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Production Function
The production function identifies the maximum quantity of good y that can be produced from any input bundle (z1, z2).
Production function is stated as: y=F(z1, z2).
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Production Functions
In a fixed proportions production function, the ratio in which the inputs are used never varies.
In a variable proportion production function, the ratio of inputs can vary.
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© 2005 Pearson Education Canada Inc.
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F(z1z2)=(1200z1z2)1/2
This is a Cobb-Douglas production function. The general form is given below where A, u and v are positive constants.
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Costs
Opportunity cost is the value of the highest forsaken alternative.
Sunk costs are costs that, once incurred, cannot be recovered.
Avoidable costs are costs that need not be incurred (can be avoided).
Fixed costs do not vary with output.
Variable costs change with output.
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Long-Run Cost Minimization
The goal is to choose quantities of inputs z1 and z2 that minimize total costs subject to being able to produce y units of output.
That is:
Minimize w1z1+w2z2 (w1,w2 are input prices).
Choosing z1 and z2 subject to the constraint y=F(z1, z2).
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Production: One Variable Input
Total production function TP (z1) (Z2 fixed at 105) defined as:
TP (z1)=F(z1, 105)
Marginal product MP(z1)the rate of output change when the variable input changes (given fixed amounts of all other inputs).
MP (z1)=slope of TP (z1)
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© 2005 Pearson Education Canada Inc.
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Diminishing Marginal Productivity
As the quantity of the variable input is increased (all other input quantities being fixed), at some point the rate of increase in total output will begin to decline.
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Figure 6.4 From total product to marginal product: another illustration
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Average Product
Average product (AP) of the variable input equals total output divided by the quantity of the variable input.
AP(Z1)=TP(Z1)/Z1
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average product
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When MP is less than AP, AP declines.
When MP=AP, AP is constant.
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The cost-minimization problem is:
Subject to constraint y=TP(z1).
The variable cost, VC(y) function is:
VC(y)=the minimum variable cost of producing y units of output.
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© 2005 Pearson Education Canada Inc.
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More Costs
Average variable cost is variable cost per unit of output. AV(y)=VC(y)/y
Short-run marginal cost is the rate at which costs increase in the short-run. SMC(y)=slope of VC(y)
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Figure 6.8 Deriving average variable cost and short-run marginal cost
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When SMC is below AVC, AVC decreases as y increases.
When SMC is equal to AVC, AVC is constant (its slope is zero).
When SMC is above AVC, AVC increases as y increases.
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AVC (y’)=w1/AP(z1’)
The average variable cost function is the inverted image of the average product function.
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SMC (y’)=(w1Δz1)/(MP(z’))
The short-run marginal cost function is the inverted image of the marginal product function.
© 2005 Pearson Education Canada Inc.
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© 2005 Pearson Education Canada Inc.
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© 2005 Pearson Education Canada Inc.
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