Production and Cost: One Variable Input

True/ False

1. If  marginal product is decreasing, then average product must also be decreasing.

2. For a fixed-proportion technology, inputs cannot be substituted for each other in production.

3. The marginal product of input 1 derived from the production function y=min[az1, bz2], diminishes for increases in input 1.

4. If the average product is declining, then average total cost must be increasing.

5. The short-run is that period of time during which some inputs cannot be varied.

6. The slope of the short-run total cost curve equals the slope of the short-run variable cost curve at every output.

7. Average fixed costs are constant for all output levels.

Answers: 1.F    2.T    3.F    4.F    5.T    6.T    7.F

3. Short Questions

1. A firm faces the following production function: y = F(z1, z2) = 100z11/2z22
The price of z1, w1, is $10, and the price of z2, w2, is $5.
a)When z1 is fixed at 100 calculate and graph the following functions: TP(z2), AP(z2), MP(z2), VC(y), AVC(y), SMC(y)
b)When z2 is fixed at 10 calculate and graph the following functions: TP(z1), AP(z1), MP(z1), VC(y), AVC(y), SMC(y)

2. Suppose a firm's short run total costs are given by: TC(y) = 3y2+y+500
a) What are the firm's variable costs? Fixed costs?
b) Calculate and graph the firm's short run average costs (SAC(y)) and short run marginal costs (SMC(y))
c) Calculate the point where SAC=SMC and mark it on your graph. What point on the SAC curve does this correspond to?

3. Hair Apparent is one of many identical firms in the highly competitive baldness treatment industry. Its cost function is given by TC = H2 + 4 where H is the number of heads treated (number of units of output). Derive an expression for and graph Hair Apparent's average cost curve, average variable cost curve, and marginal cost curve.

Answers:
1.a) TP(z2)=1000z22, AP(z2)=1000z2, MP(z2)=2000z2, VC(y)=5(y/1000)1/2, AVC(y)=5(1/1000y)1/2, SMC(y)=2.5(y/1000)1/2
   b) TP(z1)=10000z11/2, AP(z1)=10000z1(-1/2), MP(z1)=5000z1(-1/2), VC(y)=y2/107, AVC(y)=y/107, SMC(y)=2y/107
2. a) VC(y)=3y2+y, FC(y)=500
    b) SAC(y)=3y+1+500/y, SMC(y)=6y+1 . On the diagram SAC(y) is in RED (curved line) and SMC(y) is in GREEN (straight line from the origin)
   c) y=12.9, SAC(12.9)= SMC(12.9)=78.4 The point represents the MIN SAC(y).

3. SAC(H)=H+4/H, AVC(H)=H, MC(H)=2H.