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Suppose the joint cost function of a firm producing two products X and Y is given by C = 250X2 + 120Y2. Assuming that output of the two products is restricted at 4144. Using Lagrangian multiplier technique find the amounts of X and Y that will minimize cost and compute this cost. Also examine the cost implications of changing this optimal combination so as to produce equal amounts of both products.
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- Kusho Industries produces and sells computer chips. Its (hourly) production function is Q=4K 0.4L 0.6 while its (hourly) cost function is C=20L+80K. Furthermore, Kusho must produce q0=400 computer chips per hour. a. Which levels of L and K satisfy the first-order conditions for the constrained minimisation of Kusho’s cost? Use the Lagrange Multiplier (LM) method. Also, find and interpret the value of the Lagrangemultiplier b. Show that MRTS=w at the constrained cost minimising levels of L and K obtained aboveCreate a full optimization problem with constraints and employ the Lagrange multiplier technique. One on the consumer side with constraints (utility maximization) and one on the producer side (either production / cost minimization , and or profit maximization.You are the manager of a firm and you are required to optimize the Cobb Douglas function given the following parameters. The maximum amount of money available is $1600 where the price of K = 12 and the price of L=6. That is PK=12 and PL=6. The function is given as q=K0.4+L0.6. What are the optimal values of K0 and L0?
- The demand equation for a product for a product brand of mackerel is given by the equation Qx=50-7Px+0.002I+12Py Where Qx=monthly consumption per family in cartons Px=price per carton of makerel=GHC 2.00 I=median annual family income=GHC20,000 Py=price per carton of competing brand of mackerel=2.5 Required: i.At the stated values of the explanatory variables,calculate the consumption of makerel(cartons). ii.what is the relationship between Good x and Good y? iii.what is the nature of Good x ? iv.Suppose the median annual family income increases to GHC40,000,What will be your new consumptino for Good x? v.Suppose that the price per carton of competing brand increases GHC3.00, What will be your new consumption for Good x?This question illustrates the argument on p.114 of the book. A and B own neighboring properties. Beneath their properties is a common well that contains 200 units of oil. The cost to A of extracting oil from the well in period t depends on the number of units of oil in the well at the beginning of the period t, ut, and the number of units of oil A extracts in period t, xAt ; specifically, the average cost of extraction for A per unit in period t is xAt /ut. The analogous cost function for B is xBt/ut. The market price of a barrel of oil is 1, there are two periods (t = 1, 2), and the discount rate is zero. The oil is a common property resource. a) (2) Suppose that A and B "unitize" and cooperatively decide how much oil to extract, and split the profit between them. The jointly profit-maximizing policy is that each extracts 50 units of oil from her well in each of the first two periods, after which the well is dry. How much discounted profit will A and B each make? b) (2) Suppose…You are the manager of a firm and you are required to optimize the Cobb-Douglas function given the following parameters. The maximum amount of money available to spend is $340 where the price of K=8 and the price of L=4. That is Pk=8 and Pl=4. The function is given as q=K0.4L0.6 . What are the optimal values K0 and L0 ? a. None of the above b. K0≈68,L0≈34 c. K0≈72,L0≈18 d. K0≈34,L0≈68
- Two power plants provide power to all of Manila: Mapayapa plant and Hagonoy plant. Both power plants burn coal to produce electricity, and consequently produce smog as a by-product. The M power plant could reduce its smog, but at a total cost: CM(XM) = 5X2M where XM indicates the total number of units of smog abated by M. The H plant is slightly less efficient, and its total cost for cutting down on smog by XH is: CH(XH) = 7X2H + 10XH Suppose that instead of taxation, the Manila government tries to regulate quantities. However, the city of Manila cannot write a law for each firm, so it simply declares that all power plants must cut down on smog by xC1 units each year. Explain why this is not efficient.You are the manager of Taurus Technologies, and your sole competitor is Spyder Technologies. The two firms’ products are viewed as identical by consumers. The relevant total cost functions are C(Qi)=2(Qi)for each firm, and the inverse market demand curve for this unique product is given by P = 50 – Q, with Q the total quantity. Currently, you and your rival simultaneously (but independently) make production decisions. b. Suppose by making an unrecoverable fixed investment of $40, Taurus Technologies can bring its product to market before Spyder finalizes its production plans. If so, what is your quantity and profit? Should you make the investment of $40? a. What is the current optimal quantity and profit of Taurus Technologies?Q). Shown below are two alternative Cobb-Douglas production functions where Q is output per day, L is labor hours per day, and K is machine hours per day. Q = 2LK Production Method 1 Q = 10(L1/2K1/2) Production Method 2 A firm must adopt one of the two production methods in order to produce 200 units of output per day while paying employees $12 per hour and renting machinery for $48 per hour. (a) Determine the cost minimizing combination of labor and capital if the firm decides to adopt production method 1. (b) Determine the cost minimizing combination of labor and capital if the firm decides to adopt production method 2. (c) Based on your solution in part (a) and (b), which production method would you recommend to the firm? Support your recommendation with calculations.
- A company produces two products in (continuous) quantities q1 and q2. At the current production levels, the total revenue is equal to 240 monetary units. Marginal revenue with respect to q1 is 2.1 monetary units and marginal revenue with respect to q2 is 1.5 monetary units. An increase of q1 by 1 unit costs 1.5 monetary units; an increase of q2 by 1 unit costs 1 monetary units. The company will increase its budget by 10 monetary units. It intends to spend the extra budget either on a production increase of the first product only or a production increase of the second product only. Which is most profitable and what extra revenue can the company expect?After an analysis of a large number of small businesses with two to nine employees, it was determined that, in a certain market sector, the operating costs C, in thousands of dollars, could be modeled by the function(pic#2)...., where p is the number of employees working for the firm. On the other hand, the realized revenue, R, of a firm could be determined as a function of the operating costs C, where R=(pic#1)... R and C are expressed in thousands of dollars. a) Based on the analysis, what would be the operating costs for a business with three employees? b) What would be the revenue for a company with three employees? c) Determine the equation that would model the realized revenue, R, as a function of the number of employees.It is known that a certain company sells each kg of the product it manufactures at $80, it is also known that the total manufacturing cost "CT" is given by the function CT=(1/1000)x2 +100 , where "x" are the kg of product produced. a) How many units of "x" must the company sell to break even? Value = 100 points.b) How many units of "x" is the optimal quantity that should be sold to optimize the producer's profit? Value = 100 points.c) How many monetary units does that optimal profit for producers amount to? (remember, you will have to prove it mathematically either by the method of the first or by the method of the second derivative). Value = 100 points.
