Chapter 6 production and cost one variable input
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Chapter 6 Production and Cost: One Variable Input. Production Function. The production function identifies the maximum quantity of good y that can be produced from any input bundle (z 1 , z 2 ). Production function is stated as: y=F(z 1 , z 2 ). Production Functions.

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Chapter 6 Production and Cost: One Variable Input

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Chapter 6Production and Cost: One Variable Input


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).


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.


Figure 6.1 Finding a production function


From Figure 6.1

  • The production function is:

    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.


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.


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).


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)


Figure 6.3 From total product to marginal product


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.


Figure 6.4 From total product to marginal product: another illustration


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


Figure 6.5 From total product toaverage product


Figure 6.6 Comparing the average and marginal product functions


Marginal and Average Product

  • When MP exceeds AP, AP is increasing.

  • When MP is less than AP, AP declines.

  • When MP=AP, AP is constant.


Costs of Production: One Variable Input

  • The cost-minimization problem is:

    Minimize W1Z1 by choice of Z1.

    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.


Figure 6.7 Deriving the variable cost function


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)


Figure 6.8 Deriving average variable cost and short-run marginal cost


Short-run Marginal Costs and Average Variable Costs

  • 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.


Average Product and Average Cost

AVC (y’)=w1/AP(z1’)

  • The average variable cost function is the inverted image of the average product function.


Marginal Product and Marginal Cost

SMC (y’)=(w1Δz1)/(MP(z’))

  • The short-run marginal cost function is the inverted image of the marginal product function.


Figure 6.9 Comparing cost and product functions


Figure 6.10 Seven cost functions


Figure 6.11 The costs of commuting


Figure 6.12 Total commuting costs


Figure 6.13 The allocation of commuters to routes


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