Question

Suppose that Jennifer produces goody by using input xt and x2. The production function which Jennifer faces is: The cost for every unit of x; is wand the cost for every unit of x2 is w2. There is a fixed cost component F, which also forms a part of her total cost. (a) Write down the cost minimization problem. Solve this problem and express x:/X2 as a function of (b) Now that we've worked out the necessary condition for cost minimisation, use this condition to find the conditional demand for x; and x2. Once you obtain this, find how an increase in output y affects the conditional demand for x and x2. Furthermore, find how w: affects the conditional demand for X, and how w, affects the conditional demand for x. c) Now that we have found the conditional demand, substitute them back into the cost function. Find the Average Cost, Average Variable Cost, Average Fixed Cost and Marginal Cost functions assuming F=6, wi=2 and w2=4. For parts (d) and (c) assume F=6, wi=2 and wz=4. (d) 1) Sketch the AC, AVC, AFC and MC curves on the grid below which uses the cost output plane. Assume AC, AVC, AFC, MC e) Show that the MC curve intersects the AVC curve and the AC curve at the minimum point.

Answer 1

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WO . Production function. f(x, y) = y = dai + dan Cost function: (= wixit wzxz - F (a, cost minimisation problem, min C- Wix, & W, X2 + F subject to y = dai + das Forming Lagrange: L = wirrt W₂ X2 + f - x (de, & + dan - y) Taking partial derivatives: an m w . - Gate) = o parente paretise . . ) I derivatives of production function .. (21 22 da ti o constrant (3) +-yo-eustracist from!!!: Wi= - . - - from (2) W₂ - - - 22

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W2 wir - 9/2572 /250 I W₂ – x wi-x. 1 I Necessary condition for cost numinisation (6) Conditional input demands are: * ر رودا حلوا dent de if we and اره دعاره x + ( w. weigs and 21 w ways are the fienne conditional input demand functions. y de, der Janse w y Tas y sus vas ) 2 (دال + بالو)

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How increasing outputy will affect conditional demands: dag. 24 w12 demand for (Twit dus)² inputs will also increase = 2y wg 2 dy [wwit dus) 2 How w, affects : da2 dw, = ? * x2 - y2 W, Witw, + 2 w. wz dwi ng - (WitW2+2 w, W2) (242wi) - (1422) 42 W, 2 (WitW, + 2 w, wz) 2 da dwi = 224,2 + 2 y wiwg + 4yt w 2w2 - yew, 2 - 2 y 2w1w, (wit W2 + 2W, W212 y2 w12 + 4y2 w2wa (witwa + 2W, W2) 2 ] y2 W₂2 + 442 W, 2w1 (wit W₂ + 20,W2) 2 dai dal dul

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Total cost = wint w, x + F - Wi-2, w,-u, t-6. TC = 2 + 45, +6 ТС - 2 y w, 2 1чу 2 w, 2- То, 2- + Vu, ) - (Ca + Ju, ) - u 0 2 ( 2 )) 5 m) 2 (J2 + ч 22 ( 4) - / 6 2 18 тез у се счу 4 т е у дал а - - y ) , ТРЕС : Все это бу ду AVC = 842 MCZ 1 ч 9