Chapter 8, Problem 8.9P

A

To determine

Supposing the given conditions, prove that the short run total costs are equating to the given function.

Answer to Problem 8.9P

After necessary observations, it can be proved that the short run total costs are equated to the given function;

  $STC=1000+0.05q^2$

Explanation of Solution

As given, the production function is;

  $q=\sqrt{K.L}$ ,

In the short run, the K is fixed at 100, and this implies,

  $q=10\sqrt{L}$

Based on the given values, the capital rents are $10 and wages are $5;

  $\begin{array}{l}STC=vK+wL\\ =10\times 100+5L\\ =1000+5L\end{array}$

On the other hand, the short run production function will be;

  $\begin{array}{l}q=10\sqrt{L}\\ \text{Squaring on both the sides of the equation;}\\ q^2=100L\\ L=\frac{q^2}{100}\\ Thus,\\ STC=1000+5\times \frac{q^2}{100}\\ =1000+\frac{q^2}{20}\\ \text{Thus, it can be derived as,}\\ STC=1000+0.05q^2\end{array}$

Economics Concept Introduction

Introduction: The portion of the marginal cost curve lying above the average variable cost curve is the short sun supply curve.

B

To determine

Given the short run marginal cost curve, find out the quantity that firm produces at the given price, and also find the labor hours required per week.

Answer to Problem 8.9P

The quantity that could be produced is 200, and the number of labor hours required is 400.

Explanation of Solution

The function for the short run marginal cost is given as;

  $SMC=0.1q$

The price per unit is given as $20;

Now, since,

  $\begin{array}{l}P=MC\\ 20=0.1q\\ q=200\\ \text{Thus, the quantity that could be produced is 200 units}\text{.}\\ \text{Now, for labor hours, since}\\ L=\frac{q^2}{100}\\ L=\frac{{200}^2}{100}\\ L=\frac{40000}{100}\\ L=400\end{array}$

Thus, the labor hours required is 400.

Economics Concept Introduction

Introduction: The portion of the marginal cost curve lying above the average variable cost curve is the short sun supply curve.

C

To determine

Based on the given conditions, find out the quantity that could be produced and the labor hours required for the produce.

Answer to Problem 8.9P

After necessary calculations, it can be noticed that the quantity produced is 150 units and the labor hours required for this produce is 225.

Explanation of Solution

It is given that the price per unit during the recession is reduced to $15,

Now, using the relationship,

  $\begin{array}{l}P=MC\\ \text{Implies,}\\ 15=0.1q\\ q=150\\ \text{Thus, the quantity produced can be 150}\text{.}\\ \text{Now, the relation becomes;}\\ L=\frac{q^2}{100}\\ L=\frac{{(150)}^2}{100}\\ L=\frac{22500}{100}\\ L=225\end{array}$

Economics Concept Introduction

Introduction: Marginal approach stresses on the concept of analyzing the benefits of an action and comparing them with the costs incurred with the same action.

D

To determine

Describe if the proceeds said in part C will lead to a profit or loss.

Answer to Problem 8.9P

After necessary calculations and observations, it can be concluded that the current proposition is better than the previous proposition in part C.

Explanation of Solution

It is given that during the 1 week of recession, the firm incurs a loss of $1 for every reduction in the workforce.

This implies when L reduced from 400 to 225, the cost the firm incur will be 175.

This means, when q=150, the profit making will be:

  $\begin{array}{l}\pi =TR-TC\\ =150\times 15-(1000+1125)\\ =2250-2125\\ =125\end{array}$

Now, if there is a cost of 175, there will be a loss 50.

In the case of 400 labors, there will be no cost and the profit will be:

  $\begin{array}{l}\pi =TR-TC\\ =15\times 200-(1000+2000)\\ =3000-3000\\ =0\end{array}$

Thus, it can be concluded that the current proposed case is better than the previous case proposed in part C.

Economics Concept Introduction

Introduction: Marginal approach stresses on the concept of analyzing the benefits of an action and comparing them with the costs incurred with the same action.