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# Chapter 6 Production and Cost: One Variable Input - PowerPoint PPT Presentation

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 6Production and Cost: One Variable Input

• 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).

• 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.

• 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.

• 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.

• 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).

• 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)

• 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.

Average Product illustration

• 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 illustrationaverage product

Marginal and Average Product functions

• When MP exceeds AP, AP is increasing.

• When MP is less than AP, AP declines.

• When MP=AP, AP is constant.

• 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.

More Costs functions

• 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)

Short-run Marginal Costs and Average Variable Costs marginal cost

• 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 marginal 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 cost

SMC (y’)=(w1Δz1)/(MP(z’))

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