen the dollar cost you found in (c) as a lump sum (instead of subsidizing housing). Will Jen gain a higher utility from the housing subsidy or the lump-sum equivalent transfer? (compute her utility at the optimal bundle when she receives a lump sum-transfer) • Solution: (a) L = F H +λ(600−10H −F ). Solve the F.O.C: H = 400 = 40 and F = 200 = 200. 10 1 (b) The unit price subsidy eectively lower the price of housing by 60% from 10 to 4. New optimal bundle with 200 price subsidy: L = F H + λ(600 − 4H − F ). unit 400 H= = 100 and F = = 200. 4 1 (c) The subsidy paid by the employer is the price dierence between the real price and the subsidized price (∆p = 10 − 4) multiplied by the units a consumer purchases H = 100: Subsidy = (10 − 4) × 100 = 600. Level of utility U = 200 100 = 125.99. (d) A lump sum subsidy is equivalent to an increase in income: Y = Y + 600 = 1200 L=F H + λ(1200 − 10H − F ). Solution: H = 80 and F = 400. New Level of Utility when Jen receives a lump sum subsidy: U = 80 400 = 233.92. Higher than the utility level when Jen receives a unit price subsidy. 4. (Labor Supply) Suppose a person's utility for leisure (N) and consumption (Y) can be expressed as U = Y + N . The total time constraint is 24 and there is no unearned income. (a) Write out the budget constraint determining feasible allocations of leisure and consumption. (b) Compute the optimal bundle of leisure and optimal bundle of consumption. (c) Derive the consumer's labor supply function: H (w). (d) Show what happens to the person's labor supply curve when the income tax is cut from 70 % to 30 %. Denote hours worked as H and wage per hour as w. 1/3 2/3 1/3 2/3 ∗ 1/3 1/3 2/3 2/3 1/3 0.5 ∗ • Solution: (a) H + N = 24, Y = Hw, the constraint is Y + wN = 24w. (b) Lagrangian Expression: L = Y + N + λ(24w − Y − wN ). 0.5 −0.5 M RS = − M UN 0.5N =− M UY 1 = M RT = − pN = −w pY 1 1 Solve the rst order conditions: N (w) = 4w , Y = 24w − 4w . 2 6 2/3 1 (c) The labor supply function is H (w) = 24 − N (w) = 24 − 4w (d) If there is an income tax (rate) τ , Lagrangian expression becomes: L = Y + N + λ(24w(1 − τ ) − Y − w(1 − τ )N ). The solution becomes: N (w) = 4(1 −1τ ) w . 1 H (w) = 24 − N (w) = 24 − . 4(1 − τ ) w A easier way to think of this without solving the utility maximization problem again could be to simply replace w with (1 − τ )w in the labor supply function, because the invididual only makes choice based on the after-tax wage ω = (1 − τ )w one can get. Based on the labor supply function, when the income tax τ decreases, the hours worked H increase. ∗ 2 0.5 2 ∗ 2 ∗ 7 2 2 ...
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