Elasticity of scale from cost function

economics

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. ???

Best Answer

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.

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