# Afc1000 Assignments

Class Summary | Assignments | Sample Tests | Announcements

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1/8: Introduction and coverage of Ch.1; Math review of functional relationships and calculus

1/15: Completed Ch. 2 with emphasis on application involving marginal analysis; Began Ch. 3; Handout and discussion of advertsing problem

1/22: Completed Ch. 2 problems; covered up to price discrimination (p. 101) in Ch. 3

1/29: Completed optimal pricing and price discrimination in Ch. 3; covered up to returns to scale (p. 208) in Ch. 6.

2/5: Review for test, Test 1

2/12: Review tests; completed; Ch 6; began Ch. 7 through definition of short-run cost function (assignments for Ch. 7 shown below)

2/19: Finished Ch. 7.

3/5: Reviewed Chap 7 problems; Completed Ch. 10 (assignments below);

3/12: Reviewed Ch. 10 problems (on Web page); completed Ch. 11

3/19: Will handle any requests on material, chapter problems or sample test; Test 2

3/26: Reviewed Test 2; Finished Ch. 11; Ch. 12 to p. 453

4/2: Plan to finish 12 and 13 (assignments below)

4/9: Finished Chs. 13 and 8;14 (to p. 543)

4/16: Finished Ch. 14; discussed final, handled questions

TEST 1 ANSWERS

1a. TR=75Q-O.O5Q^{2}, MR=75-0.1Q

b. Q=750, P=37.5 (set MR=0)

d. Set =TR-TC=0 and solve the quadratic: Q=0 or 487

e. Q=243.3, P=62.8 (set MR=MC)

f. =8833

g. The 2^{nd} derivative of the profit function<0

h. The firm will sell to everyone willing to pay a price of 50 or higher. At P=50, Q=500. Thus the triangular area under the demand curve and above a price of 50=(0.5)(500)(75-50)=6250.

2a. Q=177,000, TR=619,500

b. E_{P}=-0.16, E_{Po}=-0.11.

c. Raise P, must be in elastic portion of demand.

d. Arc E_{P}=-0.2

3a. MRP_{L}=Price times MP_{L}=100/L^{0.5}

b. Set MRP_{L}=P_{L}. L=6.25,Q=50.

4a. Find MP's and show that they decrease as S's increases.

b. Set MP_{A}=MP_{B} and also S_{A}+S_{B}=30. Thus S_{A}=13.33,S_{B}=16.67.

c. Using formula for optimal price, P=20,000 assuming E_{P} remains at -5.

5a. -7.5%, b. 12%, c. 13.3%

TEST 2 ANSWERS

1a. Set MRTS_{LK=}=P_{L}P_{K} and solve for L=0.6K

b. TC=1000=40L+20K, Substitute for L=0.6K. Thus K=22.7, L=13.7 and Q=114

- The production function has increasing returns to scale, resulting in economies of scale (decreasing LAC)
- At P=80, Qd=8400. Because each firm produces 10 (from part c), N=840

2a. AFC=1000/Q, AVC=50-4Q+0.2Q^{2}, SAC=1000/Q+50-4Q+0.2Q^{2}, MC=50-8Q+0.6Q^{2}

b. Set the derivative of AVC=0, Q=10

c. At Q=10, AVC=30

d. The AVC and MC are U-shaped; the supply curve is the MC above the minimum point of the AVC

3a. Set P=MC, Q=15

b. TR-TC=250

c. Set the derivative of LAC=0, Q=10 and LAC=P=80

4a. MR=MC Q=950 and P=310

b. TR-TC=180,500

c. Pc=MC=120 and Qc=1900. Efficiency loss=0.5(950-120)(1900-950)=90,250

5a. Set Qd=Qs Q=112 and P=22

b. With P=22, firs are losing money. Some firms will exit reducing supply and raising price until it hits $27.

c. The supply curve shifts up by $5. The price paid by buyers increases (<5) and the net price received by sellers is reduced below 22. Quantity decreases.

6a. Set MR=LMC Q=21 and P=46

b. Set P=LAC. This yields a quadratic with 2 solutions for Q (0 and 41.7). Select the higher output and P=4.6.

## ASSIGNMENTS and PRESENTATIONS

ASSIGNMENTS

Core coverage (continued)

Chapter 7: A key chapter on costs that extends the material in Ch. 6. Probs 1-9,11-13,15.

Chapter 10: Perfect competition--the first of three important chapters on market structure. Probs 1-10

Chapter 11: Pure monopoly including cartels, natural monopoly and monopolistic competition: Probs 1-12

Chapter 12: Oligopolies including several models of behavior; a little bit on other dimensions of competition:

Probs 2-6

Chapter 13: Game theory to p. 492: Probs 3-8

Chapter 8: Decision-making under uncertainty, pp. 283-293 and 300-310 (Probs. 2-6,8)

Chapter 14: Regulation, public goods & benefit-cost to p. 552: (Probs. 2.3,8,12)

DUE DATES and PRESENTATIONS

1/29, Ch. 2: Sp-Z (Paul Sp, Barbara W, Larry S on 2/5)

2/5, Ch. 3: M-So (Andrew M, John R, David S on 2/12)

2/12, Ch. 3 or 6: H-L (Greg H, Doyle L on 2/19)

2/19: Ch. 6 or 7: C-G (Denyes K, Tricia G on 2/19

3/5: Ch. 7 or 10: A-B (John A, Lisa B, Edward B)

3/12: Ch. 10: Sp-Z (Catherine W, John W)

3/19: Test 2 (no presentations)

3/26: Ch. 11: M-So (Mark N, Becky S)

4/2: Ch. 11/12: H-L (Ken H, Tim L)

4/9: Ch. 12/13: C-G (Tim C, Andrew D)

4/16: Ch. 13/8: A-B (Craig A, John A)

Optional Permission: I authorize M. Stano to publish my work, in its original or edited form, for nonprofit, educational purposes. (Sign/Date/telephone number)

Chapter 2

Chapter 3

Chapter 6

Chapter 7

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 8

## SAMPLE TESTS

SAMPLE TEST 1 (From Winter 1996)

Value: 60 points--30% of grade. __All work must be shown to receive credit.__

1. Quopy Quat specializes in printing business cards and resumes using the latest laser technology. The manager estimates weekly demand and cost as:

Q = 25,000 - 1000P and TC = 5,000 + 13Q + 0.002Q^{2}.

a. Determine the firm's revenue maximizing price and output. (3)

b. Determine the profit function and marginal profit function. (2)

c. Determine the profit-maximizing price, output and profit level. (4)

d. Determine the price elasticity of demand at the optimal price. (2)

e. A former employee decides to sue, alleging discrimination. Management agrees to settle out of court by paying the former employee $10,000 a month for the next year. Explain what, if any, impact will this settlement have on Quopy Quat's price and output? (2)

**2. Night Timers, a small company manufacturing glow-in-the-dark products, has developed an adhesive tape that can be applied to walls and floors. The total annual cost function is TC = 50,000 + 0.25Q where Q is rolls per year.**

a. If management is considering a price of $0.65 per roll, determine the break-even quantity. (2)

b. Suppose that annual demand is estimated as Q = 350,000 - 200,000P. Determine the optimal price and quantity. (4)

c. If capacity is 140,000 rolls per year, what price should the firm set? (2)

d. If the firm can be a perfect price discriminator (ignore the capacity limit), determine the maximum profit potential. (2)

3. Nearby College estimates the demand for its MBA program as:

Q = 5,000 - 0.5P + 0.1Y + 0.2P_{c}

where Q is the number of students, P is tuition, Y is national GDP in billions, and P_{c} is the tuition charged by a competitor a few miles away.

a. Determine enrollment if P = $4000, Y = $5000, P_{c} = $6000. (2)

b. Determine the price elasticity of demand and the cross-price elasticity of demand. (4)

c. What advice do you have for Nearby's administrators. (2)

4. Big State charges instate students $2000 a term and outstate students $4500 a term. The respective demands are:

Q_{I} = 25,000 - 3P_{I} and Q_{O} = 20,000 - 1.5P_{O}.

a. If the marginal cost per student is $3000, determine Big State's optimal tuition structure (assuming no capacity limits). (4)

b. How will your answer change if Big State's capacity is 10,000 students? (3)

5. Q = 5K^{0.5}L^{0.5}

a. Explain whether the production function has increasing, decreasing or constant returns to scale. (2)

b. Fill in the following table assuming that K = 16. (3)

__L____Q____AP___{L}__MP___{L}

**1 ___ ___ ___**

**2 ___ ___ ___**

**3 ___ ___ ___**

**c. Determine the equations for the marginal product of labor and average product of labor. (3)**

d. Explain whether the production function is consistent with the Law of Diminishing Returns to labor. (2)

e. If capital (machines) is fixed at 16, but labor can be varied, determine the optimal amounts of labor if labor costs $10 per unit the price of the good is $25. (3)

f. Find MRTS_{LK} when L = 2 and K = 16. (3)

**6. Suppose that a firm's price elasticity of demand is -1.5 and its income elasticity is 2.**

a. Explain what will happen to the firm's revenues if it raises its price? (2)

b. If unit cost is constant at $20, determine the optimal price. (2)

c. If incomes are expected to rise by 5%, determine the impact on quantity demanded. (2)

Answers

1 a. Q = 12,500; P =12.5

**b. Profit = -5,000 + 12Q -0.003Q*2; Marg Profit + 12 - 0.006Q**

**c. Q = 2,000, P = 23, Profit = 7,000**

**d. Ep = -11.5**

**e. None--doesn't affect MC or MR, profit reduced by 10,000/month (2,500/week)**

2a. Q_{B} = 125,000

**b. Q=150,000, P = 1**

**c. Q=140,000, P=1.05**

**d. Maximum profit = 225,000 (i.e., triangular area above P=0.25)**

3a. Q=4,700

**b. Ep = -0.43; Epo = 0.26**

**c. raise P (currently in inelastic portion)**

4a. Q_{I}=7996, P_{I}=5670; Q_{o}=7752,P_{o}=8162

**b. Set MR**

_{I}=MR_{o}and Q_{I}+Q_{o}=10,000. Thus Q_{I}=4167,P_{I}=6945; Q_{o}=5833,P_{o}=94425a. Constant

**b. Q: 20, 28.3,34.6;AP:20,14.2,11.5; MP:--,8.3,6.3; MP (calculus):10,7.1,5.8**

**c. AP=20/L^0.5; MP=10/L^0.5**

**d. Yes--MP diminishes**

**e. Need MRPL=PL. Thus L=625**

**f. (not responsible for this test)**

6a. Decreases

**b. P = 60**

**c. 10%**

**More Detailed ANSWERS **- Updated Feb. 2 in Word 6.0 format

SAMPLE TEST 2 (from Winter 1996)

Expanded Answers

ANSWERS

1a. MRTS_{LK}=3K/L; L=1.5K (by setting 3K/L=P_{L}/P_{K}=2)

b. TC=588 (Set Q=100=10L^{0.75}K^{0.25} and substitute the optimal mix (L=1.5K). Solve for K and L (7.35 and 11.03) and substitute into isocost equation.

- Constant unit cost.
- If P=5, then Q=2,000. Does MR=MC when Q=2,000? No (MR=1 and MC=3). Thus reduce output and raise price.
- Need dP/dQ=d(LAC)/dQ. Thus Q=1,000 and P=AC=4.

2a. AVC=50+10Q; Shut-down price = 50.

b. Q=1.25 (set P=MC)

c. Q=-2.5+0.05P for al P50

3a. LAC=500 (set dLAC/dQ=0 and solve)

b. Since P<LACmin, there will be exit, a decrease in supply, rising prices until P=500 is attained.

c. N=72,000/1000=72

4a. P=10, Q=40

b. Q=4 (set P=MC), =0

c. The supply curve shifts up by $2. Thus Q_{s}'=5P-20. Set Q_{s}'=Q_{D }and solve (P=11.4,Q=37.2)

5a. Q=800,P=6,000 (set MR=MC)

b. None. No affect on MC or MR.

c. Pc=2,000 (the unit cost), thus Qc=1,600. A monopoly will set MC (2000)=MR. Thus Qm=800 and Pm=6000. The welfare loss triangle is 1,600,000.

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## Low-Cost Precursors to Novel Hydrogen Storage Materials

Suzanne W. Linehan ; Arthur A. Chin ; Nathan T. Allen ; ...

From 2005 to 2010, The Dow Chemical Company (formerly Rohm and Haas Company) was a member of the Department of Energy Center of Excellence on Chemical Hydrogen Storage, which conducted research to identify and develop chemical hydrogen storage materials having the potential to achieve DOE performance targets established for on-board vehicular application. In collaboration with Center co-leads Los Alamos National Laboratory (LANL) and Pacific Northwest National Laboratory (PNNL), and other Center partners, Dow's efforts were directed towards defining and evaluating novel chemistries for producing chemical hydrides and processes for spent fuel regeneration. In Phase 1 of this project, emphasis was placed on sodium borohydride (NaBH{sub 4}), long considered a strong candidate for hydrogen storage because of its high hydrogen storage capacity, well characterized hydrogen release chemistry, safety, and functionality. Various chemical pathways for regenerating NaBH{sub 4} from spent sodium borate solution were investigated, with the objective of meeting the 2010/2015 DOE targets ofmore » $2-3/gal gasoline equivalent at the pump ($$2-3/kg H{sub 2}) for on-board hydrogen storage systems and an overall 60% energy efficiency. With the September 2007 No-Go decision for NaBH{sub 4} as an on-board hydrogen storage medium, focus was shifted to ammonia borane (AB) for on-board hydrogen storage and delivery. However, NaBH{sub 4} is a key building block to most boron-based fuels, and the ability to produce NaBH{sub 4} in an energy-efficient, cost-effective, and environmentally sound manner is critical to the viability of AB, as well as many leading materials under consideration by the Metal Hydride Center of Excellence. Therefore, in Phase 2, research continued towards identifying and developing a single low-cost NaBH4 synthetic route for cost-efficient AB first fill, and conducting baseline cost estimates for first fill and regenerated AB using a variety of synthetic routes. This project utilized an engineering-guided R&D approach, which involved the rapid down-selection of a large number of options (chemical pathways to NaBH{sub 4}) to a smaller, more manageable number. The research began by conducting an extensive review of the technical and patent literature to identify all possible options. The down-selection was based on evaluation of the options against a set of metrics, and to a large extent occurred before experimentation was initiated. Given the vast amount of literature and patents that has evolved over the years, this approach helped to focus efforts and resources on the options with the highest technical and commercial probability of success. Additionally, a detailed engineering analysis methodology was developed for conducting the cost and energy-efficiency calculations. The methodology utilized a number of inputs and tools (Aspen PEA{trademark}, FCHTool, and H2A). The down-selection of chemical pathways to NaBH{sub 4} identified three options that were subsequently pursued experimentally. Metal reduction of borate was investigated in Dow's laboratories, research on electrochemical routes to NaBH{sub 4} was conducted at Pennsylvania State University, and Idaho National Laboratory researchers examined various carbothermal routes for producing NaBH{sub 4} from borate. The electrochemical and carbothermal studies did not yield sufficiently positive results. However, NaBH{sub 4} was produced in high yields and purities by an aluminum-based metal reduction pathway. Solid-solid reactive milling, slurry milling, and solution-phase approaches to metal reduction were investigated, and while both reactive milling and solution-phase routes point to fully recyclable processes, the scale-up of reactive milling processes to produce NaBH{sub 4} is expected to be difficult. Alternatively, a low-cost solution-phase approach to NaBH{sub 4} has been identified that is based on conventional process unit operations and should be amenable to scale-up. Numerous advances in AB synthesis have been made in recent years to improve AB yields and purities. Process analysis of several leading routes to AB (Purdue's formate-based metathesis route and PNNL's NH{sub 4}BH{sub 4}-based route) indicated the cost to produce first-fill AB to be on the order of $$9-10/kg AB, assuming a NaBH{sub 4} cost of $5/kg for a 10,000 metric tons/year sized AB plant. The analysis showed that the dominant cost component for producing first-fill AB is the cost of the NaBH4 raw material. At this AB cost and assuming 2.5 moles hydrogen released per mole of AB, it may be possible to meet DOE's 2010 storage system cost target, but the 2015 target will likely require lower cost AB and demonstrates the importance of having a low-cost route to NaBH{sub 4}. Substantial progress has also been made to define feasible pathways for the regeneration of spent ammonia borane fuel.« less

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