Supply and Demand and Marginal Cost
Demand function for air travel between the U. S. and Europe has been estimated to be: ln Q = 2. 737 – 1. 247 ln P +1. 905 ln I where Q denotes number of passengers (in thousands) per year, P the (average) ticket price and I the U. S. national income. Determine the price elasticity and income elasticity of demand (8 points). From Lecture Module 3 Equation 4 we learned the alternative formulation of elasticity. Alternative formulation of elasticity EP = dQ/dP * P/Q = dlnQ/dlnP Natural log: ln, uses the base “e” How? ?lnQ/? lnP =(d lnQ/dQ) * (dQ/dP) * (dP/dlnP) [ Note: dY/dX = 1/(dX/dY) since, dlnX/dX = 1/X, dX/dlnX = X]
Example: Q = AP-? A:Constant>0 lnQ=lnA + ln(P-? ) =lnA – ? lnP EP = dlnQ/dlnP = -? ? =? lnQ/? lnP ? =P/Q* (? Q/? K) = Elasticity The coefficients of double log model are the corresponding elasticity Price elasticity = -1. 247 Income elasticity = 1. 905 (1ii) It has been estimated that the price elasticity of demand for U. S. manufactured automobiles is -1. 2, while the income elasticity of demand is 2. 0 and the cross price elasticity of demand with respect to the foreign automobiles is 1. 5. The current volume of sales for U. S. manufactured automobiles is 10 million per year. It is expected that over the next year
Supply and Demand and Marginal Cost Essay Example
the average income of the consumers in the U. S. will increase by 2. 5 percent. It has been determined that the price of the foreign imports will increase by 6% over the next year. By how much should the U. S. automakers adjust the price of their automobiles if they wish to increase the volume of their sales by 9. 2% next year (8 points)? Price elasticity = -1. 2 Income elasticity = 2 Cross price elasticity = 1. 5 Current volume = 10 million 2. 5% Average income increase We know from Module 3 created by Dr. Ghosh that: EP = %? Qx / %? Px, where only Px changes %? Qx = EP * %? Px if only Px changes
Exy = %? Qx / %? Py, where only Py changes %? Qx = Exy * %? Py, if only Py changes EI = %? Qx / %? I, where only Income changes %? Qx = * EI %? I, if only Income changes Total % ? in Demand, %? Qx = EP * %? Px % + Exy * %? Py + EI * %? I We want the new Total % ? in Demand (%? Qx) = 9. 2%. Applying this highlighted formula we can calculate the % change in price for US automobiles to obtain a 9. 2% increase in demand: 9. 2 = -1. 2*%? Px(what we want to solve for) + 1. 5*6 + 2*2. 5 X=4% (1iii) Bright Future, Ltd (BF) is a nonprofit foundation providing medical treatment to emotionally distressed children.
BF has hired you as a business consultant to design an employment policy that would be consistent with its goal of providing the maximum possible service given its limited financial resources. You have determined that the service (Z) provided by BF is a function of its medical staff input (M) and sound service input (S) which is given by: Z = M + . 5S + . 5 MS – S2 BF’s staff budget for the coming year is $1,200,000. Annual employment costs are $30,000 for each social service staff member (S) and $60,000 for each medical staff member (M). (1iiia) Using the Lagrangean multiplier approach calculate the optimal (i.
e. , service maximizing) combination of medical and social staff. Determine the optimal amount of service provided by BF (32 points). Objective function: Z = M + . 5S + . 5 MS – S2 Constraint: 30000S+60000M = 1200000 Lagrangian: L = M + . 5S + . 5 MS – S2+? [1200000-30000S-60000M] First we need to calculate the first order conditions from the objective function: dL/dS = 0 implies 0. 5+0. 5M-2S = 30000? or 0. 5+0. 5M-2S-30000? =0 dL/dM = 0 implies 1+0. 5S=60000? or 1+0. 5S-60000? =0 dL/d? = 0 implies 30000S+60000M = 1,200,000 or 30000S+60000M-1,200,000=0 Next we need to equate dL/dS = dL/dM 60000*(0.
5+0. 5M-2S) = 30000*(1+0. 5S) 30000+30000*M-1,200,000S = 30000+15000S M = 4. 5S Substituting this value into the last equation of dL/d? 30000S+60000*4. 5S = 1,200,000 S= 4 = Social Staff M* = 4. 5*4 = 18 = Medical Staff Z = M + . 5S + . 5 MS – S2 Z= 18+0. 5*4+0. 5*18*4-42 = 40 = Optimal amount of service (1iiib) Calculate BF’s marginal cost. Explain your answer. Dr. Ghosh…I couldn’t find much in our text regarding the interpretation of this in relation to MC. I did find that the multiplier should equate to the marginal cost from research online and inferred from the definition of the multiplier
itself in your notes. Applying this, I believe the marginal cost would be calculated as follows: BF’s Marginal Cost = The reciprocal of Lagrangian multiplier Using equation calculated in problem above: ? = 1+0. 5S/60000 = (1+0. 5*4)/60000 = 0. 00005 = 1/0. 00005 = $20000 = BF’s Marginal Cost (1ciii) Using Excel-Solver verify your answer to (a). (Show your work. Show the spreadsheets in detail. Provide print outs with Solver window. To print the solver window, use print screen command on your key board and then create a MS Word document using paste. (12 points) Snapshot: Clickable Excel spreadsheet detailing work:
(1iv) The own price elasticity of demand for a pack of cigarettes is estimated to be -. 4. Current price and consumption are $4. 00 and 2 million units per year. Assuming a linear demand relationship determine the demand equation for cigarettes. Show all your calculations (12 points). From Module 3 created by Dr. Ghosh we can determine the following: If we believe that the demand equation can be captured by a straight line graph, then we can proceed as follows to estimate a linear demand function. Let the demand equation be described by the following linear equation, Q=a – bP where a>0, b>0 Our task is to estimate the parameters a and b.
From the definition of EP we can write, EP = (? Qx/? Px) * Px/Qx Current Price = P = $4 Current quantity sold = 2,000,000 Ep= -. 4 From the definition of Ep we can write, EP = (? Qx/? Px) * Px/Qx = -. 4 = -b*4/2,000,000 b = 200,000 Substituting in the equation for the demand function we obtain, Q= a – 200,000P Using Q = 2,000,000 and P = 4 2,000,000 = a- 200,000*(4) a = 2,800,000 Thus the demand equation can be determined to be Q = 2,800,000 – 200,000P (2i) The production function for a firm is given by Q = LK where Q denotes output; Land K labor and capital inputs. Wage rate and rental rate are given by w and r respectively.
(2ia) Show whether or not the above productions function exhibits diminishing marginal productivity of labor (8 points). We know from our test and from the notes by Dr. Ghosh that: Q = LK MPL= ? Q/? L = K MPL does not fall with respect to L and the second order derivative did not give a negative value thus does not reflect diminishing marginal productivity. (2ib) Determine the nature of the Return to Scale as exhibited by the above production function (12 points). We know from module 4 created by Dr. Ghosh that: Cobb-Douglas Production Function: Q = AL? K? , A > 0, ? > 0, ? > 0 if ? + ? = 1, CRS if ? + ? > 1, IRS if ? + ?
< 1, DRS Note if A =1 and ? = ? = 1, we have the production function, Q = LK As such the production function results in increasing Returns to Scale as ? + ? > 1 and 1+ 1 > 1. (2ic) Using the Lagrangean Multiplier method, calculate the least cost combinations of labor and capital and the resulting long run total cost function for the above production function. Explain the economic significance of the Lagrangean Multiplier and calculate its value (32 points). Q = LK TC = wL + rK L = wL + rK + ? (Q-LK) ?L/? L = w + ? (K) =0 therefore w = ? k ?L/? K = r + ? (L) =0 therefore r = ? L ?L/?? = 0 therefore Q=LK As such K = (w/r)L
Q=L2(w/r) and least cost value of L =(rQ/w)1/2 K = (wQ/r)1/2 least cost value of K Lagrangean multiplier: ? = (wr/Q)1/2 The Lagrange multiplier allows us to treat constrained problems as though they were unconstrained. From this we can determine the cost function as: C(Q) = wL + rK =w*[(rQ/w)1/2)] + r*[(wQ/r)1/2)] = 2vwrQ (long run total cost function) (2id) Using the above cost function, calculate the numerical value of long run total cost when Q =224, w = 16 and r = 144. Using our cost function above we can determine that: 2*(16*144*224)1/2 = 1,436. 796 (2ie) Using Excel- Solver verify your answer to (d) above (12 points).
The snapshot of the solver window minimizing the cost function is: The clickable excel workbook showing my work is: (ii) As the manager of an 80-unit motel you know that all units are occupied when you charge $60 a day per unit. Each occupied room costs $4 for service and maintenance a day. You have also observed that for every x dollars increase in the daily rate above $60, there are x units vacant. Determine the daily price that you should charge in order to maximize profit (12 points). From learning module 2 created by Dr. Ghosh we know the following: Profit = Total Revenue – Total Cost ? = TR – TC
Then we calculate the first order condition, and calculate optimal P and Q for resulting profit. Note: ? TC/? Q = dTC/dQ = Marginal Cost ?TR/? Q = dTR/dQ = Marginal Revenue From these concepts we can apply this to the problem: Fully occupied profit = 80*60 -80*4 = 4480 Price per unit = 60 + x Number of units occupied = 80 – x Cost = 4*(80-x) Revenue = P*Q= (60+x)*(80-x) Profit = TR – TC Therefore profit = (80-x) (60+x) – 4*(80-x) = (80-x) (56+x) = 4480 + 24 x – x2 In order to maximize profit we obtain the derivative of 4480 + 24 x – x2 as 24 – 2x and set it equal to 0. dQ/dY = 24 – 2x 24 – 2x = 0 2 x = 24 x = 12
Substitute the value of X into the price equation to arrive at: Price = 60 + 12 = 72 (3i) Sleak Teak builds yard furniture using hardwoods and (in a smaller shop) handcrafted knick-knacks from the same sort of wood. The hardwood usage in the two lines of product are Yard Furniture: Y = 2 Ty – . 001Ty2 Knick-knacks: Z = 20Tz – . 01 Tz2 where Y and Z are units of furniture and knick-knacks respectively. Ty and Tz are the amounts of hardwood used in Y and Z production respectively. Yard furniture can be sold at a profit (i. e. , net of costs) of $100 per unit and knick-knacks can be sold at a profit of $25 per unit.
Sleak Teak has 1300 units of hardwood available that can be allocated between these two lines of production. (3ia) Using the Lagrangean Multiplier method, determine how should the hardwood be allocated between the two lines of product so that total profit can be maximized. Also calculate the optimal amounts of Y and Z and total profit from each product line (32 points). We are provided the following information: Yard Furniture: Y = 2Ty – . 001Ty2 Knick Knacks:Z= 20Tz – . 01 Tz2 We need to obtain the derivatives of each of these functions: Y’ (derivative) = 2 -. 002Ty Marginal Profit = is 100(2-. 002Ty)
Z'(derivative) = 20 -. 02Tz Marginal Profit = 25(20-. 02Tz) Next we need to set the equations equal to each other: 100(2 – . 002Ty) = 25(20 – . 02Tz) 200 – . 2Ty = 500 – . 5Tz .5Tk – . 2Ty = 300 We know the constraint as it is given in the equation as Tz+Ty= 1300 5Tz -2Ty= 3000 2Tz +2Ty= 2600 7Tz = 5600 Tz = 800 Ty = 1300-800=500 Production should be allocated to 500 for Ty and 800 for Tz Substituting these values into the given functions we can obtain the optimal amounts of Y and Z: Yard Furniture: Y = 2Ty – . 001Ty2 Yard Furniture: Y = 2(500) – . 001(500)2 Yard Furniture: Y = 750 Knick Knacks:Z= 20Tz – . 01 Tz2
Knick Knacks:Z= 20(800) – . 01 (800)2 Knick Knacks:Z= 9,600 Profit can be calculated as = ((100*500) + (25*800)) = 315,000 (3ib) Using Excel Solver verify your answers to (a) above. (12 Points- Show work. ) A screen shot of the solver parameters can be seen below. The clickable excel workbook can be found below for equations: (3ii) You manage Dirt Diggers, an excavating firm that excavates roadside ditches for laying drainpipes. Its output follows the production function: Q = 10L -. 1L2 where L denotes labor hours and Q the length of the ditch in meters. You can hire labor at the going wage rate of $12 per hour.
As the manager of DD you want to earn as high a profit as possible. (3iia) You have received an offer to excavate 250 meters for a lump sum payment of $500. Should you accept the offer? Explain with appropriate calculations (12 points). According to Module 4 created by Dr. Ghosh: Incremental measure of productivity or Marginal Product of Labor MPL = ? Q/? L = ? Q/? L = fL = slope of total product curve in the short run when K is fixed. Application: Optimal Business Decisions: Hiring of workers Consider the short production function: Q=f(L,K) Price = P TR = P*Q = P*f(L,K) = TR(L) TC = W*L +F, where F is the fixed cost and W the wage rate
Focus on how many workers you need to use to maximize ? = TR-TC = PQ – WL = Pf(L,K)-W*L FOC for Profit Maximization: d? /dL=0 = PfL – W = 0 Marginal Revenue Productivity = Marginal Expenditure on Labor (or marginal input cost) (This is nothing but MR = MC in terms of labor and not output) Additionally from the text we know that MRPL = (MR)*(MPL) in equation 5. 2 The text goes on to detail that “to maximize profit, the firm should increase the amount of a variable input up to the point at which the input’s marginal revenue product equals its marginal cost. ” Therefore equation 5. 3 in the text states MRPL = MCL
Applying these concepts to the problem: First you can determine that the initial offer is at a Price of $2 per meter as 500/250=$2 P=$2 (current offer of $500 for 250 meters) We know that the firm maximizes profit when it equates MRPL = (MR)*(MPL) =MCL MPL = dQ/dL = 10 – . 2L. Therefore, ($2)(10 – . 2L) = 12 and L=20. In turn, Q = 10L -. 1L2 and as such Q = (10)(20) – . 1(202) and Q=160 Since the optimal quantity of meters dug at a price of $2 is 160 meters, Ditch Diggers should not accept the offer of $250 meters for $500. This is not the profit maximizing quantity at this price- reject the offer.
You can also approach this problem from another angle. Specifically, since we are provided with the production function of Q = 10L -. 1L2 and we know the offer is to dig 250 meters, we can determine the labor and cost required to achieve this result at 250 = Q = 10L -. 1L2 resulting in L of 50. Since the Marginal Cost of labor is $12 we can multiply this by the L of 50 to arrive at a cost of $600 to perform the job, while the offer is only $500 for completion. From this perspective we would still reject the offer as the revenue is less than the cost to perform $500 (revenue) < $600 (Cost).
(3iib) Suppose that instead of the previous offer, you are now offered as much or as little excavation work at a price of $2. 00 per meter dug. Should you accept the offer? Why or why not? If you accept the offer calculate DD’s resulting profit. Also, calculate the optimal level of output (meter dug) and the level of labor usage (16 points). From our calculation above we already know the following: We know that the firm maximizes profit when it equates MRPL = (MR)*(MPL) =MCL MPL = dQ/dL = 10 – . 2L. Therefore, ($2)(10 – . 2L) = 12 and L=20 (level of labor usage). In turn, Q = 10L -. 1L2 and as such Q = (10)(20) – .
1(202) and Q=160 (optimal output, meters dug) The optimal quantity of meters dug at a price of $2 is 160 meters at a level of labor usage of 20. The firms profit is = PQ – (MC)L= $2(160) – $12(20) = $80 Therefore, Ditch Diggers should accept the offer and will receive a resulting profit of $80. (3iii) As a manager of a firm you find the marginal cost of the firm to be $10 and the fixed cost $100. For the range of prices that you are planning to charge, own price elasticity of demand is believed to be –1. 5. Calculate the optimal (profit maximizing) price that you should charge. Show all calculations (8 points).
In order to solve this we need to apply the markup rule. Specifically the size of a firm’s markup (above marginal cost and expressed as a percentage of price) depends inversely on the price elasticity of demand for a good or service. According to equation 3. 13 from the text, the optimal markup rule is: P = Ep/(1+Ep)MC And from Module 3 created by Dr. Ghosh we know: Optimal Pricing (for profit maximization): Inverse Elasticity Rule Rule for profit max: Produce output up to a point so that MR = MC MR = MC P[1 – 1 / | EP |]=MC MC/P = 1-1/| EP | If adding 1 1 – MC/P = 1-[1-1/| EP |] (P-MC)/P = 1-1+1 / | EP |
(P-MC)/P (Price-Cost Margin) = 1 / | EP |: Inverse Elasticity Rule The lower the value of elasticity, the higher the price-cost margin For a competitive firm, |E|=infinity, therefore (P-MC)/P = 0 (rule for profit maximization, P=MC) Additionally we know from the text that “the logic of marginal analysis in general and the markup rule (equation 3. 13) show that optimal price and quantity depend on marginal cost (MC). Fixed Costs have no effect on the choice of optimal price and quantity. ” Therefore for this problem we do not consider the $100 fixed costs detailed. As such P = Ep/(1+Ep)MC = P = (-1. 5)/(1-1. 5)10 = $30