Managerial Economics

Supply and Demand Homework


Lecture Notes in Power Point
Lecture I and II
Production and Costs
Profit Maximization
Perfect Competition & Monopoly
Monopolistic Competition and Oligopoly
Monopolistic Competition and Oligopoly II

Introduction



Orientation/Introduction

What is managerial economics?

Managerial economics is applied microeconomics; making use of the tools of statistics, mathematics, and
decision sciences, managerial economics applies economic models and tools of analysis to (decision
making) problems faced by the managers of business firms, not-for-profit organizations and government
agencies.

Consider the following cases:

o Trying to make her sales goals of the month a department store manager needs to decide on a strategy
to increase her sales. Should she have a sale on the overstocked merchandise or should she raise the prices
on some of the more popular items?

o In an attempt to increase the market share of his company the CEO of a major beer company has to
decide whether to expand the production capacity of an existing plant or to build a new plant at another
location.

o The CEO of a small accounting firm wishes to decide whether to upgrade the firm's computer system or
to hire more accountants.

o A regional sales manager wants to decide whether to hire more sales persons or offer greater incentives
to her existing sales staff.

o The marketing manager of a computer software company wants to determine the price of an upgrade of
one of the company's popular software products.

o A pharmaceutical company wants to determine the price of a new patented cancer drug it has just had
approved by FDA.

o A local moving company is trying to decide whether it should increase the size of its fleet or replace
some of its older trucks with new ones.

o A school board wants to decide whether to buy more computers or hire more teachers.

o In reaction to an increase in the cost of claims an insurance company wants to decide whether it should
raise its premiums or it should reduce benefits.

o The director of a charity organization wants to decide between two fund-raising schemes: (1) Direct mail
to a large number of households whose addresses can be obtained at no charge from public sources; (2)
direct mail to a select number of households whose names and addresses can be purchased from a
commercial mailing list company at a relatively high price.

These are some examples of the types of managerial problems that managers in different organizational
settings face every day. Note that in all cases the managers are faced with alternative choices. Good
decision-making practices often require (appropriately) precise evaluation of alternatives. The need for
some level of precision in managerial decisions calls for precision instruments of measurement and
analysis. Aided with mathematical and statistical tools, economic models provide us with such instruments.
 

A simple example should make the above point more clear. Suppose the sales manager of an auto
dealership wishes to increase the monthly sales revenue of the dealership by $100,000. She can try to
achieve this goal by offering discounts and hoping to sell more cars. Alternatively, she can raise the prices
on certain models and hope that there won't be any reduction in the number of cars sold. Or, she can
spend more money on advertisement. Unless she knows, nearly exactly, how each of these schemes
affects her sales there is no way for her to identify and choose the best one. Let's suppose that based on
her experience, she decides to go with the discount scheme. Now the next decision that she has to make is
the size of the discount. Will a 10 percent discount on some models do it or should she offer a much larger
discount? For her to be able to make these decisions with some degree of confidence she needs to know
the demand for her cars. In other words, she needs to be able to make predictions about how different
factors affect the demand for her cars. We shall come back to this example later in this lecture.

Most managerial decisions involve making a choice from among alternative courses of action or alternative
options in order to achieve a certain objective. A manager would want to choose the best option among
the possible alternatives.

Optimization
Optimization is the process by which a desired outcome is achieved through the most efficient course of
action. In production a manager optimizes the use of a given amount of inputs to produce the greatest
amount of output possible. In consumption, a consumer with a given amount of income purchases the mix
of goods that provides him or her with greatest level of satisfaction (utility). In a not-for-profit setting, a
manager might want to accomplish a certain task at a lowest possible cost. Or, alternatively, a manager
may wish to provide the maximum amount of service with a given amount fund.

Constrained Optimization
Often managerial decisions have to be made subject to some constraints. Managers operate under a
variety of constraints including technological, financial, legal and contractual. For instance, a manager that
is trying to cut his labor costs (or optimize the size of his labor force) may be constrained by a union
contract limiting his ability to lay off workers. A farmer who wants to take advantage of good market
conditions and increase the size of his crop is limited by the amount of land that he has available.

Furthermore, managerial decisions are not made in a vacuum. Economic and market conditions constantly
change and managers must make their decisions in accordance with the dynamics of the business
environment. Especially for business firms, changing economic/market conditions present major
challenges, as their managers need to constantly make adjustments in their plans in response to changing
economic/market conditions.

Economics and Management
What does economics have to do with management?
Economics, as a discipline, is divided into two branches: microeconomics and macroeconomics.
Microeconomics is a study of the behavior of individual microeconomic units such business firms and
consumers. Macroeconomics, on the other hand, studies the economy as whole and deals with issues like
inflation, unemployment and economic growth. Managerial decisions involve identifying problems and
opportunities, examining alternative courses of actions, and making optimal choices. As complex as
managerial problems may appear, often their relevant (important) elements can nicely be fitted into
microeconomic models. That is why managerial economics is also called "applied microeconomics."
Microeconomic models help us convert seemingly complex managerial problems and solution options into
manageable forms to which quantitative tools of analysis can be applied. For example, the
profit-maximizing objective of a business firm is nicely described in the microeconomic theory of the firm.
This model provides a very good framework for analyzing many cost/revenue related managerial
questions.

A manager operates in two environments, internal and external. The internal environment is made up of
those factors over which he has at least some degree of control. What technology to use, how many
workers to hire, what method of advertisement to use are some examples of internal factors. The external
environment, however, is influenced by factors beyond a manager's control. The state of the economy,
interest rates, inflation, exchange rates and laws and regulations are among the factors that affect the
external environment for a manager.

Macroeconomics is relevant to managers as managers are often interested in knowing (and some times in
predicting) the state of the economy and the direction of macroeconomic measures such as interest rates
and inflation. Such knowledge is essential in their evaluation of opportunities and of the options they face.
In the increasingly globalized economic environment of today, accurately reading (and predicting)
macroeconomic measures such as inflation rates, interest rates, and exchange rates could be the key to
making effective managerial decisions.

Scope of the Course
Managerial economics is a growing and evolving field. It would be unrealistic for us to expect to learn all
the methods and analytical tools of managerial economics in one undergraduate course. In this course we
will attempt to learn about the application of some of the more basic microeconomic models to managerial
problems. We will start our discussion with a look at the roles and the objectives of managers and then
move on to a review of the supply and demand model. Next we will turn to the concept of elasticity.
Following a brief review of the consumer theory, through which we will learn about the concept of
optimization, we will start our discussion of the theory of the firm. It is within the framework of that
model that we will address the issue of profit maximization and discuss the behavior of business firms
under different market structures.

In MBA 531 (our graduate version of this course) we would also contain a review of regression analysis and
forecasting methods as well as topics related to factor markets and decision making under uncertainty. In
this course we will omit most these topics. We shall try, however, to address the issue of uncertainty, only
to a limited extent, in the context of our discussion of market structures.

Although many of the analytical tools of managerial economics are as useful to the managers of
government agencies and not-for-profit organizations as they are to the managers of business firms, in this
course we will do most of our analyses with business firms in mind.

What You Should Expect to Learn
One course in managerial economics is not going to make you a professional economist or a managerial
economist, for that matter. Upon successfully completing this course, however, you should expect to have
learned how relatively simple economic models and tools of analysis can be utilized to analyze (and some
times to solve) rather complex managerial problems. The course will sharpen your analytical skills,
enabling you to identify the relevant elements of a problem, fit it into a manageable (economic)
framework, and apply simple analytical tools to it to analyze and solve it.



DOCUMENT by: Said Atri
Subject: 2.1 Managerial Objectives Lecture

Managerial Objectives

Business Firms and their Functions
Before we address the objectives of a business firm let us make sure that we all know what economists mean by a business firm. Economists define a firm as an economic unit engaged in the production of one or more goods (or services) for the purpose of selling and making profits. It is generally assumed that the goals of the managers of a firm are consistent with the economic function of the firm. In other words, managers make their decisions in accordance with the profit-maximizing objective of the firm.

The economic definition of a business firm covers all sorts of business firms. IBM, GM, Microsoft, Sears, and Walmart are all business firms as are your corner service station and your favorite pizza restaurant. All these firms produce products or services to sell in the market and make profits.

Profit Maximization and the Value of a Firm
What determines the value of a business firm? Although business firms often own some physical (or intellectual; e.g. copy rights) assets (as well as liabilities), and the net worth of their assets may have some effects on their values, the real value of a business firm is likely to be more dependent on its (expected) future profitability. In other words the value of a firm is the presented value of its expected future profits. Thus a managerial decision that is expected to have a positive effect on the firm's future profits would increase the value of the firm. Conversely, an event or a decision by the firm that is perceived to have a negative effect on the firm's future profits would like cause its value to go down.

Marginal Analysis: A Precursor
Economic profit is defined as the difference between the firm’s total revenue and its total economic cost of production:

Profit = TR – TC

Both TR and TC change as the firm changes the quantity of its output (product). In other words, TR and TC are both a function of the quantity of output (Q). Thus we write:

TR = f (Q)
TC = g(Q)
Profit = TR-TC = h (Q)

Where f, g, and h denote functional relationships.

A change in a firm’s profit results from a change in its total revenue or a change in its total cost or a combination of the two. That is:

Change in Profit = Change in TR  - Change in TC

In mathematics very small changes in a variable (at the margin) are called marginal changes. A marginal change in a variable could be positive or negative.

Thus, mathematically speaking, marginal profit is equal to the difference between “marginal revenue” and “marginal cost”.

Simply put, "marginal revenue" is the change in total revenue resulting from selling one additional unit of output. Likewise, "marginal cost" is the change in the cost of production resulting from producing one additional unit of output. By the same token, "marginal profit" can be defined as the change in profit resulting from a one-unit change in the output. As long as its marginal profit is positive (greater than zero) a firm can increase its (total) profit by producing more output. The firm's profit reaches its maximum level when its marginal profit becomes zero. Thus, a firm can find the profit maximizing level of its output (product) by setting its marginal profit equal to zero (and solving for Q).

Marginal Profit = MR – MC = 0

Alternatively, the firm can find the profit maximizing level of its output by setting its MR equal to its MC.

Note that when a firm sells its product at a given market price, irrespective of how much it produces and offers to the market, its marginal revenue is simply the market price. That is because each additional unit that it sells will increase the firm’s revenue by an amount equal to the market price of the good.

An exercise:

TC = 400 + 70 Q + .002 Q^2
MC = 70 + .004 Q

TR = 250 Q
MR = 250

MR = MC

250 = 70 + .004 Q

.004 Q = 180

Q = 45,000
 

Note: The marginal value of a function (relative to a given variable) can be obtained by taking the derivative of the function (with respect to that variable). that is:
 
 
 

This subject will be addressed in more detail in future lectures.
 

The Concept of Present Value
Because assets have potentials to generate either benefits or income (if invested) for their owners, an economically "rational" person would prefer a present asset to a future asset of the same value. If you were offered a television set as a gift and are given the choice of either having it now or having it a year from now, chances are that you would want to have it now. If you were to wait a year, you would deprive yourself from its benefit for a whole year. The difference between the value of a present asset and a future asset of the same value may be more clearly observed in the case of financial assets. If you won a lottery and were given the option of receiving your prize in full now or a year from now, you would definitely demand that it be given to you now. That is because if you waited a year you would lose the opportunity to invest your prize money and earn interest from it. That is why in some lottery games players are offered the option of either receiving their prize money in installments over a period of time or receiving the discounted value of the future installments in cash. Thus we can say the present value of a future asset is equal to its discounted value. Alternatively, the future value of a present asset, say two years from now, is its present value plus the interest that would accrue over two years. So we can write:

Future value of an asset in two years = Its present value (1 + interest rate )2

From this we can generalize and write:

FVA = PVA (1+ r) n

where FVA stands for future value of asset A, PVA is the present value of asset A, r represents the interest (discount) rate, and n is the number of years into the future.

It is easy enough to rearrange the above equation to define the present value of a future asset. We simply divide both sides of the equation by  (1+ r) n .

PVA = FVA/ (1 + r) n

A simple example would make this concept more clearer. Let us suppose that when you were born, say in 1980, your Uncle Ben gave you a 2010 zero-coupon bond with a face value of $10,000.(Note: Zero coupon bonds were actually introduced in mid-1982. Zero coupon bonds, as their name suggests, have no "coupons," or periodic interest payments. Instead, the investor receives one payment (at maturity) that is equal to the principal invested plus the interest earned, compounded semiannually. In other words, when a zero coupon bond matures, the holder of the bond receives the full face amount of the bond.) Uncle Ben told your parents that they did not have to wait till year 2010 to cash the bond. He said they could go to a bank (or a brokerage firm) any time they wished and cash the bond. That made your parents very happy and they wondered how much they could get for it if they were to cash it then, although they really did not intend to do so. On your mother's insistence, your father calls his banker friend to find out. Your father was disappointedly surprised when his friend gave him the present value of your $10,000 zero-coupon bond. He said if they were to cash it then they would get about $575. When your mother heard that she said: "I knew my brother could not have become so generous all of a sudden!"

Now let see how the banker figured that out. What he did was simply calculate the present value of  $10,000 for 30 years. Those days (in late 1970s and early 1980s) the interest rates were rather high. The prime rate (the interest commercial banks charge their most creditworthy customers) reached as high as 18 percent. Thirty-year government bonds paid (or were discounted by) as much 13 percent (annually). Now let us say that your father's banker friend used a 10 percent discount rate to determine the value of your $10,000 zero-coupon bond. The future value of your 30-year zero-coupon bond on its maturity day was $10,000., as indicated right on its face. So we write:

PVA = 10,000/ (1 +.10 )^30  =  573.0855

In other words, if your uncle had given you $575 in cash and your parents had invested it for you for 30 years at a fixed interest rate of 10 percent, by the year 2010 your $575 would have grown to about $10,000. That is the power of compounding interest!

Your uncle Ben might not have been too generous, but he was a smart investor. In a few years in mid and late 1980's, the interest rate had dropped significantly. For example in 1990 the 20-year interest rate was around 8 percent. In that year the present value of your $10,000 bond, which was to mature in 20 years now, was $2145, almost 4 times what it was 10 years earlier; that means a little less than 300 percent growth in ten years.

Present Value and the Value of a Business Firm
One way to put a value on a firm is to multiply the market value of a share of its stock by the number of its outstanding shares. Outstanding shares are share issues held by investors. This value is referred to as the market value of the company. In other words that is the value that investors put on the company when they trade its share. When you buy a share of a company, in theory, you are buying a piece of the net asset or the equity of the company that could include some bolts and nuts and maybe a few bricks. But that is not really why you buy a share of a company's stock. As a small shareholder you can hardly hope to have a say in the business of the company let alone claiming your share of the bolts and nuts that the company owns. You (directly or through your investment agent) evaluate and buy a stock much the same way you would buy a bond issue. In other words, to an investor the value of a stock is a function of its (expected) future earnings. A stock is a piece of paper that gives its holder a right to a portion of the company's future profits. It is then the future profits of the company that give value to the stocks of a company. More precisely, the value of a business firm is the present value of the company's expected future profits. Using the present value formula we can write
 

where PV t  is the value of the firm in time t, r is the interest (discount) rate, and n is the number of years following time t that the firm is expected to be in operation and produce profits. Theoretically a business firm could have an infinite lifetime. One can assume a case where an investor would buy a stock with the intention of selling it after a certain period of time. In that case the last "earning" of that stock would be the market value of the stock at the time the investor intends to sell it. An investor could speculate about the value of a stock in the future and make his or her investment decision accordingly. Note that the market value of a stock at some point in the future would also depend on its expected profits from that point on. To make sure that the relationship between future profitability and the present value of a stock is well understood, let us use a simple example.

Suppose that you are considering buying a stock that is expected to yield $15 dividend (profit) per share for the next three years after which you intend to sell the stock. You also expect that at the end of year three the market value of this stock will be $200. Let us also assume that there are no transaction costs (no fees or commissions). We further assume that, considering the risk factor associated with investment in stocks, you expect at least 12 percent return on this investment. Utilizing the above present value formula, we write:

PVt  = 15/ (1+ .12)1 + 15/ (1+.12) + 15/ (1+.12)3  +200/ (1+.12)3  =  13.39+ 11.95 + 10.68 + 142.35  =  178.38

Based on your expected rate of return on this kind of investment, the present value of the expected future earnings (including the market value of the stock a the end of your intended investment period) is $178.38; that is the value you would put on this stock at this time. Note that in this simple case we assume that you adjust your expected rate of return (discount rate) for the risk you perceive in this investment, the higher the perceived risk the higher the discount rate. There are more sophisticated ways to deal with uncertainty and risk in investment decisions. At this point, however, the subject of risk management is not within the scope of our discussion.

The purpose of an investor (or an owner of a business firm) is to realize income or profit from his or her investment. As demonstrated above, there is a direct relationship between the (expected) future profits of a firm and its value. In other words, firms that are expected to be more profitable in the future would have higher market values. It is therefore expected that the manager of a firm who is supposed to act as an agent of the shareholders (owners) make his or her decisions in accordance with shareholders’ interest which is to have the value of the firm maximized.

Note: A commonly used market indicator of the expected profitability of a stock (based on its past history) is the price/earning (P/E) ratio, published daily in the W-S Journal and some other investment publications. The P/E ratio is in fact the reciprocal of the (ex post) current rate of return on the stock. For example, a P/E ratio of 20 indicates 5% return on the investment. (Visit www.buffettsecrets.com/price-earning-ratio.htm)

Generally, stocks with high P/E ratios are considered “growth stocks.” The investor is willing to pay a higher price for a growth stock with the expectation (or in the hope) that it will perform better in the future and, thus, its market value will increase.

If you have any questions about this material, please click on the  ASK A QUESTION link below. Now go to the next document to continue this module.
 


Lecture Outline

Examples of managerial problems:
   What product to produce
   What price to charge
   Where/how to get financing
   Where to locate
   How to advertise
   What method of production to use
   Whether or not to invest in new equipment

Managers’ Objectives
a. Maximizing the value of the firm (Profit maximization)
b. Possible alternative objectives:
   =>Market share maximization
   =>Growth Maximization
   =>Maximizing their own benefits

Decision Making Process
Identifying the problem or the decision to be made

Abstraction: Identifying the relevant factors
in the problem and formulating the problem into a manageable
set of questions/problems

Identifying alternative solutions to each problem
Using relevant data to evaluate alternative solutions
Choosing the best solution consistent with the firm’s objective

Consider the following news headlines:

The ups and downs:
                  High         Low         Last        PE
MSFT        118           40           62          33
IBM           134           80         109          24
Lucent          77           12           19          50
Mot             61            15           23         40
AT&T         61            16           23         14
GM             94            48           54           8
RJR             52            15           51         13

Macroeconomics and Microeconomics and Managerial Decision Making

Microeconomics: Production and cost models, price, revenue and profit, market conditions
Macroeconomcs: Economnic conditions:  Business cycles; unemployment, inflation, recession and econopmic growth and expansion

Demand and Supplied Reviewed


Demand : Definitions

Supply: Definitions Supply and Demand Schedule
   Price     Supply     Demand
 $ 0.00      ----          670
  1.00        210          470
  1.25        290          420
  1.50        370          370
  1.75        450          320
  2.00        530          270
  2.25        610          220
  2.50        690          170

Why do we study supply and demand?
We assume, generally, firms are value maximizers, realizing that the value of a firm is a function of its (expected) future profits.
                        Profit   =  TR  - TC
                        TR      =    P . Q
==> What are the factors that determine P and Q?
==> What are the elements determining a firm’s costs?

Supply and Demand Equations
  Demand:
      Qd = 670 -200 P
       P   =  3.35 -.005 Qd
  Supply:
          Qs =  - 110 + 320 P
           P  =  .34375   +  .003125 Qs

Supply and demand plotted:

An algebraic approach to supply and demand:
Qd  =  f ( Price, Income, X1, X2, ……Xn)
Qd  = 20 + .1 Income - 2 Age - 50 Price

Qs =  g( Price, W1, W2, ……. Wn )
Qs =  -40  - 5 Wage + 30 Price

If
Income = 2000
Age = 30
Wage = $8

==>Supply and demand curves:
        Qd  = 20 + .1 Income - 2 Age - 50 Price
                              ($2000)      (30)
==>  Qd   =   160 - 50P
 

         Qs = -40 - 5 Wage + 30 Price
                              ( $8)
==>   Qs  = - 80  + 30 P
 

Shifts in supply and demand curve:

Demand and Revenue
Recall that:
                      TR = Price x Quantity =   P .Q
If   P = f (Q)   =  3.2 - .02 Q,
we can write:  TR = (3.2 -.02Q).Q

Or,                 TR =  3.2 Q  -  .02 Q2


 

The case of a horizontal demand curve:
 

Marginal versus Average
Recall:   TR = P. Q  =  3.2 Q  -  .02 Q2
                          TR
               AR = ------ =   3.2  - .02 Q  =  P
                           Q

                           Change in TR      d TR
               MR = ---------------- = ------- =  3.2  - .04 Q
                           Change in Q        d Q

Demand and Revenue




Supply and demand: An exercise

(ANSWERS)
1. Plot the following supply and demand equations in a diagram measuring price on the vertical axis and quantity on the horizontal
axis.

Qs  =  - 600  +  40 P
Qd =  1200   -50 P

a.    Identify the price and quantity intercepts for each equation.
b.    Determine the slope of each line.
c.    Write each equations for price (P) in terms of quantity (Q).
d.    Determine the equilibrium price and quantity.
e.    Explain the market conditions when the price is set at  $18
f.     Explain the market conditions when the price is set at $ 25

The demand and supply functions for seats on a special shuttle flight to Orlando, Florida, have been estimated as follows:

Qd  =   900 -2 Price + .05 Income  - 5 Weather + 1.25 Pc  ( where Pc is the price offered by the competition)

Qs  =   -20 - 5 Pf + 4 Price       (where Pf is the fuel price)

Assuming:    Income( I ) = 1000,       Weather (W) = 70,         Pc =  $160,      Pf = $16

a. Write the equations for the demand curve and supply curve.
b. Carefully plot both supply and demand curves.
c. Determine the equilibrium price and quantity.
d. Determine and show on your diagram the effect of an increase in the weather temperature (W)
    from 70 to 80 on the equilibrium price and quantity.
e. Keeping the weather temperature (W) at 80, determine and  show the effect of an increase in
    the price of the competition from $160 to $200 on the equilibrium price and quantity.
f. Now keeping the weather temperature at 80, Pc at 200, and income at 1000, use your demand
    function to write the total
    revenue ( TR ) equation.
g. Using the same demand function, also write and plot the marginal revenue (MR) function.
h. Using the same demand function, determine at what price level the total revenue from this shuttle
    flight is maximized.
     Try to show your work on a diagram.

Answers:
Q1: a)  Supply: Qs = -600 + 40 P ;    P = 15 + .025 Qs      Demand:    Qd = 1200 - 50 P ;   P = 24 - .02 Q
       b)  Slope of demand curve = -.02         Slope of supply curve = +.025
       c)  P = 15 + .025 Qs ;           P = 24 -.02 Q
       d)  P = 20 ;    Q = 200
       e)  Excess demand
        f)  Excess supply
Q2: a)  Qd = 800 - 2 P  ;       Qs = -100 + 4 P
       b)  Demand: P intercept = 400 ; Q intercept = 800      Supply:  P intercept = 25 ;      Q intercept = -100
       c)  P = 150 ;     Q = 500
       d)  Shift to the left (-50)
       e)  Shift to the right (+50)
       f)   TR = 400 Q -.5 Q2
       g)  MR = 400 - Q
       h)  Q = 400 ;  P = 200



Elasticity
A general definition:
Elasticity is a standardized measure of the sensitivity of one (dependent) variable to changes in another variable.

Price elasticity of demand:
A measure of the sensitivity of the quantity demanded a good to changes in the price of that good.

 Measuring Elasticity
Elasticity is measured by the ratio between the percentage change in on variable and the percentage change in another variable:

                         Percentage change in Y
Elasticity     = ------------------------------
                         Percentage change in X

                                 Change in Y/ Y
                    =  -----------------------------
                                 Change in X/ X
 

Elasticity of Demand
The (market) demand for a good is affected by numerous factors: price, income, taste, population, weather, expectations, population demographics, etc.
The degree of sensitivity or responsiveness of the demand to changes in any of the factors affecting it can be measured in terms of “elasticity”.
            percentage change in  Qd
Ez  =  -------------------------------------
            percentage change in  X
 

Measuring a change in percentage terms:
                                Y2 –Y1                           Y1 = 80
% change in Y = ------------------                  Y2 =100
                                   Y1

                              Y1 –Y2
                        = -------------
                                 Y2
 

                                 Y2 –Y1
Arc % change = -------------------
                                 Y2 +Y1
                                -----------
                                      2

Calculating Elasticity

                  Change in Qx
              -------------------
                         Qx1
Ez  =    ---------------------
               Change in Z
              -------------
                        Z1
 


                      Change in Qx
                      --------------
                       Qx1 + Qx2
(Arc)Ez = --------------------
                      Change in Z
                    --------------
                         Z1 + Z2

Arc Price Elasticity of Demand
Definition: A measure of the responsiveness of quantity demanded of a good to changes in its price.
                           Qx2 – Qx1
                          ------------
                           Qx1 + Qx2
             Ep = -----------------------
                             P2 – P1
                         ------------
                             P1 + P2

Using the "simple" elasticity formula, the price elasticity of the demand at two different (price) ranges has been calculated. Here it is assumed that between points a and b the price has changed from 10 to 8 and between c and d it has changed  from 4 to 2.
 
 
 
 

To get the average elasticity between two points on a demand curve we take the average of the two end points (for both price and quantity) and use it as the initial value:
                Q2-Q1                   10
               (Q1+Q2)               8+18
Ea  =     --------------  =    ---------------  = -3.49
                   P2-P1                 -2
                 (P1+P2)             10+8
 



 

Production and Costs

Here again we start with our basic definition of profit:

Profit =  TR – TC  = P·Q – TC

In the previous lecture we studied the demand function to understand how the revenue of a business firm is determined and how a firm’s decisions, particularly relative to price, would affect the quantity demanded and thus its sales revenue. In this section we are going to turn our attention to the firm’s decisions relative to production and costs.

In attempting to increase the firm’s profit managers must pay attention to both the revenue and the cost side of the profit function. Let us suppose a farmer can produce wheat, barley, or corn, or a combination of them. The profitability of his operation would depend on his decisions on what to produce, how much to produce and how to produce. His revenue (P·Q) would be determined by the first two decisions (what and how much) whereas his cost would be affected by all three decisions.

Business firms operate in dynamic environments. Their both external and internal environments change all the time. Managers must adjust to their changing environments by constantly evaluating their options relative to all three questions—what, how much and how. In recent years, the emergence of new technologies, on the one hand, and increases in international trade and investment opportunities, on the other hand, have resulted in major changes in the way businesses operate. The successes and failures of all business entities have depended upon how they have reacted to their changing environments. Your textbook gives a number of examples of the structural changes that some firms have gone through during the past two decades. All of theses changes have some how affected their sales revenues or their costs or both. As read about different cases in the book be sure to think about the impact of each action that various firms have taken on their profitability.

Production Function

The production cost of the firm depends on (i) what the firm produces, (ii) how it is producing it, and (iii) what quantity it is producing. The relationship between the inputs used in the production of a good and the quantity of the output produced, assuming that the firm is using the technology available to it efficiently, can be shown in a production function. In other words a production function is a mathematical expression that shows the maximum amount of output that can be produced from a given mix of inputs.

What are inputs? Inputs are factors or materials used in the production—e.g., land, labor, capital, materials, energy, etc.

What is output? Output is a quantitative measure of the good (or service) produced.

The production process, on the one hand, generates costs— as needed inputs are purchased—and, on the other hand, generates output.
 
 
 
 

The relationship between output and cost (the cost function) is directly linked to the production process—that is the production function. We will start our analysis with the production function and then we’ll expand it to the cost function.

As said earlier the production function is a mathematical statement reflecting the relationship between inputs and the quantity of output efficiently produced from them given the available technology. For example, our farmer’s production function for wheat can be written as follows:

Q = f (K, L, N, F,W), where K is capital, L is labor, N is land, F is fertilizer, and W is whether.

This function generally implies that the quantity of the output of wheat is determined by the quantitative (or qualitative) measures of capital, labor, land, fertilizer, and weather. For example we expect the output to increase as more capital (tools and machinery) is used in the production. Or, generally, more land would also increase the output. The above function is of course in a general form—it does not show specifically by much the output would change as any or all of the inputs would change. Production functions can take different mathematical forms from simple linear functions to complex nonlinear forms. To learn about the mechanics of a production function let us use a simple two-input production function:

Q = f (K, L), assuming that K represents all capital goods and L labor.

Again this is a general production function. The following are some possible specific forms of production functions:

Q = a Lb + c KL

Q = a Kb + c KL

Q = a Kb· Lc

where a, b, and c are parameters that determine the nature of the relationship between the inputs and output.

For example, if for the first equation we a=1, b= 2 and c= 3, we can write:

Q = L2 + 3 KL

Now with 100 units of labor and 50 units of capital the output will be:

Q =  (100) 2  + 3 ( 50 x 100) =  10000 + 15000 = 25000

*A simple exercise for you: Apply the same parameters and capital and labor quantity to the third equation and determine the quantity of output. Then, double each input and recalculate the output. What kind of observation can you make about the relationship between the inputs and the output in this production function?

The production function is studied in the short run and in the long run. The long run is the period of time that is long enough for a firm to change all of its inputs or factors of production. In the short run, on the other hand, the firm can only increase or decrease some of its inputs. In our simple two-input production function, for example, in the short run labor could be considered the variable input while capital would remain constant. That means in the short run the quantity of output would be simply a function of labor.

Given K,   Q = f( L); as L changes Q changes.

Let us use our farmer’s production function to examine the relationship between the variable input and output more closely. Suppose our farmer’s variable input (in the short run) is fertilizer. All other inputs, land, labor, capital, etc., remain constant. The relationship between the amount of fertilizer and the annual size the output of wheat has been tabulated in the table below.
 

Fertilizer (Kg) 0 50 100 150 200 250 300 350 400 450
Output of wheat 100 200 280 340 380 400 360 300 200 50
 
 

Note that the addition of the first fifty kilograms of fertilizer to the land would double the annual output. As more and more fertilizer is applied the output increases, but at a decreasing rate. It peaks at 400 and then starts to fall. This relationship is depicted in the diagram below.
 

Diminishing Returns and Marginal Product of an Input
Most products are produced from a combination of inputs. In the wheat production function in addition to fertilizer we have land, labor, capital, and weather conditions. As more and more fertilizer is added to the same amounts of other inputs, beyond a certain point, its effectiveness starts to decline, and, most likely, it will eventually become negative. This phenomenon is referred to as the principle (or law) of diminishing returns (to input.) Alternatively put, as more and more of a variable input is added to fixed quantities of other inputs, beyond a certain point, the marginal product of the variable input starts to fall.

The marginal product of an input (say input X), MPx, is the change in output resulting from one additional unit of (variable) input.
 

                   ? Q
MPx  =  ----------
                   ? X
                                                                                                    ? Q
For the fertilizer in our wheat production function   MPf  =  -----------
            ? F
Fertilizer (Kg) 0 50 100 150 200 250 300 350 400 450
Output of wheat 100 200 280 340 380 400 360 300 200 50
MPf   2 1.6 1.2 0.8 0.4 -0.8 -1.2 -2 -3
 

In the table above the relationship between the variable input (fertilizer) and output are tabulated in discrete numbers. Therefore, each measure of marginal product is an average (or approximation to) the marginal product between two levels of inputs. Some textbooks call this measure arc marginal product. For example between 100 and 150 kilograms of fertilizer the marginal product (of one additional unit) of fertilizer is:

(340-280)/(150-100)   = 60/50  = 1.2

That means, other things remaining constant, between 100 and 150 units of fertilizer, on average, each kilogram of fertilizer would increase the output by 1.2 metric ton. Note that at some point between 250 and 300 kilograms the marginal product reaches zero. That is where the total product is maximized. The marginal product of (additional) fertilizer beyond that point would be negative causing reductions in the total output.
 

Average Product

Another commonly used measure of productivity of a variable input is the average product. It is simply total output divided by total variable input. This measure, as its name suggests, is the (overall) average output per unit of a variable input, again, assuming all other inputs remain constant.
                  Q
   AP f  =  -------
                   F

Fertilizer (Kg) 0 50 100 150 200 250 300 350 400 450
Output of wheat 100 200 280 340 380 400 360 300 200 50
MPf   2 1.6 1.2 0.8 0.4 -0.8 -1.2 -2 -3
Apf   4 2.8 2.27 1.9 1.6 1.2 0.86 0.5 0.11

The Marginal and Average Product: A closer Look

To understand the relationship between marginal product and average product let us use a simple production function with capital and labor as inputs.

Q =  KL2 – L3      where K is capital and L is labor.

Suppose in the sort run the capital, K, is constant at 16 units whereas labor, L, is variable. The following table shows the output for varying levels of labor as well as the marginal and average product measures corresponding to them.

Q = 16  L2 – L3            APL =  Q/L          MPL =  ?Q/?

Labor  1 2 3 4 5 6 7 8 9 10 11 12
Output  15 56 117 192 275 360 441 512 567 600 605 576
MPL 15 41 61 75 83 85 81 71 55 33 5 -29
APL 15 28 39 48 55 60 63 64 63 60 55 48
 
 

We can make the following observations:

· Given K=16, as L increases Q will increase, first at an increasing rate and then at a decreasing rate.
· The marginal product of labor peaks at about 6 units of labor at which point the law of diminishing returns goes into effect.
· As long as the MPL is greater the APL, the APL increases, but as soon as the MPL falls below the APL the APL stars to decline.
· The MPL intersects the APL at the point where the APL is at its maximum.
· Somewhere between 11 and 12 units of labor the MPL becomes zero. That is where the total output reaches its maximum.
 

Before we move onto the long-run production function let us take advantage of the above analysis and develop the short-run cost function.

The short-run cost of production in this simple case consists of two components: capital cost and labor cost.

STC =  Pk . K + W . L       where Pk is the price of capital and W is wage.
In the short run where K is constant, given the price of capital, the capital cost would be fixed. The cost of labor however would vary with the amount of labor that is variable in the short run. Assuming the price of capital is $50 per unit and the wage is $120, we write:

STC = TFC + TVC  =  50x16   +     120 L

STC = Short-Run Total Cost
TFC = Total Fixed Cost
TVC = Total Variable Cost

In the table below these three cost measures have been calculated for the labor levels 1 through 11. Three other cost measures, average fixed cost, average variable cost, average total cost, and marginal cost have also been calculated in this table.

AFC = TFC/Q       AVC = TVC/Q       ATC = TC/Q        MC = ?TC/?Q

Note that all measures of cost are calculated relative to the quantity of output, not the labor input. While the average cost measures are self-explanatory, the marginal cost might need some explanation. The marginal cost is the cost producing one additional unit of output. The marginal cost is determined by two factors: wage and the marginal product of labor. For a small firm that has no influence on the wage and, thus, the wage is given, the marginal cost is simply the wage (the change in the cost resulting from hiring one additional unit of labor) divided by the marginal product of labor.

MC =  W/MPL