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deriving cost functions from a production function aec 303

deriving cost functions from a production function aec 303...

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Unformatted text preview: - Deriving Cost Functions from Production Functions Suppose a production function is given as y =Axb where y = TPP =output and x = input. A and b are the parameters of the production function. A is generally always positive since a negative A would make the production function produce negative output quantities which is silly. For diminishing marginal product, MPP, b needs to be between zero and one. That is 0<b<l. If A is positive and b is between zero and one, MPP will always be declining, but at a diminishing rate MPP=dy/dx= b Axb'l. APP is the ratio of y/x. If we do the following division y/x= Axb/x, we discover that y/x = Axb'l MPP and APP look very similar on a graph. If 0<b<1, MPP will be below APP proportionate to the value of b. For example, if b = 0.5, then MPP will always be 1/2 of APP. The Duality of cost and production. Let us assume our production function is y =Axb . We can solve this production function for x as a function of y. This is called inverting a function. This function is quite easy to invert, but some functions cannot be inverted at all. To invert this function, first divide both sides of the function by A. That is y/A =Axb/A. Therefore xb = y/A. Then x = (y/A)(”b) To illustrate using a specific numbers, let us suppose A=10 and b= 1/3 which is approximately 0.3333. Then x”3 = y/10 and x = (y/10)3 This inverse function defines units of input as a fiJnCtion of the output level. This is not quite a cost function yet, but we can make it into a cost function by multiplying both sides of the equation by the input price. Let’s call the input price v. Then x = (y/10)3 Multiplying both sides of the equation by the input price (v) the cost function that is dual to our production function is vx = v(y/10)3 vx is actually what we have been calling Total Factor Cost. But, now that it is a function of output y not input x we could better call it Total Variable Cost. Since there are no fixed inputs in this example, it could also be Total Cost TVC=TC = v(y/10)3 Let’s make our input price to be $0.50 Then we could write TC = 0.50*(y/10)3 So long as 0<b<1 The exponent, in this example, 3, will be bigger than one. This means that if there is diminishing MPP there will be increasing Marginal Costs, and that the Law of Diminishing Returns can also be thought of as the law of rising Marginal Costs. If product (y) is rising at a diminishing rate with increased input use, then costs will be increasing at an increasing rate with respect to output. We can now differentiate our TC equation above with respect to output y not input x, and get our Marginal Cost function TC = 0.50*(y/10)3 TC = 0.50*(O.10)3y3 dTC/dy=MC = 3=I=0.50*0.001*y3'1 =3*0.50*0.001*yz Since y is raised to a power larger than 1 whenever b is between zero and one, Marginal Costs will be rising The classic rule in Economics is to find the profit maximizing output level at the point where Marginal Cost = Marginal Revenue MC :MR. We have just gotten through calculating MC for our production function. How do we get MR? Let us assume that the firm can sell as little or as much of its output y at the going market price we will call p. Further, let us make the output price equal to $4. Total revenue for the firm is simply price times output. That is TR=py In this example TR=$4*y Marginal revenue is the derivative of Total revenue with respect to output. Hence MR = dTR/dy =$4*y”= $4*y0 = $4, the product price So, equating MC and MR we get MC = $4 3*0.50*0.001*y2 = $4 This is easy to solve on a spreadsheet for y First, solve the equation for y2 y2 =$4/(3*0.50*0.001) That answer should be 2,666.67 Take the square root of that number and you have y, the profit—maximizing output level! The correct answer for the square root of 2,666.67 is 51.63978. Hence, for this example, the profit-maximizing level of y where MC=MR is 51.63978 using our numbers. Since the MR curve is horizontal and the MC curve is rising, that output level is where MC intersects MR from below. ...
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