Application of maxima
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This document provides 8 examples of applying the principles of maximizing and minimizing functions to determine optimal output levels, prices, and profits for firms producing single products, joint products, or substitute products. The examples involve solving systems of demand, marginal revenue, marginal cost, and budget constraint equations to find the profit-maximizing or utility-maximizing levels of output and prices for the products under different cost and demand conditions.
Application of maxima
- 1. Application of maxima & minima 1.If U = 35X + 95Y – 0.5X2 – 0.5Y2 and Px = Rs. 3,Py =Rs. 9 and I = Rs. 60 then find the consumption (in units) of commodities X and Y as also the numerical value of λ.. 2. If U = XY and Px = 5 and Py = 10 and I = 100, then find the consumption (in units) of commodities X and Y as also the value of λ. 3. Find the optimum commodities purchase for a consumer whose utility function and the budget constraint one U = x1.5 y and 3x + 4y = 100 respectively. 4. A tender sells hides and mutton which are jointly produced in fixed proportion. The marginal cost equation for mutton-hide-product-package is MC = 30 + 6Q.The demand and marginal revenue equations for products mutton and hides are given below. Mutton Hides P = 65 – 1Q P = 85 – 2Q MR = 65 – 2Q MR = 85 – 4Q Find out the prices to be charged for mutton and hides. Also find how many units of the product package should be manufactured. 5. Suppose Z & Co. produces X and Y that are substitutes in consumption. The demand functions of the two products were forecasted to be QX = 80 – 8PX + 6PY and QY = 40 – 4PY + 4PX where the outputs are measured in thousand units and the prices in rupees per unit. From the production manager’s estimates of incremental (marginal) cost functions were obtained. MCX = 10 +0.5 QX and MCY = 20 + 0.25 QY where output is again measured in thousand units. Find out the optimum mix of the two substitute products. 6. GB Chemicals Ltd. produces refined chemicals. In one of its divisions a joint product is produced . That is, as it refines the new chemical input , the processes will yield equal amounts of two products X and Y. The demand function for the two products had been forecasted as QX= 285 – 2PX and QY = 150- 2PY where the inputs are increased in thousands Kg. and prices in rupees per Kg. Given the existing capital stock the firm’s engineers have estimated the incremental (marginal) cost function for refining the raw material input to be MC = 10 +2Q where marginal cost is measured in rupees per Kg. and output (Q) is in thousand Kg. How much of the products X and Y should the firm produce and sell and at what prices? 7. Suppose that a firm produces two joint products in fixed ,one-to-one proportions. The demand functions for the two products have been estimated as QX= 110 – PX ;QY =10 – 0.25 PY and the incremental (marginal) cost function for the joint product has been estimated to be MC = 30 + 2Q. (i) What is the profit- maximizing level of production for the joint product? (ii) How much of each product will be sold? (iii) What are the profit-maximizing prices for the two products? 8. Assume that a firm raises sheep in order to produce two products meat and sheepskins. The firm produces sheepskins and meat according to a fixed-proportions process, with all costs shared jointly. Assume that the demand equation for meat is PM = 308 – 0.5 QM and the demand curve for sheepskins is PSS = 172 – 2QSS . The corresponding marginal revenue equations are MRM = 308- QM and MRSS= 172- 4QM . The marginal cost for raising sheep is MC = 20+ 5Q. Assume that one sheep produces one unit of meat and one sheepskin. Determine the profit maximizing quantities of sheepskins, meat and sheep and the prices of the meat and sheepskins. Suppose the marginal cost is suddenly changed to MC = 60 + 10Q. Now derive the same quantities and prices.