sol2 - Let G be the smaller number between 6 and...

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Unformatted text preview: Let G be the smaller number between 6 and 2.25+(achieved points)/8. Round G exactly to quarters of a grade to get you grade. Solution to Homework Set 2 Managerial Economics Fall 2011 Conceptual and Computational Questions 4 points 1. The X-Corporation produces a good (called X) that is a normal good. Its competitor, Y-Corp., makes a substitute good that it markets under the name "Y". Good Y is an inferior good. (a) How will the demand for good X change if consumer incomes increase? Since X is a normal good, an increase in income will lead to an increase in the demand for X (the demand curve for X will shift 1 point to the right). (b) How will the demand for good Y change if consumer incomes decrease? Since Y is an inferior good, a decrease in income will lead to an increase in the demand for good Y (the demand curve for Y will 1 point shift to the right). (c) How will the demand for good X change if the price of good Y decreases? Since goods X and Y are substitutes, a decrease in the price of good Y will lead to a decrease in the demand for good X (the 1 point demand curve for X will shift to the left). (d) Is good Y a lower-quality product than good X? Explain. No. The term inferior good does not mean inferior quality, it 1 point simply means that income and consumption are inversely related. 4 points 2. Good X is produced in a competitive market using input A. Explain what would happen to the supply of good X in each of the following situations: (a) The price of input A increases. The supply of good X will decrease (shift to the left). 1 point (b) An excise tax of $1 is imposed on good X. The supply of good X will decrease. More specifically, the supply 1 point curve will shift vertically up by exactly $1 at each level of output. 1 (c) An ad valorem tax of 5 percent is imposed on good X. The supply of good X will decrease. More specifically, the supply 1 point curve will rotate counter-clockwise. (d) A technological change reduces the cost of producing additional units of good X. 1 point The supply curve for good X will increase (shift to the right). 5 points 4. The demand for good X is given by Px Py M + - 8Pz + . 2 4 10 Research shows that the prices of related goods are given by Py = $5, 900 and Pz = $90, while the average income of individuals consuming this product is M = $55, 000. Qd = 1, 200 - x (a) Indicate whether goods Y and Z are substitutes or complements for good X. Good Y is a substitute for X, while good Z is a complement for 1 point X. (b) Is X an inferior or a normal good? X is a normal good. 1 point (c) How many units of good X will be purchased when Px = $4, 910? 1 point Qd = 1, 200 - $4,910 + $5,900 - 8 $90 + $55,000 = 5, 000. x 2 4 10 (d) Determine the demand function and inverse demand function for good X. Graph the demand curve for good X. For the given income and prices of other goods, the demand function for good X is Qd = 1, 200 - Px + $5,900 - 8 $90 + $55,000 , x 2 4 10 which simplifies to Qd = 7, 455 - 0.5Px . To find the inverse dex mand equation, solve for price to obtain Px = 14, 910 - 2Qd . The x demand function is graphed in Figure 2-2. the solution to d) consists of 3 items: -demand -inverse demand -graph 1-2 right items give 1 point, 3 right items give 2 points Figure 2-2. 2 4 points 7. Suppose demand and supply are given by Qd = 7 - x Px 2 and Qs = x Px 1 - . 4 2 (a) Determine the equilibrium price and quantity. Show the equilibrium graphically. Equate quantity demanded and quantity supplied to obtain 7 - Px Px 1 2 = 4 - 2 . Solve this equation for Px to obtain the equilibrium price of Px = 10. The equilibrium quantity is 2 units (since at 1 point the equilibrium price quantity demanded is Qd = 7 - 10 = 2). 2 The equilibrium is shown in Figure 2-3. 1 point Figure 2-3 (b) Suppose a $6 excise tax is imposed on the good. Determine the new equilibrium price and quantity. A $6 excise tax shifts the supply curve up by the amount of the tax. Mathematically, this means that the intercept of the inverse supply function increases by $6. Before the tax, the inverse supply function is P = 2 + 4Qs . After the tax the inverse supply function is P = 8 + 4Qs , and the after tax supply function (obtained by solving for Qs in terms of P ) is given by Qs = P - 2. 4 Equating quantity demanded to after-tax quantity supplied yields 7 - P = P - 2. Solving for P yields the new equilibrium price 2 4 of $12. Plugging this into the demand equation yields the new 1 point equilibrium quantity, which is 1 unit. (c) How much tax revenue does the government earn with the $6 tax? Since only one unit is sold after the tax and the tax rate is $6 per 1 point unit, total tax revenue is only $6. 3 Problems and Applications 5 points 18. From California to New York, legislative bodies across the United States are considering eliminating or reducing the surcharges that banks impose on noncustomers who make $10 million withdrawals from other banks' ATM machines. On average, noncustomers earn a wage of $20 per hour and pay ATM fees of $2.75 per transaction. It is estimated that banks would be willing to maintain services for 4 million transactions at $0.75 per transaction, while noncustomers would attempt to conduct 16 million transactions at that price. Estimates suggest that, for every 1 million gap between the desired and available transactions, a typical consumer will have tho spend an extra minute traveling to another machine to withdraw cash. Based on this information, use a graph to carefully illustrate the impact of legislation that would place a $0.75 cap on the fees banks can charge for noncustomer transactions. Figure 2-5 illustrates the relevant situation. The equilibrium price is $2.75, but the ceiling price is $0.75. Notice that, given the shortage of 12 million transactions caused by the ceiling price of $0.75, the average consumer spends an extra 12 minutes traveling to another ATM machine. Since the opportunity cost of time is $20 per hour, the nonpecuniary price of an ATM transaction is $4 (the $20 per hour wage times the fractional hour, 12/60, spent searching for another machine). Thus, the full economic price under the price ceiling is $4.75 per transaction. Figure 2-5 4 -demand and supply curves go through indicated points: 3 points -shortage is discussed or highlighted: 1 point -either nonpecuniary or economic price is discussed or highlighted: 1 point 0 points Price Ceilings and Price Floors Assume inverse demand and supply functions P = a - bQ and P = c + dQ. What are the equilibrium quantity Qeq and price Peq ? Compare your results with fields D12 and D13 in hw2.xlsx. PCF1 Sketch the situation in a graph and indicate consumer and producer surplus. 1 0 1 0 1 0 1 0 1 0 1 0 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 Consumer Surplus 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 Producer Surplus 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 11111111111111111111 00000000000000000000 1 0 1 0 1 0 1 0 P Supply Demand Q PCF2 Assume a price floor that is between a and Peq . Indicate in a separate sketch the new market price Pf and quantity Qf , consumer surplus, producer surplus, and the deadweight loss. Provide algebraic expressions for consumer surplus, producer surplus, and the deadweight loss as functions of a, b, c, d, Pf , Qf , Peq , Qeq . Use these expressions to complete the fields D40 to D42 of hw2.xlsx. What values do consumer surplus, producer surplus and deadweight loss take? 11111111111 00000000000 Supply 11111111111 00000000000 Surplus Consumer 11111111111 00000000000 11111111111 00000000000 1111111111111111111111111111111111111111111 0000000000000000000000000000000000000000000 1111111111 0000000000 11111111111 00000000000 Floor 1111111111 0000000000 11111111111 00000000000 1111111111 0000000000 11111111111 00000000000 1111111111 0000000000 11111111111 00000000000 Deadweight Loss 1111111111 0000000000 11111111111 00000000000 Producer Surplus 0000000000 1111111111 11111111111 00000000000 1111111111 0000000000 11111111111 00000000000 1111111111 0000000000 11111111111 00000000000 Demand 1111111111 0000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 11111111111 00000000000 See sol2.xls for algebraic expressions and numbers. 5 PCF3 Assume a price ceiling that is between c and Peq . Indicate in yet another sketch the new market price, the full economic price, consumer surplus and producer surplus. Provide algebraic expressions for the full economic price, consumer surplus, producer surplus, and deadweight loss. Use the expressions to complete the fields D26 to D29 of hw2.xlsx. What values do consumer surplus, producer surplus and deadweight loss take? P 111111111 000000000 111111111 000000000 Supply 111111111 000000000 111111111 000000000 PF 1 0 1111111111111 0000000000000 111111111 000000000 1 0 1111111111111 0000000000000 111111111 000000000 1 0 1111111111111 0000000000000 111111111 000000000 Deadweight 1 0 1111111111111 111111111 000000000 Consumer 0000000000000 1 0 1111111111111 0000000000000 111111111 000000000 Loss Surplus 1 0 1111111111111 0000000000000 111111111 000000000 1 0 1111111111111 0000000000000 111111111 000000000 1 0 1111111111111 0000000000000 111111111 000000000 Demand 1 0 1111111111111 0000000000000 111111111 000000000 1 0 1111111111111 0000000000000 11111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000 111111111 000000000 1 0 1111111111111 0000000000000 111111111 000000000 111111111 000000000 Producer Surplus 111111111 000000000 111111111 000000000 11111111111111111111111111111111111111111111111 00000000000000000000000000000000000000000000000 Q See sol2.xls for algebraic expressions and numbers. 15 points Activity Analysis When there are a discrete set of production technologies j, each characterized by a marginal costcj and a capacity kj , the supply curve becomes a step function corresponding to the sorted sequence of plant capacities. Consider a market in which the commodity is supplied by the following four technologies: j a b c d cj 2 5 7 10 kj 2 2 4 5 Let the inverse market demand be P = 10 - 3 Qs , where total market supply Qs is the sum of quantities qj supplied by the firms j: Qs = j qj . AA1 Draw the supply and demand curves. Indicate the market equilibrium. 6 P 10 8 6 P*=5 4 2 0 2 points 2 4 Q*=3 6 8 10 12 Q AA2 Formulate the problem of maximization of social surplus as a nonlinear program. 10 - P (qj ) Qs (qj ) 2 max P S + CS = qi j (P (qj ) - cj )qj + s.t. qj kj , 3 points 5 where P (qj ) = 10 - Qs (qj ) 3 Qs (qj ) = qj . j AA3 Solve the program in Excel or MATLAB. See activity analysis.xls and acticity analysis.m 10 points 5 points for a program that finds the intersection of supply and demand. 10 points for a program that maximizes social surplus if each firm can choose q_i, with 0 <=q_i<=k_i. 7 ...
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