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Unformatted text preview: em we know that:
( ) i.e. as the cost of production will go up (or stay the same) The FOCS are5:
[⏟ [ We can solve for from the 2nd FOC which simply reiterates the constraint that: [
If the only variable the firm wanted to calculate was
take FOCs. In fact, if we are told that then there was no need to setup the Lagrangian function and then we could’ve solved for [ directly from the constraint. However, this “no Lagrangian equation” approach works only if there is one unknown variable input. For example, if we
are told that: We would have to do the Lagrangian equation because we can’t solve for two unknowns with one equation. Returning the short run CobbDouglas CMP we can solve for (which we know is ) by substituting [ into the 1st FOC: {[
5
} Marginal product of labor
14 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. { { {
{ } } } { {} } ( } ( ) {} {} By the envelope theorem we know that [ ) {} which means that we must get the same result by differentiating the short run cost function with respect to . Let’s check if this is true (and whether the envelope
theorem works): To find the optimal cost of producing an arbitrary output level
in terms of to get: we would substitute expressions for optimal [ and Here: [ Earlier, we claimed that:
[ Is this true? From the cost function:
[ [
Which confirms that . Returning the short run CobbDouglas cost function we can express it as:
15 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ⏟ ⏟ [ ⏟ We will now do a detailed analysis of the CobbDouglas short run CMP solutions: What’s the relationship of optimal inputs and cost with output?
What happens to inputs and cost if the firm gets “better” (in the sense that
)
What happens to inputs and cost if the price of labor increases?
What happens to inputs and cost if the price of capital increases?
What is the relationship of cost, average variable cost, average fixed cost, average cost, and marginal cost with
output? In fact, we will ask these questions for all types of CMPs but this is the only time that we’ll do such a detailed analysis.
You should make sure that you understand the methods used below. First remember that for the CobbDouglas
production function
: { Remember that returns to a variable input says something about what happens to output when that variable input is
scaled up by a factor
(for example, if we doubled labor). ● Notice that the demand for labor, [ is independent of the price of labor and capital. That is because in this one variable input and one fixed input CMP, there can be only one value for the labor required to produce
capital is fixed
: given that 16
ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Target
Output Fixed 0 Here we are literally “picking off” a point on the target isoquant curve. Having picke...
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 Fall '09
 AJAZHUSSAIN
 Economics, Microeconomics

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