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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.

Chapter 6 Production and Cost: One Variable Input

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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).

### 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.

### 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)

### 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.

### 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

### 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.

### 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)

### 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.