Explanation Given: Demand function is D ( q ) = 27 − q 2 . Supply function is S ( q ) = 27 − q 2 + 1 2 q + 5 . Calculation: Consider the following demand function D ( q ) = 27 − q 2 ....... ( 1 ) And the following supply function, S ( q ) = 27 − q 2 + 1 2 q + 5....... ( 2 ) Compute the point of market equilibrium as follows, D ( q ) = S ( q ) 27 − q 2 = 1 2 q 2 + 1 2 q + 5 5 4 q 2 + 1 2 q − 22 = 0 5 q 2 + 2 q − 88 = 0 Simplify further, 5 q 2 − 22 q − 20 q − 88 = 0 q ( 5 q + 22 ) − 4 ( 5 q + 22 ) = 0 ( 5 q + 22 ) ( q − 4 ) = 0 q = − 22 5 , 4 As q cannot be negative, the point of equilibrium will be q 0 = 4 At sales level q 0 = 4 , the price will be, p 0 = D ( q 0 ) = 27 − 4 2 = 27 − 16 = 11 Use the formula for the Consumer’s surplus

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3.The demand curve of a firm is p=1200- 21q and its total cost is C(q)=2q3 – 66q2 +600q+1000 where q is the output of the firm (in thousands). Derive an expression R (q ) , for the firm' s revenue function. (ii) Derive an expression P( q) for the firm' s profit function. (iii) Is the rate of change of profit increasing or decreasing when the output level of the firm is 10,000 units? (iv) Determine the level of output at which profit is maximized.

6. In a certain firm, the total production cost function and a demand function are given by the equations TC = 0.16q³ -0.72q² +2.08q+1.46 and q=5 -0.5p. a. Is the demand function one-to-one? Explain your answer. If yes, find its inverse. b. Find the equation of the function expressing the profit P in terms of the quantity q. c. Is this profit function one-to-one? If yes, find its inverse. (Hint: use your graphing calculator.) d. Find q such that P = 6. (Hint: use your graphing calculator.)

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Chapter 6 Solutions

Calculus

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Suppose that in a certain market the demand function for a product is given by 10p + q = 1,900 and the supply function is given by 50p − q = 250. (Assume price is measured in dollars.) If the government levies a tax of $5 per item, find the equilibrium point after the tax is levied. The tax is added to the selling price of the product. (q, p) =
Q3. A firm faces a demand function of the form P = 90 – 2Q? (a) Plot the graph for their total revenue (b) If the marginal cost is given by MC(Q) = 3Q² – 8Q + 7 and TC(0) = 0, estimate the total profit if they produce 5 units (round up the answers to 2 decimals)
(4) A camera company estimates that the demand function for its new digital camera is p = 275e-0.159 and the supply function is p = 25e0.219. Here q is measured in thousands. (a) What are the price and level of sales when the market is in equilibrium? (Recall: equilibrium occurs when demand and supply are equal.) (b) Compute the consumer surplus when the market is in equilibrium. (c) Compute the producer surplus when the market is in equilibrium. (d) Compute the total surplus when the market is in equilibrium. (e) Graph the demand and supply curves. (You may use an electronic device for this if you wish.) Verify that the curves intersect at your answer to (a). Shade regions of the graph whose areas correspond to the consumer surplus and producer surplus, respectively.

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the consumer’s surplus at the point of market equilibrium.

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