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.
Edit 2: Here are also the suggested solutions.
And here are how to arrive at the solutions:
- Use Shephard's lemma
- Maximize profit and solve the first-order condition for y
- Insert the conditional demands and the supply curve into the profit maximization problem
- Use Hotelling's lemma
- ???
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.