Elasticity

EG ,S=% ∆ G% ∆ S

EG ,S=d Gd S

×SG

A measure of the responsiveness of one variable to changes in another variable; the % change in 1 variable that arises due to a given percentage change in another. IE: the elasticity of your grade w/ respect to studying, denoted EG ,Sis the % change in your grade that will result from a given %change in the time you

spend studying. If the variable G depends on S according to the functional relationship G =∫ ( S )the

elasticity of G with respect to S may be found using calculus.Own Price Elasticity

A measure of the responsiveness of the quantity demanded of a good to a change in the price of that good; the % change in quantity demanded divided by the % change in the price of the good.Positive (+) Sign determines

the relationship between G & S.

An increase in S leads to an increase in G.

Negative (-) An increase in S leads to a decrease in G.

< 1 in Absolute Value

<> 1 determines how responsive G is to changes in S.

Numerator is > the denominator in the elasticity formula, we know a small % change in S leads to a relatively large % change in G.

> 1 in Absolute Value

Numerator is < the denominator in the elasticity formula. A given % change in S will lead to a relatively small % change in G.

Own Price Elasticity of Demand

EQ x, P x=

% ∆ Qxd

% ∆ Px

Measures the responsiveness of quantity demanded to a change in price.The OPE of Demand for a good w/ a demand function Q x

d=¿ f (Px, Py, M, H) may be found using

calculus: EQ x, P x=

∂Q xd

∂ Px

×Px

Q x

Elastic Demand |EQxy Px|>1 Demand is elastic if the absolute value of the own price elasticity is greater than 1.The quantity consumed of a good is relatively responsive to a change in the price of the good. A rise in price will reduce consumption considerably.

Inelastic Demand |EQxy Px|<1 Demand is inelastic if the absolute value of the own price elasticity is less than 1.The quantity consumed of a good is relatively unresponsive to a change in the price of the good when demand is elastic. Price increases will reduce consumption very little.

Unitary Demand |EQxy Px|=1 Demand is unitary elastic if the absolute value of the OPE is equal to 1.Advertising often provides consumers with information about the existence or quality of a product, which in turn induces more consumers to buy the product.

Total Revenue Test – When demand is:

Elastic An increase (decrease) in price will lead to a decrease (increase) in total revenue.Inelastic An increase (decrease) in price will lead to an increase (decrease) in total revenue.Unitary Elasti Total revenue is maximized at the point where demand is unitary elastic.

Example: suppose the research dept. of a computer company estimates that the OPE of demand for a laptop is -1.7.

If the company cuts prices by 5 %, will computer sales increase enough to increase overall revenues?Set:-1.7 = EQx, Px

&

-5 = %∆Px

-1.7 ¿% ∆ Qx

d

−5 Solving this equation for % ∆ Qx

dyields % ∆ Qxd=¿ 8.5. The

quantity of computers sold will rise by 8.5% if prices are reduced by 5%. Since the % increase in quantity demanded is > than the % decline in prices (

|EQxy Px|>1), the price cut will actually raise the firm’s sales revenues. Expressed differently, since demand is elastic, a price cut results in a > than proportional increase in sales and thus increases the firm’s total revenues.

Perfectly Elastic Demand

Demand is perfectly elastic if the OPE is infinite in absolute value. The demand curve is horizontal. When demand is perfectly elastic, if prices rise even slightly you will find that none of the good is purchased.

Perfectly Inelastic Demand

Demand is perfectly inelastic if the own price elasticity is zero. The demand curve is vertical. When demand is perfectly inelastic, consumers do not respond at all to changes in price.

Factors affecting the OPE

Available Substitutes

The more substitutes available for the good, the more elastic the demand for it. A price leads consumers to substitute toward another product, thus considerably the quantity demanded of the good. When there are few close substitutes for a good, demand tends to be inelastic because consumers can’t readily switch to a close substitute when the price . Demand for broadly defined commodities (food) tends to be more inelastic than the demand for specific commodities (beef).

Time Demand tends to be more inelastic in the short term than in the long term. The more time consumers have to react to a price change, the more elastic the demand for the good. Conceptually, time allows the consumer to seek out available substitutes.

Expenditure Share

Goods that comprise a relatively small share of consumers’ budgets tend to be more inelastic than goods for which consumers spend a sizable portion of their incomes. In the extreme case, where a

consumer spends her or his entire budget on a good, the consumer must decrease consumption when the price rises. In essence, there is nothing to give up but the good itself.

Marginal Revenue

The change in total revenue due to a change in output. To maximize profits a firm should produce where marginal revenue equals marginal cost. Marginal revenue is < than the price for each unit sold, to induce consumers to purchase more of a good, a firm must lower its price. IE: $5 1st unit, $4 2nd unit. 1st unit has $5 revenue ($5*1), selling both revenue is $8 ($4*2). $8-$5=$3, < $4 price. (This is elastic by total revenue test)P = Price of good, E = the own price elasticity of demand for the good.

MR=P[ 1+EE ]

−∞<E<−1 Demand is elastic, and the formula implies that MR is positive.

E = −1 Demand is unitary elastic, and marginal revenue is zero, & corresponds to the output at which total revenue is maximized.

−1<E<0 Demand is inelastic, and marginal revenue is negative.

Cross-Price Elasticity

A measure of the responsiveness of the demand for a good to changes in the price of a related good; the % change in the quantity demanded of one good divided by the % change in the price of a related good.

Denoted EQx, Py & is mathematically defined as: EQ x y, P y=

%∆ Q xd

%∆ Py

When the demand function is Qdx = f (Px, Py, M, H), the cross-price elasticity of demand between goods X and Y may be

found using calculus: EQ xy P y=

∂Q xd

∂ PY

×P y

Q x

Example If the cross-price elasticity of demand between Corel WordPerfect and Microsoft Word software is 3, a 10% hike in the price of Word will increase the demand for WordPerfect by 30%, since 30%/10% = 3. This increase in demand for WordPerfect occurs because consumers substitute away from Word and toward WordPerfect, due to the price increase.

EQ xy P y>0 Whenever goods X and Y are substitutes, an increase in the price of Y leads to an increase in the demand for X.

EQ xy P y<0 Whenever goods X and Y are complements, an increase in the price of Y leads to a decrease in the demand for X.

Example Clothing and food have a cross-price elasticity of -0.18. This means that if the price of food increases by 10%, the

demand for clothing will decrease by 1.8%; food and clothing are complements. −0.18=% ∆ Qx

d

10 %Example Every item you carry is generic (generic beer, etc.). You recently read an article in The Wall Street Journal reporting that the

price of recreation is expected to increase by 15 percent. How will this affect your store’s sales of generic food products?Answer The cross-price elasticity of demand for food and recreation is 0.15. If we insert the given information into the formula for the

cross-price elasticity, we get 0.15=% ∆ Qx

d

15 % (0.15 * 15% = 0.0225) (0.0225 * 100 = 2.25%); food and recreation are

substitutes. If the price of recreation increases by 15%, you can expect the demand for generic food products to increase by 2.25%.

Revenue of X & Y

Firm’s revenues as R = Rx + Ry ;[ Rx = PxQx denotes revenues from the sale of product X ] & [ Ry = PyQy denotes revenues from the sale of product Y ]The impact of a small % change in the price of product X (%ΔPx = ΔPx/Px) on the total revenues of the firm isΔR = [Rx(1 + EQxyPx) + RyEQy, Px] * %ΔPx

Example Restaurant earns $4,000/wk in revenues from hamburger sales (product X) and $2,000/wk from soda sales (product Y). Thus, Rx = $4,000 and Ry = $2,000.

OPEoD of burgers, EQ xy Px= -1.5 & CPEoD is EQ xy Px

= -4.0 (sodas,burgers) What would happen if burger price is cut by 1%? ΔR = [$4000(1 + (-1.5)) + $2000(-4.0)] * (-1%) = $100

Income Elasticity

A measure of the responsiveness of the demand for a good to changes in consumer income; the % change in quantity

demanded divided by the % change in income. EQ x, M=

%∆ Q xd

%∆ MEQ x, M

>0 When good X is a normal good, an increase in income leads to an increase in the consumption of X.

EQ x, M<0 When good X is an inferior good, an increase in income leads to a decrease in the consumption of X.

Example IEoD for nonfed beef = -1.94

Consumer incomes are expected to rise by 10 % over the next 3 years.

As a manager of a meat-processing plant, how will this forecast affect your purchases of non-fed cattle?

Answer EQ x, M=−1.94

% ∆ M=10 −1.94=% ∆ Qx

d

10

% ∆ Qxd=−19.4

Expect to sell 19.4% less non-fed ground beef over the next 3 years. You should decrease your purchases of non-fed cattle by 19.4%, unless something else changes.

Own Advertising Elasticity for good X defines the percentage change in the consumption of X that results from a given

of Demand percentage change in advertising spent on X.Cross-Advertising Elasticity between goods X and Y would measure the percentage change in the consumption of X that results

from a given percentage change in advertising directed toward Y.Example How much should you increase advertising to increase the demand for recreation in the United States by 15%.

EQx, Ax = 0.25. Solve: 0.25=

% ∆ Qxd

% ∆ A= 15

%∆ A The % change in advertising shows that it must increase by 60% to

increase the demand for recreation by 15%.Linear Demand Functions

If the demand function is linear and given byQ x

d=α 0+α x Px+α y Py+α M M +α H H

Own Price Elasticity EQ x, P x

=α x

Px

Q x

Cross-Price Elasticity EQ x, P y

=α x

Py

Q x

Income Elasticity EQ x, M

=α MMQ x

The elasticities for a linear demand curve may be found using calculus. Specifically, and similarly for the cross-price and income elasticities.

EQ x, P x=

∂Q xd

∂ Px

×Py

Q x

=α x

Px

Q x

Q xd=100−3 Px+4 P y−0.01 M +2 Ax

Q xd=100−3 (25 )+4 (35 )−0.01 (20,000 )+2 (50 )=65 units

Suppose good X sells at $25/pair, good Y at $35/pair, Zappos utilizes 50 units of advertising, and average consumer income is $20,000. Calculate the own price, cross-price, and income elasticity of demand.

Own Price Elasticity

EQ x, P x=−3

2565

=−1.15If they raise shoe prices, the % decline in the quantity demanded of its shoes will be > in absolute value than the % rise in price. Consequently, demand is elastic: Total revenues will fall if it raises shoe prices.

Cross-Price Elasticity

EQ x, P y=4

3565

=2.15Since this is positive, good Y is a substitute for them.

Income Elasticity EQ x, M

=−0.0120000

65=−3.0 8

Zappos’ shoes are inferior goods, since this is a negative number.

Non-Linear Demand Functions

If the demand function is not a linear function & is given byQ x

d=c P xBx P y

B y M BM H BH ; C is a constantLog-linear Demand: Demand is log-linear if the logarithm of demand is a linear function of the logarithms of prices, income, and other variables. ln Q x

d=β0+βx ln Px+¿ β y ln P y+β M ln M +β H ln H ¿β0=ln(c ), & bi’s are arbitrary #’s

Own Price Elasticity

EQ x, P x=βx

Cross-Price Elasticity

EQ x, P y=β y

Income Elasticity

EQ x, M=βM

CalculusEQ xy Px

=∂ Qx

d

∂ Px( Py

Qx)=βx cPx

B x−1 PyB y M BM H BH ( Px

c PxBx Py

B y MBM H

BH )=β x & Similarly for the cross-price & income

ln Q xd=10−1.2 ln Px+¿3 ln R−2 ln A y ¿

R denotes the daily amount of rainfall and Ay represents the level of advertising on good Y.

What would be the impact on demand of a 10% increase in the daily amount of rainfall?We know that for log-linear demand functions, the coefficient of the logarithm of a variable gives the elasticity of demand with respect to that variable. Thus, the elasticity of demand for raincoats with respect to rainfall is

EQ x, R=βR=3

EQ x, R=βR=

% ∆Q xd

% ∆ R

3=%∆ Q x

d

10% ∆ Qx

d=3 0

10% increase in rainfall will lead to a 30% increase in the demand for raincoats.

What would be the impact of a 10% reduction in the amount of advertising directed toward good Y?

EQ x, A y=β A y

=−2

EQ x, A y=β A y

=% ∆ Q x

d

% ∆ Ay

−2=%∆ Q x

d

−10% ∆ Qx

d=20

10% reduction in advertising directed toward good Y leads to a 20% increase in the demand for raincoats.

Can you think of a good that might be good Y in this example?Perhaps good Y is umbrellas, for one would expect the demand for raincoats to increase whenever fewer umbrella advertisements are made.

Econometrics The statistical analysis of economic phenomena.Regression Line The line that minimizes the squared deviations btwn. the line (the expected relation) and the actual data pts.

Least Squares Regression The line that minimizes the sum of squared deviations between the line and the actual data points.

Standard Error

Of each estimated coefficient is a measure of how much each estimated coefficient would vary in regressions based on the same underlying true demand relation, but with different observations. The smaller the SE of an estimated coefficient the smaller the variation in the estimate given data from different outlets (samples of data)

The least squares regression line for the equationY = a + bX + eis given by

Y = aˆ + bˆ X

The parameter estimates, â & bˆ, represent the values of a & b that result in the smallest sum of squared errors btwn a line & the actual data.

Rule of Thumb for a 95 Percent Confidence Interval

If the parameter estimates of a regression equation are â and bˆ, the 95 percent confidence intervals for the true values of a and b can be approximated by

aˆ ± 2σ aˆ & bˆ ± 2σ bˆ where σ aˆ and σ b

ˆare the standard errors of â, and bˆ, respectively.t-statistic The ratio of the value of a parameter estimate to the standard error of the parameter estimate.

IE: If the parameter estimates are aˆ & bˆ and the corresponding standard errors are σ aˆ & σ bˆ , the t-

statistic for aˆ is t a ˆ=aˆσa ˆ

and the t-statistic for bˆ is t b ˆ=bˆσb ˆ

Rule of Thumb for Using t-Statistics

When the absolute value of the t-statistic is greater than 2, the manager can be 95% confident that the true value of the underlying parameter in the regression is not zero.

R-square (aka coefficient of determination)

Tells the fraction of the total variation in the dependent variable that is explained by the regression. It is computed as the ratio of the sum of squared errors from the regression (SSRegression) to the total sum of squared errors (SSTotal):

R2= Explained VariationTotalVariation

=SSRegres sion

SSTotal

IE :301470.89402222.50

=0.75 Value of an R-square from 0 to 1

This means that the estimated demand equation (the regression line) explains 75 percent of the total variation in TV sales across the sample of 10 outlets.

Closer the R-square is to 1, the “better” the overall fit of the estimated regression equation to the actual data. Unfortunately, there is no simple cutoff that can be used to determine whether an R-square is close enough to 1 to indicate a “good” fit. With time series data, R-squares are often in excess of .9; with cross-sectional data, .5 might be a reasonably good fit. A major drawback of the R- square is that it is a subjective measure of goodness of fit. Sometimes, the R-square is very close to 1 merely because the number of observations is small relative to the # of estimated parameters. This situation is undesirable because it can provide a very misleading indicator of the goodness of fit of the regression line.

For this reason, many researchers use the adjusted R-square as a measure of goodness of fit.

R2=1−(1−R2) (n−1)(n−k ) , where n is the total # of observations & k is the # of estimated coefficients

In performing a regression, the # of parameters to be estimated can’t exceed the # of observations. n – k represents the residual degrees of freedom (that is, estimating numerous coefficients from relatively few observations) after conducting the regression.IE: n=10,k=2 1 - (1 - .75) (9/8) = .72 =R2 ; There is little difference between the R-square and the adjusted R-square, so it does not appear that the “high” R- square is a result of an excessive # of estimated coefficients relative to the sample size.F-statistic An alternative measure of goodness of fit. Provides a measure of the total variation explained by the regression

relative to the total unexplained variation. The greater the F-statistic, the better the overall fit of the regression line through the actual data. IE: F-statistic = .0012, there is only a .12 % chance that the estimated regression model fit the data purely by accident. As with P-values, the lower the significance value of the F-statistic, the more confident you can be of the overall fit of the regression equation. Regressions that have F-statistics with significance values of 5% or less are generally considered significant. Based on the significance value reported in cell 26-F of Table 3–8, our regression is significant at the .12 percent level. The regression is therefore highly significant.

Sometimes, a plot of the data will reveal nonlinearities in the data. It appears that price and quantity are not linearly related: The demand function is a curve. The log-linear demand curve we examined earlier has such a curved shape.

To estimate a log-linear demand function, the econometrician takes the natural logarithm of prices and quantities before executing the regression routine that minimizes the sum of squared errors (e):Q’ = β0 + βPP’ + e ; where Q’ = Q ln and P’ = P ln