Rose-Hulman Institute of Technology / Department of Humanities & Social Sciences / K. Christ

Fall Quarter, 2009 – 2010 / SL 351, Managerial Economics; EMGT 531, Economics for Technical Managers

Problem Set 2 -- Solutions

Textbook Problems:

Hirschey, Chapter 3: P3.5, P3.7, P3.8, P3.10

Hirschey, Chapter 4: P4.6, P4.7, P4.8, P4.9, P4.10

Hirschey, Chapter 5: P5.3, P5.8, P5.9, P5.10

Hirschey, Chapter 6: P6.3, P6.4, P6.5, P6.6, P6.8

Extra Problems:

1. Diversified Products. You are the manager of a diversified products firm that received revenues

of $30,000 per year from product X and $70,000 per year from product Y. The own-price

elasticity of demand for product X is – 2.5, and the cross-price elasticity of demand between

product X and product Y is 1.1. How much will your firm‟s total revenues (combined revenues

from both products) change if you increase the price of good X by 1%?

2. Kodak. You are a manager in charge of monitoring cash flow at Kodak (in 2002). Traditional

photography equipment comprises 80% of Kodak‟s revenues, which grow about 2% annually.

You recently received a preliminary report that suggest consumers take three time more digital

photographs than photos with traditional film, and that the cross-price elasticity of demand

between digital and disposable cameras is – 0.2. Over the last several years, Kodak has invested

over $5 billion to develop and begin producing digital cameras. If the own price elasticity of

demand for disposable cameras is -2.5, how will a 1% decrease in the price of disposable

cameras affect Kodak‟s overall revenues from both disposable and digital camera sales?

3. Use data set „rhit pizza‟. This data set contains hypothetical demand data for an unidentified

pizza supplier to the Rose-Hulman campus over the 2001-2002 academic year.

a. Regress Q1 on P1 and P2.

b. Regress lnQ1 on lnP1 and lnP2. Interpret your regression results in light of this data

transformation.

c. Plot Q1 against the date variable and see if you can determine what other factors might

usefully be included as explanatory variables in a demand model.

d. Modify the regression specification from part (b) to include the following additional right

hand side variables: FINALS, INSESSION, MON, TUE, WED, THU, FRI, SAT (not

SUN!). Compare your results with those you obtained in part (b).

4. Use data set „export good‟. This data set contains hypothetical historical sales data (variable

q1) for an unspecified producer that sells in both domestic and foreign markets.

a. Generate a chart showing the month/year on the horizontal axis and q1 on the vertical

axis.

b. Regress q1 on the trend variable.

c. Regress lnq1 on lnp1, lnp3, lnip, and lntwex.

d. Regress lnq1 on lnp1, lnp3, lnip, lntwex, and trend.

e. Comment on your regression results.

5. Use the following data to generate the forecast MAPE, RMSE, and FA:

Month Forecast Actual

January

4,532

4,268

February

4,634

4,573

March

4,737

5,024

April

4,839

5,214

May

4,942

4,969

June

5,044

5,121

July

5,147

4,898

August

5,249

5,047

September

5,352

5,136

October

5,454

5,372

November

5,556

5,702

December

5,659

5,821

P3.5

A. The demand faced by CPC in a typical market in which P = $10, Pop = 1,000,000

persons, I = $60,000, and A = $10,000 is:

Q = 5,000 - 4,000P + 0.02Pop + 0.25I + 1.5A

= 5,000 - 4,000(10) + 0.02(1,000,000) + 0.25(60,000) + 1.5(10,000)

=15,000

B. If advertising rises from $10,000 to $15,000, CPC demand rises to:

Q = 5,000 - 4,000P + 0.02Pop + 0.25I + 1.5A

= 5,000 - 4,000(10) + 0.02(1,000,000) + 0.25(60,000) + 1.5(15,000)

= 22,500

C. The effect of an increase in advertising from $10,000 to $15,000 is to shift the

demand curve upward following a 7,500 unit increase in the intercept term. If

advertising is $10,000, the CPC demand curve is:

Q = 5,000 - 4,000P + 0.02(1,000,000) + 0.25(60,000) + 1.5(10,000)

= 55,000 - 4,000P

Then, price as a function of quantity is:

Q = 55,000 - 4,000P

4,000P = 55,000 - Q

P = $13.75 - $0.00025Q

If advertising is $15,000, the CPC demand curve is

Q = 5,000 - 4,000P + 0.02(1,000,000) + 0.25(60,000) + 1.5(15,000)

= 62,500 - 4,000P

Then, price as a function of quantity is:

Q = 62,500 - 4,000P

4,000P = 62,500 - Q

P = $15.625 - $0.00025Q

P3.7

A. With quantity expressed as a function of price, the industry supply curve is:

Q = -59,000,000 + 500,000P - 125,000PL - 500,000PK + 2,000,000W

= -59,000,000 + 500,000P - 125,000(8) - 500,000(10) + 2,000,000(20)

= -25,000,000 + 500,000P

With price expressed as a function of quantity, the industry supply curve is:

Q = -25,000,000 + 500,000P

500,000P = 25,000,000 + Q

P = $50 + $0.000002Q

B. Industry supply at each respective price is:

P = $50: Q = -25,000,000 + 500,000($50) = 0

P = $60: Q = -25,000,000 + 500,000($60) = 5,000,000

P = $70: Q = -25,000,000 + 500,000($70) = 10,000,000

C. The price necessary to generate each level of supply is:

Q = 4,000,000: P = $50 + $0.000002(4,000,000) = $58

Q = 6,000,000: P = $50 + $0.000002(6,000,000) = $62

Q = 8,000,000: P = $50 + $0.000002(8,000,000) = $66

P3.8

A. Each company will supply output to the point where MR = MC. Because P = MR in

this market, the supply curve for each firm can be written with price as a function of

quantity as:

Olympia

MRO = MCO

P = $350 + $0.00005QO

Yakima

MRY = MCY

P = $150 + $0.0002QY

When quantity is expressed as a function of price:

Olympia

P = $350 + $0.00005QO

0.00005QO = -350 + P

QO = -7,000,000 + 20,000P

Yakima

P = $150 + $0.0002QY

0.0002QY = -150 + P

QY = -750,000 + 5,000P

B. The quantity supplied at each respective price is:

Olympia

P = $325: QO = -7,000,000 + 20,000($325) = -500,000 0

(because Q < 0 is impossible)

P = $350: QO = -7,000,000 + 20,000($350) = 0

P = $375: QO = -7,000,000 + 20,000($375) = 500,000

Yakima

P = $325: QY = -750,000 + 5,000($325) = 875,000

P = $350: QY = -750,000 + 5,000($350) = 1,000,000

P = $375: QY = -750,000 + 5,000($375) = 1,125,000

For Olympia, MC = $350 when Q0 = 0. Because marginal cost rises with output,

Olympia will never supply output unless a price in excess of $350 per unit can be

obtained. Because negative output is not feasible, Olympia will simply fail to supply

output when P < $350. Similarly, MCY = $150 when QY = 0. Thus, Yakima will

never supply output unless a price in excess of $150 per unit can be obtained.

C. When P < $350, only Yakima can profitably supply output. The Yakima supply

curve will be the industry curve when P < $350:

P = $150 + $0.0002Q

or

Q = -750,000 + 5,000P

D. When P > $350, both Olympia and Yakima can profitably supply output. To derive

the industry supply curve in this circumstance, simply sum the quantities supplied by

each firm:

Q = QO + QY

= -7,000,000 + 20,000P + (-750,000 + 5,000P)

= -7,750,000 + 25,000P

To check, at P = $375:

Q = -7,750,000 + 25,000($375)

= 1,625,000

This answer is supported by the answer to part B, because QO + QY = 500,000 +

1,125,000 = 1,625,000

(Note: Some students mistakenly add prices rather than quantities in an attempt to derive the

industry supply curve. To avoid this problem, it is important to emphasize that industry supply

curves are found through adding up output (horizontal summation), not by adding up prices

(vertical summation).)

P3.10

A. When quantity is expressed as a function of price, the demand curve for Eye-de-ho

Potatoes is:

QD = -1,450 - 25P + 12.5PW + 0.2Y

= -1,450 - 25P + 12.5($4) + 0.1($15,000)

QD = 100 - 25P

When quantity is expressed as a function of price, the supply curve for

Eye-de-ho Potatoes is:

QS = -100 + 75P - 25PW - 12.5PL + 10R

= -100 + 75P - 25($4) - 12.5($8) + 10(20)

QS = -100 + 75P

B. The surplus or shortage can be calculated at each price level:

Price

Quantity

Supplied

Quantity

Demanded

Surplus (+) or

Shortage (-)

(1)

(2)

(3)

(4) = (2) - (3)

$1.50:

QS = -100 + 75($1.50)

= 12.5

QD = 100 - 25($1.50)

= 62.5

-50

$2.00:

QS = -100 + 75($2)

= 50

QD = 100 - 25($2)

= 50

0

$2.50:

QS = -100 + 75($2.50)

= 87.5

QD = 100 - 25($2.50)

= 37.5

+50

C. The equilibrium price is found by setting the quantity demanded equal to the quantity

supplied and solving for P:

QD = QS

100 - 25P = -100 + 75P

100P = 200

P = $2

To solve for Q, set:

Demand: QD = 100 - 25($2) = 50 (million bushels)

Supply: QS = -100 + 75($2) = 50 (million bushels)

In equilibrium QD = QS =50 (million bushels).

P4.6 εP = ΔQ/Q † ΔP/P

= 10%/-1%

= -10 (Highly elastic)

The profit-maximizing price can be found using the optimal price formula:

P* = MC/(1 + 1/εP)

= ($23,500 + $350)/[1 + 1/(-10)]

= $26,500

P4.7

A.

B.

C. Yes, the |EP| = 2.75 > 1 calculated in part A implies an elastic demand for appetizers

and that an additional price reduction will increase appetizer revenues. EPX = -3.67 <

0 indicates that beverages and appetizers are complements. Therefore, a further

decrease in appetizer prices will cause a continued growth in beverage unit sales and

revenues. Alternatively, If P = a + bQ, then $12 = a + b(90) and $10 = a +

b(150). Solving for the demand curve gives P = $15 - $0.033Q. At P = $12, total

revenue is $1,080 (= $12 × 90). If P = $10, total revenue is $1,500 (= $10 × 150). At

P = $8, total revenue is $1,680 (= $8 × 210). In any case, to determine the profit

effects of appetizer price changes it is necessary to consider revenue and cost

implications of both appetizer and beverage sales.

75.2- = 0)9 + 05(1

)2$1 + 10($

)2$1 - 10($

0)9 - 50(1 =

Q + Q

P + P

P

Q = E

12

12P

67.3- = 0)30 + 600(

)2$1 + 10($

)2$1 - 10($

0)30 - 600( =

Q + Q

P + P

P

Q = E

12

1X2X

X

PX

P4.8

A.

B. Without a price increase, sales this year would total 50 million units. Therefore, it is

appropriate to estimate the arc price elasticity from a before-price-increase base of 50

million units:

C. Lower. Since carpet demand is in the elastic range, EP = -8, an increase (decrease) in

price will result in lower (higher) total revenues.

9.5 =

30 + 50

$55,500 + $58,500

$55,500 - $58,500

30 - 50 =

Q + Q

I + I

I

Q = E

12

12I

(Elastic) 8- =

50 + 30

$15.50 + $16.50

$15.50 - $16.50

50 - 30 =

Q1

+ Q2

P1 + P2

P

Q = EP

P4.9

A. EPX = Q + Q

P + P

P - P

Q - Q

1Y2Y

1X2X

1X2X

1Y2Y

= 10,000 + 4,800

$137 + $85

$137 - $85

10,000 - 4,800

= 1.5 (Substitutes)

B. EP = Q + Q

P + P

P - P

Q - Q

12

12

12

12

= 4,800 + 6,000

$140 + $130

$140 - $130

4,800 - 6,000

= -3 (Elastic)

C. EP = Q + Q

P + P

P - P

Q - Q

12

12

12

12

-3 = 6,000 + 10,000

$130 + P x

$130 - P

6,000 - 10,000 2

2

-3 = $130) - P4(

$130 + P

2

2

-12P2 + $1,560 = P2 + $130

13P2 = $1,430

P2 = $110

This implies a further price reduction of $20 because:

ΔP = $130 - $110 = $20

P4.10

A. EP = Q + Q

P + P

P

Q

12

12 =

16,200 + 9,000

$9 + $12

$9 - $12

16,200 - 9,000

= -2

B. The effective price reduction is $2 since 40% of sales are accompanied by a coupon:

ΔP = -$5(0.4) or P2 = $12 - $5(0.4)

= -$2 = $10

ΔP = $10 - $12

= -$2

C. To calculate the arc advertising elasticity, the effect of the $2 price cut implicit in the

coupon promotion must first be reflected. With just a price cut, the quantity

demanded would rise to 13,000, because:

EP = Q + Q*

P + P

P - P

Q - Q*

1

12

12

1

-2 = 9,000 + Q*

$12 + $10

$12 - $10

9,000 - Q*

-2 = 9,000) + (Q*

9,000) - 11(Q*-

-2(Q* + 9,000) = -11(Q* - 9,000)

-2Q* - 18,000 = -11Q* + 99,000

9Q* = 117,000

Q* = 13,000

Then, the arc advertising elasticity can be calculated as:

EA = Q* + Q

A + A

A - A

Q* - Q

2

12

12

2

=13,000 + 15,000

$3,250 + $3,750

$3,250 - $3,750

13,000 - 15,000

= 1

D. It is important to recognize that a coupon promotion can involve more than just the

independent effects of a price cut plus an increase in advertising as is implied in Part

C. Synergistic or interactive effects may increase advertising effectiveness when the

promotion is accompanied by a price cut. Similarly, price reductions can have a

much larger impact when advertised. In addition, a coupon is a price cut for only the

most price sensitive (coupon-using) customers, and may spur sales by much more

than a dollar equivalent across-the-board price cut.

Synergy between advertising and the implicit price reduction that accompanies

a coupon promotion can cause the estimate in Part C to overstate the true advertising

elasticity. Similarly, this advertising elasticity will be overstated to the extent that

targeted price cuts have a bigger influence on the quantity demanded than similar

across-the-board price reductions, as seems likely.

P5.3

A. To find the revenue-maximizing price-output rental rate, set MR = 0, and solve for Q.

TR = P × Q

= ($1,200 - $0.04Q)Q

= $1,200Q - $0.04Q2

MR = ∂TR/∂Q

MR = $1,200 - $0.08Q = 0

0.08Q = 1,200

Q = 15,000

At Q = 15,000, P = $1,200 - $0.04(15,000) = $600

Total revenue at a price of $600 is TR = P × Q

= $600 × 15,000

= $9 million per week

π = TR - TC

= $1,200Q - $0.04Q2 - $800Q

= $1,200(15,000) - $0.04(15,0002) - $800(15,000)

= -$3 million per week (loss)

(Note: ∂2TR/∂Q

2 < 0. This is a revenue-maximizing output level because total

revenue is decreasing for output beyond Q > 15,000 units.)

B. To find the profit-maximizing output level analytically, set MR = MC, or set Mπ = 0,

and solve for Q. Because

MR = MC

$1,200 - $0.08Q = $800

0.08Q = 400

Q = 5,000 At Q = 5,000, P = $1,200 - $0.04(5,000) = $1,000

Total revenue at a price of $1,000 is TR = P × Q:

= $1,000 × 5,000

= $5 million per week

π = TR - TC

= $1,200Q - $0.04Q2 - $800Q

= $1,200(5,000) - $0.04(5,0002) - $800(5,000)

= $1 million per week

(Note: ∂2π/∂Q

2 < 0. This is a profit maximum because total profit is falling for

Q > 5,000.)

P5.8

A. Demand

QD = 4,000 - 200P + 2,000T

200P = 4,000 - QD + 2,000T

P = $20 - $0.005QD + $10T

Supply

QS = -2,000 + 200P

200P = 2,000 + QS

P = $10 + $0.005QS

B.,C. D. This problem illustrates the identification problem. If either the demand or supply

function is shifting while the other is stable, then the price/output data can be used to

trace out the stable curve. Here the supply curve is stable while demand is growing

rapidly (shifting to the right). Therefore, the price/output data given in the problem

can be used to trace out the relevant supply curve.

$0

$10

$20

$30

$40

$50

$60

$70

$80

$90

$100

0 1 2 3 4 5 6 7 8 9 10 11 12

Ho

url

y R

ate

Billable Hours (000)

Consulting Services, Inc., Demand and Supply Curve Analysis

D5

D1 D2D3

D4

D6

Supply Curve

P5.9

A. (i) Coefficient of determination = R2 = 93%, implying that 93% of demand

variation is explained by the regression model.

(ii) Corrected coefficient of determination = 2R = R

2 - (k - 1/n - k)(1 - R

2) = 0.93

- (4/28)(1 - 0.93) = 0.92, implying that 92% of demand variation is explained

by the regression model when both coefficient number, k, and sample size, n,

are controlled for.

(iii) F statistic = (n - k/k - 1)(R2/1 - R

2) = (28/4)(0.93/0.07) = 93 > F*4, 28, α = 0.01 =

4.07 implying one can reject the null hypothesis H0: b1 = b2 = b3 = b4 = 0 and

conclude with 99% confidence that the dependent variables as a group explain

a significant share of demand variation.

(iv) Standard error of the estimate = SEE = 6 implying that

Q = Q̂ ± 2.048 × 6 with 95% confidence.

Q = Q̂ ± 2.763 × 6 with 99% confidence.

B. To determine whether quantity demanded depends upon “own” price, the question

must be asked: is bP ≠ 0? If bP ≠ 0, then evidence exists that sales do indeed depend

upon price. For testing purposes, the null hypothesis one seeks to reject is the

converse of the above question:

H0: bP = 0 (Two-tail test)

where |t| = 2.763 = t > 2.78 = 1.8

5 = |

b| 0.01=28,

b

P

P

Therefore, it is possible to reject H0: bP = 0 with 99% confidence and conclude that

demand is sensitive to price.

C. Because Q̂ = 4 - 5P + 2A + 0.2I + 0.25HF

= 4 - 5(5) + 2(30) + 0.2(55) + 0.25(40) = 60(000)

The point advertising elasticity is calculated as:

εA = ∂Q/∂A × A/Q = 2 × 30/60 = 1

Because εA = 1, a 1% increase in advertising will lead to commensurate percentage

increase in demand.

D. Pr = 0.5 or 50%. To generate breakeven revenues of $300,000, Colorful Tile would

have to sell Q = TR/P = TC/P = $300,000/$5 = 60,000 cases.

From part C, Q̂ = 60(000). Because there is a 50/50 chance that actual sales

will be above or below this level, there is a 50/50 chance that the Austin store will

make a profit when TC = $300,000.

P5.10

A. The exponents of multiplicative demand functions are elasticity estimates.

Therefore, tour demand is elastic with respect to price provided

1 > |b|Py

or 0. > 1 - |b|Py

For testing purposes, the null hypothesis to reject is:

This means it is possible to reject H0 with 95% confidence and conclude that tour

demand is elastic with respect to price.

B. Because exponents are elasticity estimates in a multiplicative demand model, tours

will be a normal good provided bI > 0. For testing purposes, the null hypothesis to

reject is:

H0: bI < 0 (One-tail test)

where

This means it is reasonable to reject H0 with 99% confidence and conclude tours are

a normal good.

C. Because exponents are elasticity estimates in a multiplicative demand model, tours

and X will be substitutes provided 0, > bPX and complements if 0. < bPX

One may

first wish to test the substitute good hypothesis. For testing purposes, the hypothesis

to reject is:

H0: 0 < bPX (One-tail test)

where

This means it is reasonable to reject H0 with 90% confidence and conclude tours and

X are substitutes.

D. Both “own” and competitor advertising appear to increase sales. Although relatively

uncommon, this is not a rare occurrence. Advertising of substitutes can sometimes

raise sales for competitor products due to beneficial spillover effects following

increased customer awareness.

test)tail(One- 0 < 1 - |b|or 1 < |b| :H PP0 YY

t = 1.812 > 2.5 = 0.04

1 - 1.10 =

|b|

1 - |b| =t *

0.05=10,

P

P

Y

Y

t = 2.764 > 4.11 = 0.45

1.85 =

b =t *

.01=10,

b

I

I

t = 1.372 > 1.43 = 0.35

0.50 =

b =t *

0.10=10,

P

P

X

X

P6.3

A. St = S0(1 + g)t

$65,000,000 = $25,000,000(1 + g)10

2.6 = (1 + g)10

ln(2.6) = 10 × ln(1 + g)

0.956/10 = ln(1 + g)

e(0.0956)

- 1 = g

g = 0.100 or 10.0%

B. Five-Year Sales Forecast

St = S0 (1 + g)t

= $65,000,000 (1 + 0.10)5

= $65,000,000 (1.611)

= $104,715,000

Ten-Year Sales Forecast

St = S0 (1 + g)t

= $65,000,000 (1 + 0.10)10

= $65,000,000 (2.594)

= $168,610,00

P6.4

A. Ct = C0egt

$100 = $80e3g

1.25 = e3g

ln(1.25) = 3g

g = 0.223/3

= 0.074 or 7.4%

B. Import Cost = C0egt

$115.90 = $100e(0.074)t

1.159 = e(0.074)t

ln(1.159) = 0.074t

t = 0.148/0.074

= 2 years

P6.5

A. At = At-1 + ΔAt-1

At = At-1 A 1 - B

B - 1t-

2t-

1t-

B. At = At-1 A 1 - B

B - 1t-

2t-

1t-

= 100 100 1 - 75

90 -

= 80.

P6.6

A.

St+1 = St + S 1 - A

A 2 + S 1 -

Y

Y 2 t

1t-

tt

1t-

t

- S 1 - CA

CA 0.5 t

1t-

t

= St + S2 - A

A S2 + S2 -

Y

Y S2 t

1t-

ttt

1t-

tt

- S 2

1 +

CA

CA S0.5 t

1t-

tt

= CA

CA S

2

1 -

A

A S2 +

Y

Y S2

1t-

tt

1t-

tt

1t-

tt

- 2.5St

B.

St+1 = 2($500,000)(1.02) + 2($500,000)(0.80)

- 0.5 ($500,000)(1.10) - 2.5 ($500,000)

= $1,020,000 + $800,000 - $275,000 - $1,250,000

= $295,000

P6.8

A. St+1 = St + ΔS

= St - ΔSP + ΔST

= St - 2(Pt+1/Pt - 1)St + 3(Tt+1/Tt - 1)St

= -2(Pt+1/Pt)St + 3(Tt+1/Tt)St

B. St+1 =-2(16.5/15)10,000 + 3(1.15)10,000

= -22,000 + 34,500

= 12,500 games

Extra Problems

1. Diversified Products.

Use :

= 0.01[30,000(1-2.5) + 70,000(1.1)] = 0.01[-45,000 + 77,000] = + 320

2. Kodak.

Use :

= -0.01[600(1-2.5) + 400(-0.2)] = -0.01[-900 – 80] = + 9.8 million.

3. Pizza. MSExcel regression output for parts (a) and (d):

P2 P1 Constant

1.788 -2.850 17.276

1.091 1.164 7.813

0.020 6.657 #N/A

3.064 301.000 #N/A

271.532 13,338.205 #N/A

SAT FRI THU WED TUE MON INSESS FINALS ln(P2) ln(P1) Constant

-0.310 0.207 -0.188 -0.676 -0.204 -0.706 1.407 0.956 1.022 -1.270 1.420

0.100 0.100 0.099 0.099 0.100 0.100 0.061 0.116 0.748 0.804 1.269

0.726 0.463 #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A

77.711 293 #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A

167.155 63.02 #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A #N/A

4. Export Good. MSExcel Regression output for part (d):

TREND TWEX IP P3 P1 Constant

0.006 -0.207 0.519 0.147 -0.536 9.563

0.001 0.097 0.149 0.041 0.363 2.157

0.975 0.055 #N/A #N/A #N/A #N/A

1,279.364 162.000 #N/A #N/A #N/A #N/A

19.319 0.489 #N/A #N/A #N/A #N/A

5.158 -2.139 3.487 3.624 -1.475 4.433

5. Forecast Metrics.

Month Forecast Actual e abs(e) e^2 max abs(e/y) January 4,532 4,268 264 264 69696 4532 0.0619 February 4,634 4,573 61 61 3721 4634 0.0133 March 4,737 5,024 -287 287 82369 5024 0.0571 April 4,839 5,214 -375 375 140625 5214 0.0719 May 4,942 4,969 -27 27 729 4969 0.0054 June 5,044 5,121 -77 77 5929 5121 0.0150 July 5,147 4,898 249 249 62001 5147 0.0508 August 5,249 5,047 202 202 40804 5249 0.0400 September 5,352 5,136 216 216 46656 5352 0.0421

October 5,454 5,372 82 82 6724 5454 0.0153 November 5,556 5,702 -146 146 21316 5702 0.0256 December 5,659 5,821 -162 162 26244 5821 0.0278

12 61,145 61,145

2148 506814 62219 0.4263

MAPE 0.0355 RMSE 205.5103 FA 0.965 BIAS 0