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3. Mathematical Versions of Simple Growth Models
Functional FormsExamples of Choosing
Functional FormsBasic Theory of Econ ModelsThe Error Term
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3. Functional FormsAccurately describing the mathematical
relationship between 2 variables is ESSENTIAL to studying human behavior
One variable’s increase can cause the other variable to:
-increase,
-decrease
-not change
This impact can be non-constant and change over time:
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3.1.2.1 – Linear Functional Form
Linear:
Yt= β1 + β2X
Constant slope: β2
-straight line relationship
-same increase every period (β2>0)
-same decrease every period (β2<0)
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3.1.2.1 – Positive Linear Growth
Linear Time Trend (β2 >0):x=7+2t
0
5
10
15
20
25
1 2 3 4 5 6 7 8
t
x x
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3.1.2.1 – Negative Linear Growth
Linear Time Trend (β2 <0):
x=250,000-10,000t
0
50000
100000
150000
200000
250000
300000
1 2 3 4 5 6 7 8
t
x x
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3.1.2.1 – Linear Growth Examples
Positive Slope Examples:
-Simple saving ($500/year, put into a matress)
-Age: Starting age +1 every 365.25 days
Negative Slope Example:
-40 Day Christmas countdown
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3.1.2.2 –Quadratic Functional Form
Quadratic :
Yt= β1 + β2 X + β3 X2
Slope: Changing (see graphs)
-U-shaped (β3 >0) or inverted U (β3 <0)
-negative growth, then no growth, then positive OR
-positive growth, then no growth, then negative
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3.1.2.2 U-shaped Quadratic Model
Quadratic Valley (β3 >0):
x=15-10t+t*t
-15
-10
-5
0
5
10
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Inverse U Quadratic Model
Quadratic Hill (β3 <0):
x=15+10t-t*t
0
10
20
30
40
50
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 –Quadratic Model
U-shaped Examples:
-Introvert meeting someone new: less comfortable then more comfortabled
-Investing: Decreased disposable income now for increased in future
Inverted U-shaped Examples:
-Working out: increases health before decreasing it from overwork
-Studying late at night: Improves mark before decreasing it
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3.1.2.2 Lin-Log Functional Form
Lin-Log:
Yt= β1 + β2 *ln(X)
Slope: Changing, positive or negative according to β2
-if β2 is positive, increases at a decreasing rate
-if β2 is negative, decreases at a decreasing rate
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3.1.2.2 Increasing Lin-Log Model
Lin-Log Trend (β2 >0):
x=7+40ln(t)
0
20
40
60
80
100
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Decreasing Lin-Log Model
Lin-Log Trend (β2 <0):x=100-40ln(t)
0
20
40
60
80
100
120
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Reciprocal Functional Form
Reciprocal:
Yt= β1 + β2(1/X)Slope: Changing and tricky:
-if β2 is negative, increases at a decreasing rate
-if β2 is positive, decreasing at a decreasing rate
-(sharper jumps than lin-log)
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3.1.2.2 Reciprocal Model
Reciprocal (β2<0):x=50-8(1/t)
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40
42
44
46
48
50
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Reciprocal Model
Reciprocal (β2>0):x=50+8(1/t)
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48
50
52
54
56
58
60
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Log-log Functional Form
Log-log:
ln(Yt)= β1 + β2 ln(X)
Slope: Changing, positive or negative according to β2
-shape depends on β2 (< or >1)
-more gradual/smooth than lin-log or reciprocal
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3.1.2.2 Increasing Log-log Model
Log-log (β2 >0):
x=5+8ln(t)
0
5
10
15
20
25
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Slightly Increasing Log-log Model
Log-log (0< β2 <1):
x=5+1/20ln(t)
4.9
4.95
5
5.05
5.1
5.15
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Slightly Decreasing Log-log Model
Log-log (-1< β2 <0):
x=5-1/20ln(t)
4.8
4.85
4.9
4.95
5
5.05
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Decreasing Log-log Model
Log-log (β2 <-1):
x=5-17ln(t)
-35-30-25-20-15-10
-505
10
1 2 3 4 5 6 7 8
t
x x
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3.1.2.2 Positive Slope ModelExamples:
-Studying: each hour yields less as you approach perfect (diminishing marginal returns)
-Pizza: Your enjoyment decreases after each piece (if enjoyment can become negative, u-shaped curve is appropriate)
-Race Dilemma: Keep running ½ of remaining distance to finishing line, never actually reaching it
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3.1.2.2 Negative Slope Models
Examples:
-Drugs: fries a lot of brain cells to start, then when your whole brain is fried, few left to fry
-Taxes: Relatively large when you’re rich, relatively small when you’re poor
-Earthquake: Each aftershock is less than the previous
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3.1.2.2 Logistic Functional Form
Logistic:
Xt eY
32)( 121
21
-if all β’s are positive and β2 > β1, slanted-S shape
-you can create or find a functional form to model any relationship
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3.1.2.2 Logistic Model
Logistic:
Xt eY
32)( 121
21
Examples:
Investing in undervalued stock. Little return, then huge increase (stock realized), then little return.
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3.1.2.2 Cyclical Functional Form
Cyclical:
Yt= β1 + β2 sin(2πX/p)
+ β3 cos(2πX/p)
-alternating negative and positive growth
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3.1.2.2 Cyclical Model
Cyclical
x=0+2sin(2pi*t/14)+2cos(2pi*t/14)
-4
-3
-2
-1
0
1
2
3
4
1 3 5 7 9 11 13 15 17 19
t
x x
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3.1.2.2 Cyclical Model
Examples:
Housing markets
Tech markets
Oil markets
Yearly seasonal markets (fruit, ice cream, etc.)
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3.2.1 Mathematical Models of Economic Relationships #1
Example 1 – Consumption Function
-consumption is based on income
-even with zero income, some consumption (autonomous consumption) occurs
-as income rises, consumption rises
-out of every new dollar earned, a fraction, the marginal propensity to consume (mpc) is spent on consumption – remainder is saved
-the mpc determines the slope of the graph
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3.2.1 Mathematical Models of Economic Relationships #1
constant slope/mpc – is this realistic?
Linear Consumption Function
0
100
200
300
400
500
600
700
0 100 200 300 400 500 600 700 800 900
Income
Co
ns
um
pti
on
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3.2.1 Mathematical Models of Economic Relationships #1
decreasing slope/mpc –is this realistic?
Quadratic Consumption Function
050
100150200
250300350400450
0 100 200 300 400 500 600 700 800 900
Income
Co
ns
um
pti
on
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3.2.1 Mathematical Models of Economic Relationships #1
Consumption Function – slope = mpc
Linear:
Consumption = 100+0.5income
mpc = dc/di = 0.5
Non-Linear
Consumption = 100+0.95income-0.001income2
mpc = dc/di = 0.95-0.002income
Are any other functional forms viable for consumption?
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3.2.2 Mathematical Models of Economic Relationships #2
Example 2 – Short-run Phillips Curve
-if no excess demand in the economy, the economy will be at the natural rate of unemployment
-if unemployment falls, wages and prices will tend upwards (hard to find workers)
-if unemployment rises, wages and prices will fall (easier to find workers)
-these changes are asymmetric (excess labour demand has a bigger effect than excess labour supply)
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3.2.2 Mathematical Models of Economic Relationships #2
Short-run Phillips Curve – lin-log function
Short-run Phillips Curve
-2
-1
0
1
2
3
4
5
6
1 2 3 4 5 6 7 8 9 10
Unemployment
Infl
ati
on
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3.2.2 Mathematical Models of Economic Relationships #2
Example 2 – Short-run Phillips Curve
Slope = short-run response of inflation to a change in unemployment
Inflation = 5 –ln(unemployment)
Slope = -1/unemployment
-as unemployment increases, a change in unemployment has less effect on inflation
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3.2.3 Mathematical Models of Economic Relationships #3
Example 3 – Demand for Xbox Gamma’s
-price of new game systems is often a hot topic
-Xbox’s sell for LESS than the Playstation
-lose MORE money on system in order to make more money on more games
-all else held equal, as price decreases, quantity demand increases
-Xbox sacrificing gain on each system to sell more systems (and later sell more games)
-downward sloping demand curve
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3.2.2 Mathematical Models of Economic Relationships #3
Example 3 – Linear demand for Xbox Gamma’s
Xbox Demand
0
1000000
2000000
3000000
4000000
5000000
6000000
0 100 200 300 400 500 600 700 800 900
Xbox Price
Xb
ox
De
ma
nd
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3.2.3 Xbox Math
Note that while demand functions are generally of the form:
Price21 dQ
Since price is on the y-axis, to graph this function we need to solve for price:
dQ2
1
2
1 1Price
But does a linear graph make the most sense? Will no one buy an expensive Xbox?Will someone say no to a free Xbox?
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3.2.3 Mathematical Models of Economic Relationships
Example 3 – Lin-Log demand for Xbox Gamma’s
Xbox Demand
0
1000000
2000000
3000000
4000000
5000000
6000000
0 100 200 300 400 500 600 700 800 900
Xbox Price
Xb
ox
es
De
ma
nd
ed
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3.2.5 Interpreting Parameters
Economists need to EXPLAIN MATHEMATICAL RELATIONSHIPS by explaining:
1) Intercepts
2) Slopes (first derivative)
3) Long-Run Effects (second derivative)
4) Elasticities
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3.2.5 Simple ExampleMark = 60 + 4 studyMark = percentage mark on midtermStudy = hours of study (up to 10 – it’s the
night before)Parameter Explanation:1) 60 = intercept – without studying, you’d
get a 60% on the exam - you genius you!2) 4 = coefficient of study (first derivative)–
every extra hour spent studying increases your mark by 4%
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3.2.5 Simple ExampleMark = 60 + 4 study3) D2mark/dstudy2=0, this function is a
straight line; each hour of studying is equally beneficial
4) Elasticity=dM/dSt(St/M)Elasticity=4(Study/Mark)a 1% increase in studying has a varying
impact on mark (depending on how much studying has occurred)
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3.2.5 Another Demand ExampleLet q=100-p/2
Parameter Explanation:1) The intercept (100) expresses the quantity
demanded when the good is free (p=0)
2) The slope (-1/2) indicates how much q changes as p increases by 1.
3) The second derivative is zero, we have another straight line
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3.2.5 Another Demand ExampleLet q=100-p/2
Parameter Explanation:4) Multiplying the slope by (p/q) gives elasticity
that varies over the function:Elasticity = (-1/2)(p/q)@ p,q=(10,95), elasticity =(-1/2)(10/95)=-0.05
-inelastic@ p,q=(120,40), elasticity=(-1/2)(120/40)=-1.5
-elastic
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3.2.5 Example with LogsLet ln(q)=3.912 -0.1ln(p)Parameter Explanation:1) The intercept:When ln(p)=0 (p=1), ln(q)=3.912 (q=50)2) The slope:
P
Q
dQ
Pd
Pd
Qd
Qd
dQ
dP
dQ1.0
)ln(
)ln(
)ln(
)ln(
We see here that at a low price, a price changecauses a large quantity change, whereas at ahigh price, a price change causes a smallquantity change
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3.2.5 Example with LogsLet ln(q)=3.912 -0.1ln(p)
Parameter Explanation:
3) The Second Derivative:d2Q/dP2=0.11Q/P2
Is always positive; the slope is always increasing
4) Elasticity-0.1, ln(p)’s parameter, is the elasticity of
demand with respect to priceThe elasticity is always inelastic.
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3.3 The error termAlthough economists try to model real behavior,
their attempts are not always 100% accurate, for a variety of valid reasons:
1)Excluded variables
2)Random events (shocks)
3)Error in data collection
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3.3 The ERROR termTo account for this, models always include an
error term:
Y= β1 + β2 X + β3 Z + β4 W + Ɛ
The error term, Ɛ, accounts for all of these discrepancies. We now have an econometric model!
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For a simple two variable model:
Yi= β1+ β2Xi+ei
Yi: value of the OBSERVABLE explanatory variable for observation i
Xi: value of the OBSERVABLE explanatory variable for observation I
β1 and β2: UNOBSERVABLE parameters or coefficients of the model
ei: or εi is the UNOBSERVABLE random error
3.3 The ERROR term
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3.3 Erroneous Example 1Iphone 7 Demand
ln(Qi) = β1 + β2ln(Pi) + ei
Qi = Iphone 7’s sold in state/province i
Pi = Price of Iphone 7 in state/province i
(cross sectional data)
Error: price of Androids, price/availability of itune Aps, price of phone and data plans, shipping constraints, shift in tastes (Huge Anti-Apple Surge)
![Page 51: 1 3. Mathematical Versions of Simple Growth Models Functional Forms Examples of Choosing Functional Forms Basic Theory of Econ Models The Error](https://reader030.vdocuments.net/reader030/viewer/2022032709/56649eb25503460f94bb8d5a/html5/thumbnails/51.jpg)
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3.3 Erroneous Example 2
Weight
Weighti = β1 + β2ln(dieti) + ei
Weighti = a given person’s weight
Diet = a given person’s diet
(a cross sectional study)
Error: differences in metabolism, exercise, height, liposuction
![Page 52: 1 3. Mathematical Versions of Simple Growth Models Functional Forms Examples of Choosing Functional Forms Basic Theory of Econ Models The Error](https://reader030.vdocuments.net/reader030/viewer/2022032709/56649eb25503460f94bb8d5a/html5/thumbnails/52.jpg)
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3.3 Erroneous Example 3
Depression
Depressioni = β1 + β2Econmarki + ei
Depressioni = level of depression at any point in the course
Econmarki = current econ mark
(a time series study)
Error: quality of last Walking Dead episode, social life, sport team standings, weather, success of Economics pick-up lines