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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