Economics of the Firm

Economics of the Firm

Consumer Demand Analysis

Demand relationships are based off of the theory of consumer choice. We can characterize the average consumer by their utility function.

HLU ,

“Utility” is a function of lemonade and hot dogs

Consumers make choices on what to buy that satisfy two criteria:

L

L

H

H

P

MU

P

MU

ILPHP LH

Their decision on what to buy generates maximum utility

Their decision on what to buy generates is affordable

IPPDQ LHH ,,These decisions can be represented by a demand curve

Example: Suppose that you have $10 to spend. Hot Dogs cost $4 apiece and glasses of lemonade cost $2 apiece.

# Hot Dogs MU (Hot Dogs)

# Lemonade MU (Lemonade)

1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

2

4

4

8 ILPHP LH

210,2,4 DQH

L

L

H

H

P

MU

P

MU 101224

This point satisfies both conditions and, hence, is one point of the demand curve

Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still have $10 to spend)



1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

2

4

6

8 ILPHP LH

L

L

H

H

P

MU

P

MU 101226

Your decision at the margin has been affected. You need to buy less hot dogs and more lemonade (Substitution effect)

You can’t afford what you used to be able to afford – you need to buy less of something! (Income effect)

Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still have $10 to spend)



1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

2

3

6

9 ILPHP LH

L

L

H

H

P

MU

P

MU 102216

110,2,6 DQH


Demand curves slope downwards – this reflects the negative relationship between price and quantity. Elasticity of Demand measures this effect quantitatively

Quantity

Price

$4.00

2

10$ID

$6.00

1

%50100*2

21

%50100*4

46

150

50

%

%

P

QD

Arc Elasticity vs. Price Elasticity

At the heart of this issue is this: “How do you calculate a percentage change?

Suppose that a variable changes from 100 to 125. What is the percentage increase?

%25100*100

100125%

%20100*125

100125%

- OR -

Note: This discrepancy wouldn’t be a big deal if these two points weren’t so far apart!

Consider the following demand curve:

Quantity

Price

$10.00

40

10$ID

$12.00

32

PQ 480

P

QD

%

%

Arc Elasticity

%22100*36

4032%

Q

%18100*11

1012%

P

2.118

22

D

36

11

Q

P

Consider the following demand curve:

Quantity

Price

$10.00

40

10$ID

$12.00

32

PQ 480

P

QD

%

%

Point Elasticity

%20100*40

4032%

Q

%20100*10

1012%

P

120

20

D

If demand is linear, the slope is a constant, but the elasticity is not!!

Quantity

Price

$10.00

40

10$ID

$18.00

8

PQ 480

Q

P

P

Q

P

QD %

%

$2.00

72

98

184

D

140

104

D

11.72

24

D

If demand is linear, the slope is a constant, but the elasticity is not!!

Quantity

Price

$10.00

40

10$ID

$18.00

8

PQ 480

$2.00

72

Low prices = Low Elasticities

High prices = High Elasticities

Unit Elasticity

9D

1D

11.D

PQ 480

If you are interested in maximizing revenues, you are looking for the spot on the demand curve where elasticity equals 1.

PQPTRPQTR %1%%% R

even

ues

Price

1D 1D

Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to spend.



1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

2

4

4

8 ILPHP LH

L

L

H

H

P

MU

P

MU 201224

Your decision at the margin is unaffected, but you have some income left over (this is a pure income effect)

Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to spend.



1 9 1 4

2 8 2 3

3 7 3 1.5

4 6 4 1

5 5 5 .5

2

3

4

6 ILPHP LH

L

L

H

H

P

MU

P

MU 202244

420,2,4 DQH


For any fixed price, demand (typically) responds positively to increases in income. Income Elasticity measures this effect quantitatively

Quantity

Price

$4.00

2

10$ID

4

%100100*2

24%

Q

%100100*10

1020%

I

1100

100

%

%

I

QD

20$ID

Cross price elasticity refers to the impact on demand of another price changing

Quantity

Price

$4.00

2

2$LPD

6

%200100*2

26%

Q

%100100*2

24%

LP

2100

200

%

%

L

HL P

Q

4$LPD

Note: These numbers aren’t coming from the previous example!!

A positive cross price elasticity refers to a substitute while a negative cross price elasticity refers to a compliment

Time

Dem

and

Fact

ors

t t+1t-1

Cross Sectional estimation holds the time period constant and estimates the variation in demand resulting from variation in the demand factors

For example: can we predict demand for Pepsi in South Bend by looking at selected statistics for South bend

Estimating Cross Sectional Demand Curves

Lets begin by estimating a basic demand curve – quantity demanded is a function of price.

XX PDQ

Next, we need to assume a functional form. For simplicity, lets start with a linear model

XX PaaQ 10

0

0.5

1

1.5

2

2.5

3

3.5

4

0 20 40 60 80 100 120 140 160 180

Pric

e

Quantity

Next, Collect Data on Prices and Sales

Regression Results

Variable Coefficient Standard Error t Stat

Intercept 47.996 3.004 15.977

Price (X) -10.04 .774 -12.967

XX PQ 1048 That is, we have estimated the following equation

Regression Statistics

R Squared .782

Standard Error 10.02

Observations 250

XX PQ 1048

ValuesPrice of X $2.50

Average Price of X $5

(3.004) (0.774) (10.02)

90.902.10774.5)50.2(2)50.2()774(.)004.3( 22222 StdDev

23)50.2(1048 Mean

Our forecast of demand is normally distributed with a mean of 23 and a standard deviation of 9.90.

XX PQ 1048

xxx

x

x

x

x

p

x

p

p

x

p

x

px

10

10

10

If we want to calculate the elasticity of our estimated demand curve, we need to specify a specific point.

XP

XQ 2350.21048

50.2$

XQ

P$2.50

23

09.123

50.210

x

Given our model of demand as a function of income, and prices, we could specify a variety of functional forms

xpax 40

Linear Demand Curves Here, quantity demanded responds to dollar changes in price (i.e. a $1 increase in price lowers demand by 4 units.

x

p

x

p

p

x

p

x

px

xx

x

x

x

x

4

4

4

XP

XQ

xd pax ln4ln 0


Semi Log Demand CurvesHere, quantity demanded responds to percentage changes in price (i.e. a 1% increase in price lowers demand by 4 units.

xxp

x

p

x

px

xx

x

x

14

1

%

4%

ln4

xx pp %ln

XP

XQ

xd pax 4ln 0


Semi Log Demand CurvesHere, percentage change in quantity demanded responds to a dollar change in price (i.e. a $1 increase in price lowers demand by 4%.

14

1

%

4%

4ln

xx

x

x

x

x

pp

p

x

p

x

px

XP

XQ

xd pax ln4lnln 0


Log Demand CurvesHere, percentage change in quantity demanded responds to a percentage change in price (i.e. a 1% increase in price lowers demand by 4%.

4%

%

ln4ln

xx

x

p

x

px

XP

XQ

Log Linear demands have constant elasticities!!

One Problem

x

xp

Suppose you observed the following data points. Could you estimate a demand curve?

D

Estimating demand curves

x

xp

Market prices are the result of the interaction between demand and supply!!

dxd Iapaax 210

A problem with estimating demand curves is the simultaneity problem.

D

S

xp

sd xx

Estimating demand curves

x

xp

Case #1: Both supply and demand shifts!!

D’’

S’S

S’’

D’D

x

xp

D

S’S

S’’

Case #2: All the points are due to supply shifts

An example…

dxd Iapaax 210

sxs pbbx 10Supply

Demand

Equilibrium ds xx

Suppose you get a random shock to demand

The shock effects quantity demanded which (due to the equilibrium condition influences price!

Therefore, price and the error term are correlated! A big problem !!

sxdx pbbIapaa 10210

Suppose we solved for price and quantity by using the equilibrium condition

ds xx

ssd

sdx

abbI

ab

ab

ab

babbx

abI

ab

a

ab

bap

111

11

21

11

0010

1111

2

11

00

We could estimate the following equations

232

110

Ix

Ipx

11

213

11

21

ab

ab

ab

a

The original parameters are related as follows:

1

31

b

We can solve for the supply parameters, but not demand. Why?

xpbxb 10

dxd Iapaax 210

sxs pbbx 10

x

xpS

D

D

D

By including a demand shifter (Income), we are able to identify demand shifts and, hence, trace out the supply curve!!

Time

Dem

and

Fact

ors

t t+1t-1

Time Series estimation holds the demand factors constant and estimates the variation in demand over time

For example: can we predict demand for Pepsi in South Bend next year by looking at how demand varies across time

Time series estimation leaves the demand factors constant and looks at variations in demand over time. Essentially, we want to separate demand changes into various frequencies

Trend: Long term movements in demand (i.e. demand for movie tickets grows by an average of 6% per year)

Business Cycle: Movements in demand related to the state of the economy (i.e. demand for movie tickets grows by more than 6% during economic expansions and less than 6% during recessions)

Seasonal: Movements in demand related to time of year. (i.e. demand for movie tickets is highest in the summer and around Christmas

Suppose that you work for a local power company. You have been asked to forecast energy demand for the upcoming year. You have data over the previous 4 years:

Time Period Quantity (millions of kilowatt hours)

2003:1 11

2003:2 15

2003:3 12

2003:4 14

2004:1 12

2004:2 17

2004:3 13

2004:4 16

2005:1 14

2005:2 18

2005:3 15

2005:4 17

2006:1 15

2006:2 20

2006:3 16

2006:4 19

0

5

10

15

20

25

2003-1 2004-1 2005-1 2006-1

First, let’s plot the data…what do you see?

This data seems to have a linear trend

A linear trend takes the following form:

btxxt 0

Forecasted value at time t (note: time periods are quarters and time zero is 2003:1)

Time period: t = 0 is 2003:1 and periods are quarters

Estimated value for time zero

Estimated quarterly growth (in kilowatt hours)

Regression Results


Intercept 11.9 .953 12.5

Time Trend .394 .099 4.00


R Squared .53

Standard Error 1.82

Observations 16txt 394.9.11

Lets forecast electricity usage at the mean time period (t = 8)

50.3ˆ

05.158394.9.11ˆ

t

t

xVar

x

0

5

10

15

20

25

2003-1 2004-1 2005-1 2006-1

Here’s a plot of our regression line with our error bands…again, note that the forecast error will be lowest at the mean time period

T = 8

0

10

20

30

40

50

60

70

Sample

We can use this linear trend model to predict as far out as we want, but note that the error involved gets worse!

7.47ˆ

85.4176394.9.11ˆ

t

t

xVar

x

Time Period Actual Predicted Error

2003:1 11 12.29 -1.29

2003:2 15 12.68 2.31

2003:3 12 13.08 -1.08

2003:4 14 13.47 .52

2004:1 12 13.87 -1.87

2004:2 17 14.26 2.73

2004:3 13 14.66 -1.65

2004:4 16 15.05 .94

2005:1 14 15.44 -1.44

2005:2 18 15.84 2.15

2005:3 15 16.23 -1.23

2005:4 17 16.63 .37

2006:1 15 17.02 -2.02

2006:2 20 17.41 2.58

2006:3 16 17.81 -1.81

2006:4 19 18.20 .79

One method of evaluating a forecast is to calculate the root mean squared error

n

FARMSE tt

2

Number of Observations

Sum of squared forecast errors

70.1RMSE

0

5

10

15

20

25

2003-1 2004-1 2005-1 2006-1

Lets take another look at the data…it seems that there is a regular pattern…

Q2

Q2Q2

Q2

We are systematically under predicting usage in the second quarter

Time Period Actual Predicted Ratio Adjusted

2003:1 11 12.29 .89 12.29(.87)=10.90

2003:2 15 12.68 1.18 12.68(1.16) = 14.77

2003:3 12 13.08 .91 13.08(.91) = 11.86

2003:4 14 13.47 1.03 13.47(1.04) = 14.04

2004:1 12 13.87 .87 13.87(.87) = 12.30

2004:2 17 14.26 1.19 14.26(1.16) = 16.61

2004:3 13 14.66 .88 14.66(.91) = 13.29

2004:4 16 15.05 1.06 15.05(1.04) = 15.68

2005:1 14 15.44 .91 15.44(.87) = 13.70

2005:2 18 15.84 1.14 15.84(1.16) = 18.45

2005:3 15 16.23 .92 16.23(.91) = 14.72

2005:4 17 16.63 1.02 16.63(1.04) = 17.33

2006:1 15 17.02 .88 17.02(.87) = 15.10

2006:2 20 17.41 1.14 17.41(1.16) = 20.28

2006:3 16 17.81 .89 17.81(.91) = 16.15

2006:4 19 18.20 1.04 18.20(1.04) = 18.96

Average Ratios

•Q1 = .87

•Q2 = 1.16

•Q3 = .91

•Q4 = 1.04

We can adjust for this seasonal component…

10

11

12

13

14

15

16

17

18

19

20

2003-1 2004-1 2005-1 2006-1

Now, we have a pretty good fit!!

26.RMSE

0

10

20

30

40

50

60

70

52.4304.185.4176394.9.11ˆ tx

Recall our prediction for period 76 ( Year 2022 Q4)

We could also account for seasonal variation by using dummy variables

33221100 DbDbDbtbxxt

else

iquarterifDi ,0

,1

Note: we only need three quarter dummies. If the observation is from quarter 4, then

tbxx

DDD

t 00

321 0

Regression Results


Intercept 12.75 .226 56.38

Time Trend .375 .0168 22.2

D1 -2.375 .219 -10.83

D2 1.75 .215 8.1

D3 -2.125 .213 -9.93


R Squared .99

Standard Error .30

Observations 16

321 125.275.1375.2375.75.12 DDDtxt

Note the much better fit!!

Time Period Actual Ratio Method Dummy Variables

2003:1 11 10.90 10.75

2003:2 15 14.77 15.25

2003:3 12 11.86 11.75

2003:4 14 14.04 14.25

2004:1 12 12.30 12.25

2004:2 17 16.61 16.75

2004:3 13 13.29 13.25

2004:4 16 15.68 15.75

2005:1 14 13.70 13.75

2005:2 18 18.45 18.25

2005:3 15 14.72 14.75

2005:4 17 17.33 17.25

2006:1 15 15.10 15.25

2006:2 20 20.28 19.75

2006:3 16 16.15 16.25

2006:4 19 18.96 18.75

26.RMSE

Ratio Method

25.RMSE

Dummy Variables

10

11

12

13

14

15

16

17

18

19

20

2003-1 2004-1 2005-1 2006-1

Dummy Ratio

A plot confirms the similarity of the methods

0

10

20

30

40

50

60

70

Recall our prediction for period 76 ( Year 2022 Q4)

25.4176375.75.12 tx

btxxt 0

Recall, our trend line took the form…

This parameter is measuring quarterly change in electricity demand in millions of kilowatt hours.

Often times, its more realistic to assume that demand grows by a constant percentage rather that a constant quantity. For example, if we knew that electricity demand grew by g% per quarter, then our forecasting equation would take the form

t

t

gxx

100

%10

tt gxx 10

If we wish to estimate this equation, we have a little work to do…

Note: this growth rate is in decimal form

gtxxt 1lnlnln 0

If we convert our data to natural logs, we get the following linear relationship that can be estimated

Regression Results


Intercept 2.49 .063 39.6

Time Trend .026 .006 4.06


R Squared .54

Standard Error .1197

Observations 16

txt 026.49.2ln

Lets forecast electricity usage at the mean time period (t = 8)

0152.ˆ

698.28026.49.2ˆln

t

t

xVar

xBE CAREFUL….THESE NUMBERS ARE LOGS !!!

0152.ˆ

698.28026.49.2ˆln

t

t

xVar

x

The natural log of forecasted demand is 2.698. Therefore, to get the actual demand forecast, use the exponential function

85.14698.2 e

Likewise, with the error bands…a 95% confidence interval is +/- 2 SD

945.2,451.20152.2/698.2

00.19,60.11, 945.2451.2 ee

0

5

10

15

20

25

30

2003-1 2004-1 2005-1 2006-1

Again, here is a plot of our forecasts with the error bands

T = 8 70.1RMSE

0

1

2

3

4

5

6

7

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97

When plotted in logs, our period 76 ( year 2022 Q4) looks similar to the linear trend

207.ˆln

49.476026.49.2ˆln

t

t

xVar

x

0

100

200

300

400

500

600

1 13 25 37 49 61 73 85 97

Again, we need to convert back to levels for the forecast to be relevant!!

8.221,8.352/

22.8949.4

SD

eErrors is growth rates compound quickly!!

Quarter Market Share

1 20

2 22

3 23

4 24

5 18

6 23

7 19

8 17

9 22

10 23

11 18

12 23

Consider a new forecasting problem. You are asked to forecast a company’s market share for the 13th quarter.

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12

There doesn’t seem to be any discernable trend here…

Smoothing techniques are often used when data exhibits no trend or seasonal/cyclical component. They are used to filter out short term noise in the data.


MA(3) MA(5)

1 20

2 22

3 23

4 24 21.67

5 18 23

6 23 21.67 21.4

7 19 21.67 22

8 17 20 21.4

9 22 19.67 20.2

10 23 19.33 19.8

11 18 20.67 20.8

12 23 21 19.8

A moving average of length N is equal to the average value over the previous N periods

N

ANMA

t

Ntt

1

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12

Actual

MA(3)

MA(5)

The longer the moving average, the smoother the forecasts are…


MA(3) MA(5)

1 20

2 22

3 23

4 24 21.67

5 18 23

6 23 21.67 21.4

7 19 21.67 22

8 17 20 21.4

9 22 19.67 20.2

10 23 19.33 19.8

11 18 20.67 20.8

12 23 21 19.8

Calculating forecasts is straightforward…

MA(3)

33.213

231823

MA(5)

6.205

1722231823

So, how do we choose N??


MA(3) Squared Error

MA(5) Squared Error

1 20

2 22

3 23

4 24 21.67 5.4289

5 18 23 25

6 23 21.67 1.7689 21.4 2.56

7 19 21.67 7.1289 22 9

8 17 20 9 21.4 19.36

9 22 19.67 5.4289 20.2 3.24

10 23 19.33 13.4689 19.8 10.24

11 18 20.67 7.1289 20.8 7.84

12 23 21 4 19.8 10.24

Total = 78.3534 Total = 62.48

95.29

3534.78RMSE 99.2

7

48.62RMSE

Exponential smoothing involves a forecast equation that takes the following form

ttt FwwAF 11

Forecast for time t+1

Actual value at time t

Forecast for time t

Smoothing parameter

Note: when w = 1, your forecast is equal to the previous value. When w = 0, your forecast is a constant.

1,0w


W=.3 W=.5

1 20 21.0 21.0

2 22 20.7 20.5

3 23 21.1 21.3

4 24 21.7 22.2

5 18 22.4 23.1

6 23 21.1 20.6

7 19 21.7 21.8

8 17 20.9 20.4

9 22 19.7 18.7

10 23 20.4 20.4

11 18 21.2 21.7

12 23 20.2 19.9

For exponential smoothing, we need to choose a value for the weighting formula as well as an initial forecast

Usually, the initial forecast is chosen to equal the sample average

8.216.205.235.

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12

Actual w=.3 w=.5

As was mentioned earlier, the smaller w will produce a smoother forecast

Calculating forecasts is straightforward…

W=.3

04.212.207.233.

W=.5

45.219.195.235.

So, how do we choose W??


W=.3 W=.5

1 20 21.0 21.0

2 22 20.7 20.5

3 23 21.1 21.3

4 24 21.7 22.2

5 18 22.4 23.1

6 23 21.1 20.6

7 19 21.7 21.8

8 17 20.9 20.4

9 22 19.7 18.7

10 23 20.4 20.4

11 18 21.2 21.7

12 23 20.2 19.9


W = .3 Squared Error

W=.5 Squared Error

1 20 21.0 1 21.0 1

2 22 20.7 1.69 20.5 2.25

3 23 21.1 3.61 21.3 2.89

4 24 21.7 5.29 22.2 3.24

5 18 22.4 19.36 23.1 26.01

6 23 21.1 3.61 20.6 5.76

7 19 21.7 7.29 21.8 7.84

8 17 20.9 15.21 20.4 11.56

9 22 19.7 5.29 18.7 10.89

10 23 20.4 6.76 20.4 6.76

11 18 21.2 10.24 21.7 13.69

12 23 20.2 7.84 19.9 9.61

Total = 87.19 Total = 101.5

70.212

19.87RMSE 91.2

12

5.101RMSE

Economics of the Firm

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