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Unformatted text preview: put:
( ) ( ) Notice that: That is, doubling all inputs exactly doubles output. As such, we’d expect to be always constant since doubling inputs always doubles cost, and with constant RTS, doubles output, so that the ratio will be constant: 10
ECO 204 Chapter 11: Practice Problems & Solutions for Producer Theory – The Basics in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. AC Q (c) Derive ABC’s (longrun) cost function. Please show all calculations below and state any assumptions.
Answer:
This problem cannot be solved by the Lagrangian method since the production function is not differentiable everywhere.
Instead, note that at the optimal choice of inputs it must be that:
( ) And: This implies:
( ) And:
( ) That is: 11
ECO 204 Chapter 11: Practice Problems & Solutions for Producer Theory – The Basics in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. This makes sense: 100 passengers required 1 bus ( ) and two workers ( ) (11.9) A company uses labor and capital as complements to produce output according to the production function:
(
Here, is a parameter for technological progress and ) is a parameter. Assume . (a) What is this company’s “returns to scale”?
Answer:
We want to check when there are decreasing, constant or increasing returns to scale: strictly speaking we should be checking:
( ( ) ( ) ) ( ( ) ) ( ) Provided the company is using some labor and capital: (
Since ) (scaling up):
⇔ Alternatively by comparing the outputs returns to scale is presented as “compare the output with doubled inputs with
double the output with initial inputs”.
The initial output is:
( ) If all inputs are doubled, the company’s output is:
12
ECO 204 Chapter 11: Practice Problems & Solutions for Producer Theory – The Basics in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ( ) ( ) There are decreasing, constant or increasing returns to scale when: ( ( ) ( ) ) ( ( ) ( ) ( ) ( ) ) Assuming the company is producing output, this becomes: Hence, the company has : Decreasing returns to scale (
Constant returns to scale (
Increasing returns to scale ( ) if
) if
) if (b) Solve for the optimal (long run) labor and capital demands. Show all calculations.
Answer:
The CMP is:
( ) We cannot solve this by the Lagrangean method because you cannot differentiate the production function at the
“corner”. Use the fact that at the optimum, the company won’t waste inputs so that: Thus:
( ) ( ) 13
ECO 204 Chapter 11: Practice Problems & Solutions for Producer Theory – The Basics in ECO 204 (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. And since : (c) Derive the (long run) cost function.
Answer:
We had: The optimal cost of producing target...
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 Fall '14

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