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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 proﬁtmaximizing 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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 Spring '10
 DrCarlDillion

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