Demand curve: Difference between revisions

Revision as of 14:04, 14 August 2010

Definition

The demand curve for a good, service, or commodity, is defined with the following in the background:

The specific good, service, or commodity.
A unit for measuring the quantity of that commodity.
A unit for measuring price.
A certain set of economic actors who are the potential buyers of that commodity.
A time frame within which the demand is measured.
An economic backdrop that includes all the determinants of demand other than the unit price of that commodity.

The demand curve is a curve drawn with:

The vertical axis is the price axis, measuring the price per unit of the commodity.
The horizontal axis is the quantity axis, measuring the quantity of the commodity demanded in total by all the economic actors chosen above.

Note that the demand curve makes sense only ceteris paribus -- all other determinants of demand being kept constant.

A demand schedule is a discrete version of the demand curve, specifying demand values for a number of different prices.

Individual versus aggregate demand curve

Terminology

Individual demand curves are demand curves for a single economic actor. This actor could be an individual, a household, or a firm (where the firm may be a for-profit, a non-governmental non-profit, or a governmental agnecy).

Aggregate demand curves aggregate (or add up) the demand curves for a number of economic actors. For instance, the aggregate household demand curve for a good in a town is obtained by adding up the demand curves for all the households in the demand curve. The aggregate demand curve for steel in the automobile industry is obtained by adding up the demand curves for steel for all firms in the automobile industry.

Qualitative distinctions

Here individual could mean individual, household, or firm -- any single economic actor.

A lot of interesting and quirky phenomena may be obtained at the level of individual demand curves but may become less visible (due to smoothing and averaging out) at the aggregate level because of the canceling out or smoothing out effects. Some examples are discussed below:

For items where purchase quantities are discrete, individual demand curves are by nature discontinuous, while aggregate demand curves are likely to be continuous given sufficient heterogeneity among individuals. Note that individual demand quantities could be fractional even with discrete purchase quantities -- for instance, my weekly number of loaves of bread purchased could be $3.5$ if I purchase a loaf of bread every second day.
Individual demand curves are more likely to exhibit sharp discontinuities for other reasons: Individuals may use threshold prices and reference prices to determine which item to purchase and how much. For instance, if, for me, $A$ and $B$ are equivalent goods (i.e., they are perfect substitute goods for each other, then I buy none of $A$ when its price exceeds that of $B$ , but I shift my entire consumption to $A$ when its price drops below that of $B$ . The price of $B$ is thus a point of discontinuity in the demand curve. In the aggregate, the heterogeneity of individuals ensures that they do not all perceive the same pairs of goods as perfect substitutes, and hence these jumps are less likely to occur.
Various violations of the law of demand, both rational and irrational, are more likely to be seen at the individual level than at the aggregate level: For instance, the Giffen good phenomenon and the Veblen good phenomenon may play an important role in the consumption behavior of one individual or household, but because of differing incomes and differing tastes and preferences that lead individuals to value substitutes differently, the phenomena would not apply to all economic actors. Since an aggregate Giffen good phenomenon depends on the phenomenon affecting a large number of individuals, aggregate Giffen good phenomena may be much rarer than individual Giffen good phenomena. The same holds for various forms of mild irrationality and idiosyncratic behavior.

Examples of demand curves

This gives some mathematical possibilities for the demand curve. The list is illustrative and not intended to be exhaustive.

Demand curve for fixed total budget: reciprocal relationship between price and quantity

Here, the individual or household spends a fixed amount of money on the commodity regardless of its price. Thus, the quantity purchased is inversely proportional to the unit price, i.e., the demand curve equation is given by:

$q = \frac{b}{p}$

where $b$ is the total budget amount that the household spends. Here are some properties of such a demand curve:

The price-elasticity of demand for this demand curve is $- 1$ .
As the price falls to $0$ , demand goes to infinity. An actual demand of $\infty$ (infinity) may be infeasible, hence the above expression may not work for very low prices. However, such a demand curve may well explain what appears to be a zero price effect.
For the quantity demanded to become zero, the price needs to be raised to infinity. Again, this may not be the case in reality, since with a sufficiently high price, the quantity demanded may be so low that the transaction costs dominate the picture.

A slight variant of the above is:

$q = \frac{b}{p + p_{t}}$

where $p_{t} > 0$ is the transaction cost (which we assume as a cost per unit quantity purchased). This variant avoids the problem of infinities at one end: as the price falls to $0$ , demand goes to $b / p_{t}$ . In particular, for small transaction costs, this demand is extremely high.

Yet another variant involves two kinds of transaction costs: a transaction cost per unit, and a transaction cost that is constant for all nonzero amounts purchased and $0$ if nothing is purchased. If we denote these two transaction costs as $p_{t}$ (per unit) and $p_{f}$ (fixed) then:

$q = \frac{b - p_{f}}{p + p_{t}}$

Yet another variant is where there is a minimum quantity that can be demanded if any nonzero quantity is demanded, which we call $q_{min}$ . In this case, we first use the formula to determine quantity demanded. If the quantity demanded is less than $q_{min}$ , we replace it by $0$ .

Linear demand curve

Fill this in later

Curve characteristics

Slope or first derivative

The slope (or rate of change) of the demand curve is inversely related of what is termed the price-elasticity of demand. The price-elasticity measures the total change in quantity demanded per unit change in price, made dimensionless by dividing by the quantity-price ratio. In other words, at a price $p$ and quantity $q$ , the price-elasticity of demand is:

$\frac{d q / d p}{q / p}$

The slope in mathematical jargon would be:

$\frac{d p}{d q}$

Sign of slope

Further information: Law of demand

Typically, the price-elasticity of demand is negative, which is equivalent to saying that the slope of the demand curve is negative, or the demand curve is downward-sloping. In other words, ceteris paribus:

A decrease in the price per unit leads to an increase in the total quantity demanded.
An increase in the price per unit leads to a decrease in the total quantity demanded.

This is termed the law of demand. There are two broad explanations for the law of demand:

Concavity of the utility functions of individuals and households: the marginal utility of an individual or household per unit of a good decreases with the amount already purchased. Also, the income effect and the substitution effect. Further information: Law of demand for individual buyers follows from diminishing marginal utility
Heterogeneity in households, i.e., differences in the reservation prices across households. Further information: Law of demand for multiple buyers follows from differences in reservation prices

An exception to the law of demand is Veblen goods, whose demand is related to conspicuous consumption. Other goods that appear to violate the law of demand (though they do not directly violate it) are Giffen goods and certain kinds of goods where there is inadequate information about quality and a higher price may be taken as a signal of higher quality.

Sign of second derivative

There is no general principle that uniformly predicts the sign of the second derivative; in other words, there is no overarching rule about whether the price-elasticity of demand rises or falls with price.

In general, there are certain price ranges where the price-elasticity of demand is particularly high. These, in general, tend to be the price ranges where the substitution effect operates most strongly, i.e., the price ranges where there could be significant changes in the extent to which to substitute towards and away from substitutes. Alternatively, these are the price ranges where the reservation prices of many households are clustered.

In general below these price ranges, demand for the commodity is high, but largely price-inelastic, while above these price ranges, demand for the commodity is low, but again largely price-inelastic.

In this model of some critical price ranges, the sign of the second derivative is not constant.

In general, the behavior of the second derivative depends on the following:

At the level of individual households, it depends on the second derivative of the marginal utility curve, or the third derivative of the utility curve.
At the level of an economy, it depends on the distribution of income as well as of tastes and preferences in the economy.

Movement along the curve

Any particular price corresponds to a point on the demand curve: the price coordinate of the point is that price, while the quantity coordinate is the quantity demanded at the price. Changes in price, while keeping other factors constant, is termed movement along the demand curve. Decreasing price with time is termed riding down the demand curve (also called price skimming) while increasing price with time is termed riding up the demand curve.

In a free market, suppliers are expected to vary their price until demand equals supply. In other words, demand moves along the demand curve until the price reaches the market price.

Movement of the curve

The demand curve changes when one (or more) of the determinants of demand other than price changes.

Outward shift of the demand curve

An outward shift of the demand curve (also called an expansion of the demand curve) indicates an increase in the demand at every price, or equivalently, an increase in the price needed to restrict demand to a particular level.

Inward shift of the demand curve

An inward shift of the demand curve (also called a contraction of the demand curve) indicates a decrease in the demand at every price, or equivalently, a decrease in the price needed to restrict demand to a particular level.

Other shape changes of the demand curve

The demand curve may change in shape in a way that moves it inward for some price levels and outward for other price levels. This may happen, for instance, with changes in the income distribution.

@@ Line 7: / Line 7: @@
 * A unit for measuring price.
 * A certain set of economic actors who are the potential buyers of that commodity.
+* A time frame within which the demand is measured.
 * An economic backdrop that includes all the [[determinants of demand]] ''other than'' the unit price of that commodity.
@@ Line 18: / Line 19: @@
 A [[demand schedule]] is a discrete version of the demand curve, specifying demand values for a number of different prices.
+==Individual versus aggregate demand curve==
+===Terminology===
+Individual demand curves are demand curves for a ''single'' economic actor. This actor could be an individual, a household, or a firm (where the firm may be a for-profit, a non-governmental non-profit, or a governmental agnecy).
+Aggregate demand curves aggregate (or add up) the demand curves for a number of economic actors. For instance, the aggregate household demand curve for a good in a town is obtained by adding up the demand curves for all the households in the demand curve. The aggregate demand curve for steel in the automobile industry is obtained by adding up the demand curves for steel for all firms in the automobile industry.
+===Qualitative distinctions===
+Here ''individual'' could mean individual, household, or firm -- any single economic actor.
+A lot of interesting and quirky phenomena may be obtained at the level of individual demand curves but may become less visible (due to smoothing and averaging out) at the aggregate level because of the canceling out or smoothing out effects. Some examples are discussed below:
+* For items where purchase quantities are discrete, individual demand curves are by nature discontinuous, while aggregate demand curves are likely to be continuous given sufficient heterogeneity among individuals. Note that individual demand quantities ''could'' be fractional even with discrete purchase quantities -- for instance, my weekly number of loaves of bread purchased could be <math>3.5</math> if I purchase a loaf of bread every second day.
+* Individual demand curves are more likely to exhibit sharp discontinuities for other reasons: Individuals may use threshold prices and [[reference price]]s to determine which item to purchase and how much. For instance, if, for me, <math>A</math> and <math>B</math> are equivalent goods (i.e., they are perfect [[substitute good]]s for each other, then I buy none of <math>A</math> when its price exceeds that of <math>B</math>, but I shift my entire consumption to <math>A</math> when its price drops below that of <math>B</math>. The price of <math>B</math> is thus a point of discontinuity in the demand curve. In the aggregate, the heterogeneity of individuals ensures that they do not all perceive the same pairs of goods as perfect substitutes, and hence these jumps are less likely to occur.
+* Various violations of the law of demand, both rational and irrational, are more likely to be seen at the individual level than at the aggregate level: For instance, the [[Giffen good]] phenomenon and the [[Veblen good]] phenomenon may play an important role in the consumption behavior of one individual or household, but because of differing incomes and differing tastes and preferences that lead individuals to value substitutes differently, the phenomena would not apply to all economic actors. Since an aggregate Giffen good phenomenon depends on the phenomenon affecting a large number of individuals, aggregate Giffen good phenomena may be much rarer than individual Giffen good phenomena. The same holds for various forms of mild irrationality and idiosyncratic behavior.
+==Examples of demand curves==
+This gives some mathematical possibilities for the demand curve. The list is illustrative and not intended to be exhaustive.
+===Demand curve for fixed total budget: reciprocal relationship between price and quantity===
+Here, the individual or household spends a fixed amount of money on the commodity regardless of its price. Thus, the quantity purchased is inversely proportional to the unit price, i.e., the demand curve equation is given by:
+<math>\! q = \frac{b}{p}</math>
+where <math>b</math> is the total budget amount that the household spends. Here are some properties of such a demand curve:
+* The [[price-elasticity of demand]] for this demand curve is <math>-1</math>.
+* As the price falls to <math>0</math>, demand goes to infinity. An actual demand of <math>\infty</math> (infinity) may be infeasible, hence the above expression may not work for very low prices. However, such a demand curve may well explain what appears to be a [[zero price effect]].
+* For the quantity demanded to become zero, the price needs to be raised to infinity. Again, this may not be the case in reality, since with a sufficiently high price, the quantity demanded may be so low that the [[transaction cost]]s dominate the picture.
+A slight variant of the above is:
+<math>\! q = \frac{b}{p + p_t}</math>
+where <math>p_t > 0</math> is the transaction cost (which we assume as a cost per unit quantity purchased). This variant avoids the problem of infinities at one end: as the price falls to <math>0</math>, demand goes to <math>b/p_t</math>. In particular, for small transaction costs, this demand is extremely high.
+Yet another variant involves two kinds of transaction costs: a transaction cost per unit, and a transaction cost that is constant for all nonzero amounts purchased and <math>0</math> if nothing is purchased. If we denote these two transaction costs as <math>p_t</math> (per unit) and <math>p_f</math> (fixed) then:
+<math>\! q = \frac{b - p_f}{p + p_t}</math>
+Yet another variant is where there is a minimum quantity that can be demanded if any nonzero quantity is demanded, which we call <math>q_\min</math>. In this case, we first use the formula to determine quantity demanded. If the quantity demanded is less than <math>q_\min</math>, we replace it by <math>0</math>.
+===Linear demand curve===
+{{fillin}}
 ==Curve characteristics==
 ===Slope or first derivative===
-The slope (or rate of change) of the demand curve is the reciprocal of what is termed the [[price-elasticity of demand]]. The price-elasticity measures the total change in quantity demanded per unit change in price. (The reciprocal needs to be taken because slope, as defined mathematically, measures the rate at which the vertical coordinate changes with respect to the horizontal coordinate, while price-elasticity measures the rate at which the horizontal coordinate changes with respect to the vertical coordinate).
+The slope (or rate of change) of the demand curve is inversely related of what is termed the [[price-elasticity of demand]]. The price-elasticity measures the total change in quantity demanded per unit change in price, made dimensionless by dividing by the quantity-price ratio. In other words, at a price <math>p</math> and quantity <math>q</math>, the price-elasticity of demand is:
+<math>\! \frac{dq/dp}{q/p}</math>
+The slope in mathematical jargon would be:
+<math>\! \frac{dp}{dq}</math>
 ===Sign of slope===
@@ Line 35: / Line 91: @@
 This is termed the [[law of demand]]. There are two broad explanations for the law of demand:
-* Concavity of the utility functions of individuals and households: the marginal utility of an individual or household per unit of a good decreases with the amount already purchased. {{further|[[Law of demand for individual buyers follows from diminishing marginal utility]]}}
+* Concavity of the utility functions of individuals and households: the marginal utility of an individual or household per unit of a good decreases with the amount already purchased. Also, the [[income effect]] and the [[substitution effect]]. {{further|[[Law of demand for individual buyers follows from diminishing marginal utility]]}}
 * Heterogeneity in households, i.e., differences in the [[reservation price]]s across households. {{further|[[Law of demand for multiple buyers follows from differences in reservation prices]]}}
@@ Line 67: / Line 123: @@
 ===Outward shift of the demand curve===
-An ''outward'' shift of the demand curve indicates an increase in the demand at every price, or equivalently, an increase in the price needed to restrict demand to a particular level.
+An ''outward'' shift of the demand curve (also called an ''expansion'' of the demand curve) indicates an increase in the demand at every price, or equivalently, an increase in the price needed to restrict demand to a particular level.
 ===Inward shift of the demand curve===
-An ''inward'' shift of the demand curve indicates a decrease in the demand at every price, or equivalently, a decrease in the price needed to restrict demand to a particular level.
+An ''inward'' shift of the demand curve (also called a ''contraction'' of the demand curve) indicates a decrease in the demand at every price, or equivalently, a decrease in the price needed to restrict demand to a particular level.
 ===Other shape changes of the demand curve===
 The demand curve may change in shape in a way that moves it ''inward'' for some price levels and outward for other price levels. This may happen, for instance, with changes in the income distribution.