Chapter 4. the Theory of Demand

Chapter 4. The Theory of Demand

Part A, © 2002, Kwan Choi

Optimal consumption bundle

A consumer’s problem is to choose chose x1 and x2 to maximize U(x1,x2) subject to the budget constraint, M = p1x1 + p2x2. Recall from the previous chapter that an optimal consumption bundle satisfies the condition that the slopes of the budget constraint and the highest attainable indifference must be the same. From this condition, we obtain demand functions for the two goods, which can be written:

and

Note that there are three parameters, p1, p2, and M in the budget constraint, but no other parameters appear in the utility function. Lack of parameters of utility function indicates that the demand functions are derived from a general utility function. This is why only these three parameters appear in the above demand functions. If a special utility function were used, its parameters will appear in the demand functions.

Next, we see how these parameters affect demand curves.

Income Consumption Curve.

An Engel Curve shows the relationship between demand and income.

A good is normal if an increase in income raises its consumption.

A good is inferior if an increase in income lowers its consumption.

Income elasticity of demand for x

Normal goods: income elasticities are positive.

Inferior goods: income elasticities are negative.

Normal good: As M increases, X increases.

Inferior good: Not all goods can be inferior goods.

Price consumption Curve and Demand Curve

Income and Substitution Effects

Hicksian Decomposition

Initial equilibrium is at point a. Suppose the price of good 1 declines, which causes a counterclockwise rotation of the budget line.

The substitution effect measures the change in consumption, holding utility constant, in response to a change in the relative price, This is measured by Arrow #1 in Figure 5.

Because of this decline in the relative price, it does not cost as much to maintain the same level utility as before. Since the actual money income M has not changed, this price decline results in an increase in a sort of “real income.” The fictitious budget line at point b, is parallel to the new budget line. The gap between the two budget line shows the increment of real income. Given this increment, the consumer moves from point b to c. This movement is the income effect. The total effect is the sum of the two effects.

Slutsky Decomposition of the price change.

Δx1/Δp1 = Δx1/Δp1│u - x1(Δx1/ΔM)

In the case of an inferior good, the increment in “real income” causes a decline in consumption. In a rare case, this negative income effect can more than offset the initial substitution effect. Figure 6 illustrates such a case where consumption of the good actually declines (point c is to the left of point a) in response to a decline in its price.

Compensated and Marshallian (uncompensated) demand curves

Marshallian demand: M is fixed

Compensated demand: U is fixed.

In Figure 7a, Marshallian demand curve is denoted by DM and the compensated demand curve by Dc. The substitution effect is always negative. Thus, the compensated demand curve is always negatively sloped. Marshallian demand curve is not always negatively sloped. If the demand curve for a good is positively sloped, we call it a Giffen good. Thus, we can say that the Marshallian demand curve is negatively sloped, except for the Giffen good. However, Giffen good is rarely observed. There was a great famine in Ireland during the 19th century. Their staple diet consisted mainly of potato and meat. During the famine, the price of potato increased and the Irish farmers consumed more potatoes. We assume that demand curve is negatively sloped from now on.

In the case of a normal good, Marshallian demand curve is always flatter than the compensated demand curve. For an inferior good, the reverse relationship is true.

Cost of Living Adjustment

Prices change over time. Sometimes a manager or a vice president of a multinational company assigned to move to a foreign city receives a long term contract which includes a cost of living adjustment based on the consumer price index (CPI).

Managers of multinational firms are often reluctant to move to foreign cities. In addition to coping with a different culture, they are facing different prices in another city or country. The idea is to allow managers to maintain their lifestyle and induce them to accept assignments in foreign cities. It is necessary then to adjust their income so that with the new income the manager is able to afford the previous consumption bundle at the time the contract is made.

Price decline

The initial equilibrium is at point a. If the price of good 1 declines, a cost of living adjustment requires a pay cut, after which the consumer is still able to purchase the original consumption bundle a. But once the compensation is made, he has no obligation to stay at the initial equilibrium. He does not stay there, but moves to point b, which is on a higher indifference curve U '.

Price increase

The initial equibrium  a. An increase in the price of good 1 rotates the budget line from 1 to 2. In this case, budget line 2’ is requires to purchase the original consumption bundle a. This difference between the two budget lines (2’ – 2) is the cost of living adjustment. But will the consumer stay at point a? No. he moves to a better point, b.

These two diagrams demonstrate that cost of living adjustment overcompensates the consumer, and makes him better off.

Price Decline  Income may be cut for the cost of living adjustment.

Price Increase  Income must be increased for the cost of living adjustment.

Market Demand Curve

Market Demand: a horizontal summation of individual demand curves.

Normally, a business firm does not care whether more demand comes from James Bond or from Lady Bush. The firms are concerned with the market demand curve.

Market demand is also negatively sloped, since the individual demand curves are negatively sloped. At lower prices consumers demand more.

Now let us turn our attention to the sensitivity of the relationship. Price sensitivity of demand may be greater for some goods than for others. Even for the same product, sensitivity changes along the demand curve.

ΔQ cannot be compared among different demands. For example, as price declines by $1, ΔQ = 2 for automobiles, but ΔQ = 3 bushels for corn. We cannot say that ΔQ = 2 cars is bigger than ΔQ = 3 bushels. To compare sensitivity of changes in demand, we need a relative concept.

PED relates to percentage change in quantity demanded to percentage change in price. It is a unit free measure, and it can be compared across commodities. For this reasons, we can compare the elasticities of demand for automobiles in Germany and in Japan, although their demand curves are expressed in terms of mark and yen prices.

Measurement of PED

Point elasticity (Є)  use initial price and quantity

Є  =

Next,

Є  =

Thus, PED can change dramatically, depending on what point is used as the basis. This example shows that point elasticity is reliable only for small price changes, but not for large price changes. To avoid this problem, we sometimes use arc elasticity, which is measured in a price interval.

Arc elasticity  use average price and average quantity.

Using, the above example, we have

Є =

This means that revenue remains unchanged by price changes.

DEMAND FOR MODEL T

In 1905, the average automobile produced in the US cost more than the average Japanese car in 1980. Many automobile firms hired skilled mechanics and produced handcrafted automobiles. However, simplicity and automation were the key to the automobile industry’s future. Henry Ford introduced assembly line to automobile production. He introduced model T in 1909. The following table shows the success of Model T. (Source lost)

Fill out the blanks:

The record sale between 1910 and 1920 confirmed the wisdom of Chinese saying (Low margin sells volumes) That is, it is better to sell many cars at a small margin than to sell few cars at a high margin.

This statement is true only if demand is sufficiently elastic.

Elasticity Categories:

Є = .5 means that as price declines by 1%, quantity demanded increases by 0.5%.

Actual Quantity changed

Q = Є P.

Special Cases:

Є > 1  demand is price elastic.

Є < 1  demand is price inelastic.

Є = 1 demand is unitary price elastic. (In this case, the demand curve is a rectangular hyperbola. The area of the square generated by any point on the demand curve is the same.)

Є = demand is perfectly elastic, horizontal demand curve.

Є = 0  demand is perfectly inelastic, a vertical demand curve.

Graphical Measurement of Point Elasticity of Demand

To measure the price elasticity of demand at a given point C on the demand curve, draw a tangent line AB at point C.

ΔQ = AQ*, ΔP = OP*.

ΔQ/ΔP = AQ*/OP*.

P/Q = OP*/OQ*

Then Є = AQ*/OQ* = AC/BC

Linear Demand and PED

A point on a linear demand curve bisects it. When the top and the bottom segments are of equal length, (e.g., AC = CE), PED = 1. As one moves upward from that point on the demand curve, the bottom segment becomes longer than the top, and hence PED increases, approaching  at point A. On the other hand, a downward movement along the demand curve reduces the bottom segment, approaching zero at point E. (The book expresses elasticities as negative numbers.)

TOTAL REVENUE AND ELASTICITY

There is a straightforward relationship between TR and PED. TR is defined by PQ, and is described by the rectangular area generated by a point on a given demand curve.

Useful Formula:

Let Z = XY, and let denote a percentage change in X.

Next, Z changes to Z '. Then

Thus,

If the percentage changes and are small, then is even smaller. For instance: if and = 0.03, then Thus, for small percentage changes, it is safe to ignore the term, Then we have the following formulas, which are useful for ready estimation.

Using these formulas, we have

(1) Inelastic Demand (є < 1)  and have the same sign. That is, price and revenue move in the same direction.

Example: є = 0.5, and = - 10%. Then

(2) Elastic Demand (є > 1)  and have the opposite signs. Price and revenue move in the opposite direction.

Example: є = 2 = - 10%. Then

(3) Unitary Elastic Demand (є = 1).

= 0. That is, revenue is unaffected by price change.

Gains and losses from a price decline

In Figure 10, at point A, demand is infinitely elastic. Thus, a price decline will increase revenue.

At point B, PED is 3, and demand is elastic. A price decline still increases revenue.

At point C, demand is unitary elastic. At this point revenue is maximized.

At point D, demand is inelastic, and a price decline reduces revenue.

At point E, demand is perfectly inelastic. Hence, a price increase will increase revenue.

OVER A BUCK FOR DINNER? OUTRAGEOUS

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