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I have a problem that I just dont seem to figure out how to solve.

Find the elasticity of scale when the following cost function is given:

3(n)^(2/3)(m)^(1/3)y^(4/3)

Where n and m are input prices of the two input factors and y is output.

The definition of elasticity of scale is:

E = dY/dS * S/Y where S is the scale parameter.

But we dont have a production function so how am I supposed to get there?

I can solve for the conditional input demands, the supply function and the profit function. But how in the world do I get E?

The elasticity of cost w.r.t. output should be 4/3 if we take dC/dY * Y/C but this doesnt help much.

The given solution I have is that E = 3/4. But how?

Much appreciated!

Edit: Here is the entire problem. Im stuck at number 5. The answer is supposed to be 3/4. enter image description here

Edit 2: Here are also the suggested solutions. enter image description here

And here are how to arrive at the solutions:

  1. Use Shephard's lemma
  2. Maximize profit and solve the first-order condition for y
  3. Insert the conditional demands and the supply curve into the profit maximization problem
  4. Use Hotelling's lemma
  5. ???
Johanna W
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  • Please provide more context. What is the problem word by word? Also don't forget to accept nice answers. It shows that you appreciate the people, who take the time to answer your questions. – callculus42 Oct 06 '22 at 18:51
  • What do you mean by accept nice answers? Have I forgot to upvote some answers I have got before? It always says that I need at least 15 points to be able to vote. So I have not been able to do that. There is no more context to the problem. The cost function is given. Calculate the elasticity of scale. Thanks anyway! – Johanna W Oct 06 '22 at 19:34
  • You can accept an answer by clicking on the check mark $\color{limegreen}{\checkmark}$. – callculus42 Oct 06 '22 at 20:12
  • Maybe you can uoload the exercise. – callculus42 Oct 06 '22 at 20:12
  • Oh, I see. Ive now accepted the best answers from my previous questions. I have also uploaded the problem above. – Johanna W Oct 06 '22 at 20:56
  • Great. I've thought that your exercise has something to do with the marshallian demand. But it doesn't lead me in the right direction. So, I don't have a useful idea. – callculus42 Oct 06 '22 at 21:00
  • Allright, thanks for trying. – Johanna W Oct 06 '22 at 21:01
  • And thanks for the upload. I will have a look at the exercise in a few hours. – callculus42 Oct 06 '22 at 21:07
  • If you would happen to make another try, I have now uploaded the suggested solutions above. – Johanna W Oct 06 '22 at 22:59

1 Answers1

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I have found the answer.

Elasticity of Scale is the inverse of the cost elasticity of output. So that

1/(dC/dY * Y/C) = dY/dC * C/Y = dY/dS * S/Y

where S is the scale parameter. Thus when inputs are scaled up by s, then so is cost, as the cost function is homogeneous of degree 1 in prices.

Thus 1/(4/3) = 3/4, which is the correct solution.

Johanna W
  • 45
  • It seems right, what you have done. You can mark your own answer as accepted $\left( \color{limegreen}{\checkmark} \right)$. – callculus42 Oct 07 '22 at 17:55