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 6 production and cost one variable input

Chapter 6Production and Cost: One Variable Input


Production function

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

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

Figure 6.1 Finding a production function


From figure 6 1

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

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

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

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

Figure 6.3 From total product to marginal product


Diminishing marginal productivity

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

Figure 6.4 From total product to marginal product: another illustration


Average product

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 to average product

Figure 6.5 From total product toaverage product


Figure 6 6 comparing the average and marginal product functions

Figure 6.6 Comparing the average and marginal product functions


Marginal and average product

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

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

Figure 6.7 Deriving the variable cost function


More costs

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

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


Short run marginal costs and average variable costs

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

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

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 6.9 Comparing cost and product functions


Figure 6 10 seven cost functions

Figure 6.10 Seven cost functions


Figure 6 11 the costs of commuting

Figure 6.11 The costs of commuting


Figure 6 12 total commuting costs

Figure 6.12 Total commuting costs


Figure 6 13 the allocation of commuters to routes

Figure 6.13 The allocation of commuters to routes


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