Chapter 9, Problem 3E

a)

To determine

To evaluate the long run average variable cost function for electricity generation.

Explanation of Solution

Following is the average variable cost (AVC):

  $AVC=1.24+0.0033Q+0.0000029Q^2-0.000046QZ-0.26Z+0.0018Z^2$

Total Variable cost (TVC)

  $=\text{Average variable cost }\left(\text{AVC}\right)\times Quantity$

  $\begin{array}{l}=\left(1.24+0.0033Q+0.0000029Q^2-0.000046QZ-0.026Z+0.00018Z^2\right)Q\\ =1.24Q+0.0033Q^2+0.0000029Q^3-0.000046Q^{2}Z-0.026ZQ+0.0018Q{Z}^2\end{array}$

Economics Concept Introduction

Introduction: The long-run average cost (LRAC) curve shows the lowest cost per unit for the business at each output point, assuming all production factors are variable. The LRAC curve assumes the firm has selected the optimal mix of factors.

b)

To determine

To evaluate the long run marginal cost function for electricity generation.

Explanation of Solution

  $TVC=1.24Q+0.0033Q^2+0.0000029Q^3-0.000046Q^{2}Z-0.026ZQ+0.0018Q{Z}^2$

Differentiating the total variable cost with respect to Q:

  $\begin{array}{c}MC=\frac{\partial }{\partial Q}\left(1.24Q+0.0033Q^2+0.0000029Q^3-0.000046Q^{2}Z-0.026ZQ+0.00018Q{Z}^2\right)\\ =1.24+0.0066Q+0.0000087Q^2-0.0000092QZ-0.026Z+0.0018Z^2\end{array}$

Economics Concept Introduction

Introduction: Long-run marginal cost is an enterprise metric that represents the long-run average cost per unit of output, where all inputs are considered variable and the scale of production is variable. The long-run average cost curve displays the lowest overall cost for long-run generating a given production point.

c)

To determine

To evaluate the short run average variable cost and marginal cost functions for electricity generation while holding plant size constant at 150,000 kilowatts

Explanation of Solution

  $\begin{array}{c}SRAVC=1.24+0.0033Q+0.0000029Q^2-0.000046QZ-0.26Z+0.0018Z^2\\ =1.24+0.0033Q+0.0000029Q^2-0.000046Q\left(150\right)-0.026\left(150\right)+0.0018{\left(150\right)}^2\\ =1.24+0.0033Q+0.0000029Q^2-0.000046Q\left(150\right)-0.26\left(150\right)+0.0018{\left(150\right)}^2\\ =1.24+0.0033Q+0.0000029Q^2-0.0069Q-3.9+4.05\\ =1.39-0.0036Q+0.0000029Q^2\end{array}$

Size of plant constant at 150,000 kilowatt and marginal cost:

  $\begin{array}{c}MC=1.24+0.0066Q+0.0000087Q^2-0.0000092QZ-0.26Z+0.00018Z^2\\ =1.24+0.0066Q+0.0000087Q^2-0.0000\left(150\right)Q-0.26\left(150\right)+0.0018{\left(150\right)}^2\\ =1.24+0.0066Q+0.000087Q^2-0.0138Q-3.9+4.05\\ =1.39-0.0072Q+0.0000087Q^2\end{array}$

Economics Concept Introduction

Introduction: Average variable cost is the overall variable cost per unit of output incurred when a business participates in the manufacture of short runs. It can be observed in two ways. Because average variable cost is total variable cost per output unit, this can be detected by dividing total variable cost by output quantity.

d)

To determine

To evaluate the output level that minimizes short run average variable costs for a plant size equal to 150,000 kilowatts

Explanation of Solution

  $SRAVC=1.39-0.0036Q+0.0000029Q^2$

On differentiating the short run average variable cost and equating it with zero:

  $\begin{array}{c}\frac{d}{dQ}\left(1.39-0.0036Q+0.0000029Q^2\right)\\ =-0.0036+0.0000048Q\\ 0.0000058Q=0.0036\\ Q=\frac{0.0036}{0.0000058}\\ =620.6\end{array}$

Economics Concept Introduction

Introduction: Average variable cost is the overall variable cost per unit of output incurred when a business participates in the manufacture of short runs. It can be observed in two ways. Because average variable cost is total variable cost per output unit, this can be detected by dividing total variable cost by output quantity.

d)

To determine

To evaluate the output level that minimizes short run average variable costs for a plant size equal to 150,000 kilowatts

Explanation of Solution

Output that minimize the short run average variable cost:

Following is the average variable cost:

  $\begin{array}{l}SRAVC=1.39-0.0036Q+0.0000029Q^2\\ \end{array}$

On differentiating the short run average variable cost and equating it with zero:

  $\begin{array}{c}\frac{d}{dQ}\left(1.39-0.0036Q+0.0000029Q^2\right)\\ =0.0036+0.0000048Q\\ =0.0000058Q=0.0036\\ Q=\frac{0.0036}{0.0000058}\\ =\boxed{620.6}\end{array}$

Economics Concept Introduction

Introduction: Average variable cost is the overall variable cost per unit of output incurred when a business participates in the manufacture of short runs. It can be observed in two ways. Because average variable cost is total variable cost per output unit, this can be detected by dividing total variable cost by output quantity.

e)

To determine

To evaluate the short run average variable cost and marginal cost at the output level obtained in part (d).

Explanation of Solution

Short run average cost:

  $\begin{array}{c}SRAVC=1.39-0.0036Q+0.0000029Q^2\\ =1.39-0.0036\left(620.7\right)+0.0000029{\left(620.7\right)}^2\\ =1.39-2.234+1.117\\ =\boxed{0.273}\end{array}$

  $\begin{array}{c}MC=1.39-0.0072Q+0.0000087Q^2\\ =1.39-0.0072\left(620.7\right)+0.0000087{\left(620.7\right)}^2\\ =1.39-4.46+3.35\\ =\boxed{0.28}\end{array}$

Economics Concept Introduction

Introduction: Average variable cost is the overall variable cost per unit of output incurred when a business participates in the manufacture of short runs. It can be observed in two ways. Because average variable cost is total variable cost per output unit, this can be detected by dividing total variable cost by output quantity.