Ch 13 SR CMP

# Its short run cmp is noting that the

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Unformatted text preview: [{ the production function { } ⏟ } implies that: By the way, we know that . ____________________________________________________________________ ______________________________ 3. The “ECO 100 Optimal Labor Rule” Before we solve the short run CMPs for specific production functions, we can use the general short run CMP to derive the ECO 100 rule for the optimal amount of a variable input such as labor2. ECO 100 argues that the optimal amount of labor is when: Marginal Benefit of an additional unit of labor = Marginal Cost of an additional unit of labor Let price of a unit if labor so that: ⏟ The marginal benefit of another unit of labor is the “value of the output produced by this additional unit of labor”: 2 In the lecture slides, we used the Cobb-Douglas CMP to derive the ECO 100 optimal labor rule. Here, we derive this rule for a general CMP. 11 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ⏟ The value of the output produced by this additional unit of labor is the output produced by this additional unit of labor times the price of the output: ⏟ The output produced by this additional unit of labor is in fact the “marginal product of labor” price of the output by . Denoting we get: Can we get this condition from the general short run CMP? Sort of. Consider the following variable general short run CMP with ( ): ( { { Assume that to produce so that: ( } } [ the firm must use labor, i.e. ( ) ( ⏟ ) ⏟ (perhaps it uses a Cobb-Douglas production function) { Look at the FOCS3 (remember ) and fixed } [ ( ) ( ) )): ( ) ⏟ [ [ ( ) Now, by the envelope theorem we know that: 3 Marginal product of labor 12 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ( That is, Marginal cost ) . As such, the 1st FOC becomes: That is: Price of Labor = Marginal Cost optimal labor: Marginal Product of Labor. This equation almost looks like the ECO 100 rule for Comparing with we see that ECO 100 assumes that which is true if the output is sold in a competitive market. That is, the ECO 100 rule is a consequence of the optimal amount of labor where labor must be used for production and the output is sold in a competitive market. If the output is not sold in a competitive market (such as a monopoly) then the optimal rule is: Note to self: For next version, derive expressions comparing this FOC in competition vs. monopoly. 4. Cobb-Douglas Short Run Cost Minimization Problem (1 Variable input and 1 Fixed input) Having seen how to setup, solve and apply the envelope theorem to the general short run CMP we now consider short run CMPs for specific production functions, starting with the with variable and fixed Cobb-Douglas production 4 function where tells us the “returns” to the variable input labor : ⏟ { } { } [ ⏟ As we saw earlier: As such, the short run Cobb-Douglas CMP becomes: 4 Do not confuse “returns” with “returns to scale” 13 ECO 204 Chapter 13: The Short Run Cost Minimization Problem (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. { } [ Where by the envelope theor...
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