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The Relationship between the long-run and short-run (average) cost curves Cen Hui, June 5, 2007(05Economics Department)
Background: In chapter 21, we have studied the relationship between relationship between the long-run and short-run (average) cost curves. Precisely speaking, we have learned the following two conclusions: 1. The long-run cost curve is the lower envelope of the short-run cost curves. 2. The long-run average cost curve is the lower envelope of the short-run average cost curves. Abstract: Here I will use a way which does not require too much mathematical tools to prove Conclusion One. (I think Conclusion Two can be proved in the similar method.) First I will divide Conclusion One into two aspects: Any one short-run cost curve must be tangent to the long-run cost curve at some one point. Any one short-run cost curve must be above the long-run cost curve.

Proof: If above two aspects can be proved, then the problem facing to us can be solved at the same time. Before I do it, I think it is necessary for us to review the process to attain the short-run and long-run cost functions. As I have told you, cost minimization problem is not our purpose, what we want to do is to analyze the behavior of the typical firm. In order to obtain the law for the behavior of the typical firm, we should get the cost function, which is defined as

the function which gives the minimum cost to produce the given any level of production. According to the definition of cost function and the definitions of long-run and short-run, we can solve for the short-run and long-run cost functions from different cost minimization problems.

For the short-run cost function,

min {ω x x1 1 1

+ ω 2 x2 ,


s.t. y = f x1 , x 2 s 1




⇒ x1 = x (ω1 , ω 2 , x 2 , y ), x 2 = x 2 → c s ( y, x 2 ) = ω1 x1s (ω1 , ω 2 , x 2 , y ) + ω 2 x 2

For the long-run cost function,

min {w x x1 , x2

1 1

+ w2 x2 }

s.t. y = f ( x1 , x2 ) ⇒
* * * * MP ( x1 , x2 ) MP2 ( x1 , x2 ) 1 = =λ w1 w2


* * * * → x1 = x1 ( w1 , w2 , y ), x2 = x2 ( w1 , w2 , y ) * * → c( y ) = w1 x1 ( w1 , w2 , y ) + w2 x2 ( w1 , w2 , y )

Note that the objective function is not the cost function required. Take the long-run minimization problem for example, when y = y * ,the minimum of the objective function g ( x x , x 2 ) = w1 x1 + w2 x 2 at point
* * * * * * ( x1 , x 2 ) is c( y * ) , where x1 = x1 ( y * ), x 2 = x 2 ( y * ) . Then we have * * * ∂g ( x1 , x 2 ) ∂C ( y*, x 2 ( y * )) = = 0 ,in accordance with the first order * * ∂x 2 ∂x 2

condition for the extreme. The above analysis gives us the relationship between the objective function and the cost function.

Now we begin to prove the first aspect. (Tangency is defined as the crossing case with the same slope.) For a specific short-run cost function, x 2 is fixed at some constant value x 2 . For
* the above two cost minimizing problems, when x 2 equals x 2 which is the

optimal choice in the long-run problem, these two problems have the same solution, such that the short-run and long-run cost functions give the same value at point y = y * . So there exists one point this short-run cost curve and

the long-run cost curve cross. At the same point, we always have c( y ) = c s ( y, x 2 ( y )) . (Remember that the right

side of the equation is denoting the long-run cost function by means of the form of short-run cost function, so the equation stands absolutely.)
Differentiating both sides with respect to y yields to

dc ∂Cs( y, x 2 ( y )) ∂Cs ( y, x 2 ( y )) dx 2 = + ⋅ dy dy ∂y ∂x 2
When y = y * , thus
* * * ∂g ( x1 , x 2 ) ∂C ( y*, x 2 ( y * )) = = 0. * * ∂x 2 ∂x 2


That is to say,

dc ∂Cs ( y, x 2 ( y )) holds for y = y * . At the crossing point, the = dy ∂y

slopes for the short-run and long-run cost curves are equal. The first aspect is true for any one short-run cost curve.

For the second aspect, we have known at the point y = y * , the two cost minimization problems have the same solution. But if

∀y = y ' , y ' ≠ y * , these two cost minimization problems will have different solutions, which are
* * * * x1 = x * ( y ' , x2 ( y * )), x2 = x2 ( y * ) = x2 , for the short-run; x
* * * * * x1 = x1 ( y ' , x 2 ( y ' )), x 2 = x 2 ( y ' ) , for the long-run.

Then we can attain that
* * * w1 x1 ( y ' , x 2 ) + w2 x 2 f w1 x1 ( y ' ) + w2 x 2 ( y ' ) .

This inequality holds by the property of cost minimization problem.
* That is to say C s ( y' , x2 ) = C s ( y ' , x2 ( y * )) f C ( y ' ), ∀y ' ≠ y * .

This specific short-run cost function is above the long-run cost function. The second aspect is true for the related one short-run cost curve.

Conclusion: Notice that what I have done above focuses on only one short-run cost function and the related long-run cost function, so you can apply the above

method to accomplish the proof for any one short-run cost function, and the Conclusion One stands.
Conclusion Two is left for your job!

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