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Chapter 6 Production and Cost: One Variable Input

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- 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 (AP) of the variable input equals total output divided by the quantity of the variable input.
AP(Z1)=TP(Z1)/Z1

- 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.

- 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)

- 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.

AVC (y’)=w1/AP(z1’)

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

SMC (y’)=(w1Δz1)/(MP(z’))

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