intermediate microeconomics (consumer model) · pdf file! 1—3! 1...
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Intermediate Microeconomics 1 Economic Circumstances .................................................................................................. 1—3 1.1 Notes from lecture – 30th July ................................................................................. 1—3 1.1.1 What are economic circumstances? ............................................................ 1—3 1.1.2 How Economic Circumstances affect Choice (1) – Exogenous Income 1—3 1.1.3 How Economic Circumstances affect Choice (2) – Endogenous Income .................................................................................................................................... 1—9
2 Indifference Curves .......................................................................................................... 2—14 2.1 Notes from lecture – 6th August .......................................................................... 2—14 2.1.1 Notation ................................................................................................................ 2—14 2.1.2 Assumptions in Tastes ................................................................................... 2—15 2.1.3 Mapping Tastes ................................................................................................. 2—16 2.1.4 Utility functions – creating indifference curves mathematically 2—20
3 “Doing the Best We Can” – budgets and preferences ........................................ 3—22 3.1 Preferences Cont’d ................................................................................................... 3—22 3.1.1 Essential Goods ................................................................................................. 3—24
3.2 Choice ............................................................................................................................. 3—24 3.2.1 Cobb-‐Douglas functions ................................................................................ 3—25 3.2.2 Special Cases: Indifference Curves (Perfect Complements, Substitutes and Quasilinear) .............................................................................................................. 3—26 3.2.3 Special Cases: kinked lines ........................................................................... 3—26
4 Income and Substitution Effects ................................................................................. 4—27 4.1 Extending last week’s theory ............................................................................... 4—27 4.1.1 Cobb-‐Douglas Preferences ........................................................................... 4—27 4.1.2 Quasilinear preferences ................................................................................ 4—28
4.2 Creating the model – Changing Income .......................................................... 4—29 4.2.1 Changes in Consumption associated with a change in Income (Normal, neutral and inferior goods) ..................................................................... 4—29 4.2.2 Relative consumption – luxuries and necessities .............................. 4—32
4.3 Creating the model – Changing prices ............................................................. 4—34 4.3.1 Example: A carbon tax (let’s talk it through) ....................................... 4—34 4.3.2 Example – A Carbon Tax (in pictures!) ................................................... 4—35 4.3.3 Separating the Income and Substitution Effects ................................. 4—37 4.3.4 The substitution effect in more detail ..................................................... 4—39
4.4 The Wealth effect ...................................................................................................... 4—41 4.4.1 What’s the deal with Income? ..................................................................... 4—41
1—2
5 Demand for Goods, Supply of Labor and Capital ................................................. 5—42 5.1 Last week ...................................................................................................................... 5—42 5.1.1 How? ...................................................................................................................... 5—42 5.1.2 Goal for this week: ........................................................................................... 5—42
5.2 Demand and Supply Curves ................................................................................. 5—43 5.2.1 Demand for Goods ........................................................................................... 5—43 5.2.2 Supply of Labor ................................................................................................. 5—46 5.2.3 Supply of Capital ............................................................................................... 5—48
5.3 Consumer Welfare .................................................................................................... 5—49 5.3.1 Method 1: step by step ................................................................................... 5—49 5.3.2 Method 2: MWTP and the Compensated Demand Function ......... 5—49 5.3.3 My idea: ................................................................................................................ 5—49
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1 Economic Circumstances Chapters 2/3 of the Textbook
1.1 Notes from lecture – 30th July
1.1.1 What are economic circumstances? • The reasons for economic decision-‐making outside of consumer
preferences • For Example
o Prices o Income o Endowments (goods/assets, time)
• For the first part of the class, we will discuss choice with an endogenous income, and then we will move to an exogenous model.
1.1.2 How Economic Circumstances affect Choice (1) – Exogenous Income Here, income is given externally, and there is no way for the participant to modify it.
1.1.2.1 Example 1: Introducing the model, and changing Income • Let us give Patrick $2, and Johanan $4 • The price of goods are:
o Chocolate -‐ $2 o Cereal Bar -‐ $1
• Original Choices o Patrick chose: 2 Cereal bars o Johanan chose: 2 Cereal bars and 1 chocolate
• Let us plot their choices on a (Cereal, Chocolate) axis:
1—4
• But this is just one choice of many: what are the extreme choices? o Patrick
§ (2,0) or (0,1) o Johanan
§ (4,0) or (0,2) o Let us graph these
• This assumes that they spend all of their money – this is called the Budget
Line: the line representing the maximum they could possibly purchase • But, it is possible that Johanan might purchase (2,0) instead of (2,1): he
can still afford it. Thus, to represent all possible choices, we need a region – let us shade the appropriate region
o The blue region represents Johanan’s possible choice set – limited
by his income, and the purple region represents Patrick’s possible choice set
1—5
• Now, let us try model the choice set algebraically: first, let me introduce the notation
o C(P1,P2, I ) -‐ The Choice Set in this case is dependent on the Prices and the Income: these are the “Economic Circumstances”
o C(P1,P2, I ) = (x1, x2 )∈ R+2{ }
§ The Choice Set may be valid for any value of both goods (x1, x2 )
§ As long as they are both positive rational numbers ∈ R +2
o However, this isn’t correct yet: yes, both goods choices have to be positive and rational, but that isn’t enough – we have to include a separate condition that limits the choices to the budget
o C(P1,P2, I ) = (x1, x2 )∈ R+2 | P1x1 +P2x2 ≤ I{ } This is read as:
§ The choice set is dependent on the prices in the market, and the income of the choice-‐maker
§ The possible choices for good 1 and 2 must be positive rational numbers
§ However, they must also satisfy the following condition: P1x1 +P2x2 ≤ I , that is:
• The price of the first good multiplied by the respective quantity, added to the price of the second good multiplied by the respective quantity must be less than the budget (I)
• Or: you can’t spend more in total than you have. • To model the budget line, we do the same, except the inequality
P1x1 +P2x2 ≤ I (representing the fact that you must spend less than what you have) becomes an equation P1x1 +P2x2 = I (representing that you must spend all that you have). In its full form, it is as follows:
o B(P1,P2, I ) = (x1, x2 )∈ R+2 | P1x1 +P2x2 = I{ }
§ As above § BUT price of the first good multiplied by the respective
quantity, added to the price of the second good multiplied by the respective quantity must be equal to the budget (I)
• Or: you must spend every cent you have.
1—6
• This is the formal way of looking at it, but, for simplicity, we just write it as an equation of the budget line:
o I = P1x1 +P2x2 • But, it since I,P1,P2 are given, it is best to make it an equation of x2 in
terms of x1 (so that it is in the easily workable form of y =mx + b )
o I = P1x1 +P2x2−P2x2 = P1x1 − I
x2 =IP2−P1P2x1
o Thus, we can see that the gradient of the line is dependent on the
prices in the market P1P2 (representing the opportunity cost of
chosing one good over the other), and the intercept of the line is
dependent on both price and income IP2
As an example, let us model the line discussed above, for Patrick and Johanan • Patrick’s Choice Set
o CPatrick (P1,P2, I ) = (x1, x2 )∈ R+2 | P1x1 +P2x2 ≤ I{ }
o CPatrick (1, 2, 2) = (x1, x2 )∈ R+2 | (1)x1 + (2)x2 ≤ 2{ }
• Patrick’s Budget Line o BPatrick (1, 2, 2) = (x1, x2 )∈ R+
2 | (1)x1 + (2)x2 = 2{ } o BPatrick ⇒ (1)x1 + (2)x2 = 2
o x2 =1−12x1
• Johanan’s Choice Set o CJohanan (1, 2, 2) = (x1, x2 )∈ R+
2 | (1)x1 + (2)x2 ≤ 4{ } • Johanan’s Budget Line
o BJohanan ⇒ x2 =IP2−P1P2x1
o x2 = 2−12x1
• As expected, the gradient didn’t change, since there is no change in prices in the market: all that changed between Johanan and Patrick is the Income, and, therefore, the intercept of the graph – in line with what we showed earlier, graphically.
• In an exogenous income model, a change in income alone will shift the budget line, parallel to the original.
1—7
1.1.2.2 Example 2: Changing prices • Let us compare the above with JJ, who has received $4, but has different
prices assigned to her P1 = 2;P2 = 2 • First, let us compare the budget lines of Johanan and JJ:
o BJJ ⇒ x2 =
IP2−P1P2x1
x2 =42−22x1
o BJJ ⇒ x2 = 2−1x1
o BJohanan ⇒ x2 = 2−12x1
• Johanan and JJ have the same x2 intercept, but Johanan has a flatter gradient – what does this mean for the available choice set? To see, let us graph this using JJ’s two extremes (0,2) and (2,0).
• As we can see, Johanan and JJ can afford the same amount of chocolate, but, due to the price difference, and thus, the change in gradient, Johanan can afford more cereal in the alternative (there is a smaller opportunity cost for choosing Cereal) – thus, he has the more diverse choice set.
• In an exogenous income model, a change in prices changes the gradient of the budget line (and, perhaps, the intercept too).
1—8
1.1.2.3 More than two goods • Modeling a choice between two goods is convenient, since it is easy to
graph (you only need two dimensions). • However, it is possible to algebraically model a choice between any
number of goods: o C(P1,P2,....,Pn, I ) = (x1, x2,..., xn )∈ R+
2 | P1x1 +P2x2 +...+Pnxn ≤ I{ } • For the sake of ease, we usually synthesise all other choices (against the
good we are focusing on) into a composite good “C”. Thus, the above becomes:
o C(P1,PC, I ) = (x1,C)∈ R+2 | P1x1 +PCC ≤ I{ }
• This allows us to use the same graphical model, in two dimensions, which we are used to.
1—9
1.1.3 How Economic Circumstances affect Choice (2) – Endogenous Income Here, the participant may modify his income by (e.g.) selling goods to buy others; trading off leisure time for consumption; saving today to spend more tomorrow; etc.
1.1.3.1 Example 1: Endogenous Income via a Goods Endowment Say that instead of money, Henry was given (2,2) at prices P1 = 2;P2 =1 (he receives two chocolates worth $2 each, and two cereal bars worth $1 each).
• Let us start by plotting the endowment
• And the extremes of the budget line
o If he sells both chocolates, he would receive $4, allowing him to buy 4 additional cereal bars, totaling 6 cereal bars and 0 chocolates
o If he sells both cereal bars, he would receive $2, allowing him to buy 1 additional chocolate, totaling 0 cereals and 3 chocolates
• This is just like what we are used to: and, like we are used to, a change in
the endowment will shift the graph without pivoting it
1—10
• Let us look at what happens when we change the price levels in the economy – let us start by looking at it intuitively
o Say that the prices changed to P1 = 2;P2 = 2 o He was endowed with 2 cereal bars and 2 chocolate, so that point
would remain unchanged o However, the extreme points change: if he sold both cereal bars, he
would be able to buy 2 additional chocolate bars (0,4); if he sold both chocolates, he would be able to buy 2 additional cereal bars (4,0)
Thus, a change in prices in an exogenous (endowment) situation will cause a pivot of the graph around the endowment point (you will always be able to afford the endowment, since it is yours already). Now, let us model this algebraically:
• Before, our budget line was as follows o B(P1,P2, I ) = (x1, x2 )∈ R+
2 | P1x1 +P2x2 = I{ } • But that was when our income was exogenous: how are we to model the
sale of goods as a generator of income? By replacing the Income term with a potential for sale
o Income now becomes the total price for the sale of the endowment o B(P1,P2,e1,e2 ) = (x1, x2 )∈ R+
2 | P1x1 +P2x2 = e1P1 + e2P2{ } o Where the “e” terms are the endowment
Now, let us put this in the form y=mx+b • B⇒ P1x1 +P2x2 = P1e1 +P2e2
• x2 =P1e1 +P2e2
P2−P1P2x1
• There is no longer income, there is wealth (the total value of your endowment, sold):
Now, a change in Prices affects both the intercept and the gradient
1—11
1.1.3.2 Example 2: Endogenous Income – workers work and change of wages Work is a tradeoff between leisure time (l) and consumption (C).
• L = time endowment (total hours the worker is able to work in a week) • W = wage rate (the gain from work, or the economic cost of leisure) • C = Consumption (an aggregate of consumption of a composite good) • l = hours of leisure
The following graph shows the choice for a person with 6 possible work hours (or, an endowment of (6,0)).
• As we can see, the more leisure time, the less total consumption, as could
be guessed. • Further, since the endowment point is at the x intercept, (the “leisure”
intercept), a change in wages will not increase the leisure hours available, merely modify the gradient of the line representing an increased or decreased opportunity cost of leisure.
Let us now fit this into our algebraic model: • C(W,L) = (l,C)∈ R+
2 |Wl +C ≤ LW{ } o Or, the choice set is dependant on the amount of hours (L)
available, and the wage level o Further, the choice between leisure and consumption must satisfy
the following: § The opportunity cost of leisure (Wl) § Added to the level of consumption (C) § Must be less than the total possible wages available (LW)
1—12
Let us now try to model this same equation, but on a different set of axis (Hours Worked, Consumption). Whilst all the above graphs plotted one good against another, here, we are plotting a good against a “bad”.
• Here, as hours work increase, consumption can increase, thus, there is a
positive gradient Let us try to put this algebraically:
• h – Hours worked • H – Total possible hours • W – wage rate • C – Consumption
C(W,H ) = (h,C)∈ R+2 |C ≤ hW | h ≤ H{ }
• Or, the formula to be satisfied is Consumption has to be equal to or less than wages earned (hW).
• And, you must work less than the total hours available
0 1 2 3 4 5 6 7
1
2
3
4
5
Hours Worked
Con
sum
ptio
n
E(5,6)
1—13
1.1.3.3 Example 4: Endogenous Income – Borrowing v saving Here, we have an inter-‐temporal model, where we plot consumption in Period 1 (C1 ) against consumption in Period 2 (C2 ), giving us the plane (C1,C2 ).
• The terms for this example are as follows o C1 -‐ consumption in the first period o C2 -‐ consumption in the second period o r – interest rate (to be earned by saving, to be lost by borrowing) o Y1 -‐ income in period 1 o Y2 -‐ income in period 2
• Note, the maximum amount of money that can be borrowed in the first period is the amount that can be paid back in the second period (i.e. the second amount discounted to today) à Y2 •(1− r) , allowing a maximal expenditure in the first period of Y1 +Y2 (1− r)
• The gains from saving today’s income for expenditure tomorrow are Y1 •(1+ r) , allowing for a maximal expenditure in period 2 (assume no expenditure in period 1) of Y2 +Y1(1+ r)
Graphing this, we get:
And, plugging this into our algebraic form, we get:
• C(r,Y1,Y2 ) = (C1,C2 )∈ R+2 |Y1(1+ r)+Y2 ≤C1(1+ r)+C2{ }
Remember: the point of endowment (the pivot point for a change in interest rates), is full consumption of Y1 in P1 (i.e. C1 =Y1 ) and full consumption of Y2 in P2 (C2 =Y2 ), which is at the center of the graph – any pivot happens around that point.
0 1 2 3 4 5
1
2
3
4
5
Consumption in Period 1
Con
sum
pti
on in
Per
iod 2
E(0,Y1(1+r)+Y2)
E(Y1+Y2(1-r),0)
More saving
2—14
2 Indifference Curves Chapters 4/5 of the Textbook
2.1 Notes from lecture – 6th August Last week we discussed what certain economic circumstances allowed a person to afford: either via exogenous income or endowment, via endogenous returns to labor or via spending or saving. What we mapped were possible consumption points. This week, we look at the other side of the equation, we map what the consumer would most like to consume, as apposed to what is possible. It is important to note that different persons have different tastes/preferences, but we are trying to map a generality. We will use the terminology “indifference” to map all points of consumption where the consumer is equally satisfied.
2.1.1 Notation • A(x1
A, x2A ) -‐ Bundle “A” which consists of x1
Aunits of consumption of good “1” and x2
A of good 2 • B(x1
B, x2B ) -‐ Bundle “B”
• Squiggly ≥ -‐ at least as good as • -‐ Strictly preferred
o ≠> -‐ Strictly better • ~ -‐ As good as – “indifferent”
2—15
2.1.2 Assumptions in Tastes 1. Complete tastes
a. The decider can always make a choice – they never “can’t choose” between options
2. Transitive Tastes a. The decider’s tastes are always syllogistically sound b. i.e. a < b < c not a < b;b < c;c < a
3. Monotonicity a. More is better, or at least not worse b. i.e.
i. Given two bundles, A(x1A, x2
A ) and B(x1B, x2
B ) ii. Where x1
A > x1B and x2
A > x2B
iii. Then A(x1A, x2
A ) B(x1B, x2
B ) c. This is called strict monotonicity d. Where the is replaced with a squiggly ≥ , there is simple
monotonicity 4. Convexity
a. Averages are better than extremes, or at least not worse b. i.e.
i. Assuming A ~ B and, given that 0 <α <1(alpha is just a term used to average A and B)
ii. αA+ (1−α)B A and αA+ (1−α)B B c. This means that, drawing a line between any two indifferent points
on a graph, any points on the line are at least as good, if not better, than the points at the extremes
d. This is because they lie on “further” indifference curves 5. Continuity
a. There are no sudden “jumps” in satisfaction
2—16
2.1.3 Mapping Tastes Let us take a point A, and divide the plane into quadrants around it.
Using our assumptions, we can model where the indifference curve must lie:
• Monotonicity excludes the NE and SE quadrants o In the SW quadrant, any point has strictly less of both goods,
therefore the chooser will be less satisfied with a point in that quadrant
o In the NW quadrant, any point has strictly more of both goods, therefore the chooser will be more satisfied with a point in that quadrant
o Therefore, the “satisfaction” change in both quadrants excludes an indifference curve from passing through either quadrant.
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
X1(Meusli)
X2(Chocolates)
A(2,7)
NE
SESW
NW
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
X1(Meusli)
X2(Chocolates)
A(2,7)
NE
SESW
NW
2—17
To further refine our search for an indifference curve, we need a point at least one more point of indifference: B, where A~B. The line between A and B represents the “averages” between them.
Convexity tells us that each point on the line between A and B is preferred to both A and B.
• Here, point F is preferred to A and B because of Convexity. Therefore every point on the line is excluded
• Further, any point above the line is excluded, since monotonicity states it is preferable to a point on a line, which is, in turn, preferable to both A and B.
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
X1(Meusli)
X2(Chocolates)
A(2,7)
B(7,2)
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
X1(Meusli)
X2(Chocolates)
A(2,7)
B(7,2)
F(5,4)
G(7,6)
2—18
From this, we can see that the plane for drawing the indifference curve has been drastically reduced. Following the rule of convexity – the closer to the middle of A and B, the more preferred the point would be. Thus, our indifference curve would look something like as follows:
Points G and F would appear on “further” indifference curves, making them preferred, rather than indifferent to points A and B.
Note, the slope of the indifference curve at any point is the Marginal Rate of Substitution (MRS): the willingness to trade one less unit of x1 for an additional unit of x2 (or, the number of x2 's needed to convince me to trade one x1 ).
• Because of strict convexity, there is a diminishing marginal rate of substitution – as you have more of x1 , we are willing to trade less for an additional unit of it.
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
X1(Meusli)
X2(Chocolates)
A(2,7)
B(7,2)
F(5,4)
G(7,6)
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
X1(Meusli)
X2(Chocolates)
A(2,7)
B(7,2)
F(5,4)
G(7,6)
2—19
Note 2, whilst indifference curves do not have to be parallel, they can never cross, since that would violate transitivity
• Here, point A ~ B, and point B ~ C • But, because of monotonicity A C • This violates transitivity, since transitivity would require A ~ B ~ C
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
X1(Meusli)
X2(Chocolates)
A(2,7)
B(7,2)
C(4,7)
2—20
2.1.4 Utility functions – creating indifference curves mathematically Utility functions assign a value to the “utility” received from each bundle of goods. To illustrate this, let us start with a simple utility function: u(x1, x2 ) = x1 • x2
• This is read as: the utility from consumption of both goods is equal to the product of both goods
o Given point A, where A(4,8) § The utility from this point would be: u(x1, x2 ) = (4•8) = 32
o Now, given point B, where B(2, 9) § The utility from this point would be: u(x1, x2 ) = 2•9 =18
o HereuA > uB , therefore bundle A is to be preferred (uA > uB →∴A B )
• This is a two way relationship: o If (x1
A, x2A ) (x1
B, x2B )→ uA > uB
o If uA > uB → (x1A, x2
A ) (x1B, x2
B ) • That is, if the utility of bundle A is greater than the utility of bundle B,
bundle A is to be preferred o And, if bundle A is preferred to bundle B, it is indicative that
bundle A has a higher utility Now, let us try to derive from a utility function and a point an indifference curve.
• The indifference curve exists where the utility is unchanged: therefore, du = 0 (there is no change to u).
• But how do we find this?
o du = ∂u(x1, x2 )∂x1
•dx1 +∂u(x1, x2 )∂x2
•dx2 = 0
o This looks complicated, but isn’t
§ ∂u(x1, x2 )∂x1
•dx1Means, simply, what is the change in utility (
∂u ) for a given change in x1 (∂x1 ). Or, what is the marginal utility of x1 -‐ how much extra utility do I get for each additional unit of x1 .
§ Concurrently, ∂u(x1, x2 )∂x2
•dx2 is the marginal utility of x2
o Now, let us rearrange a little, we get:dx2dx1
= −∂u(x1, x2 ) /∂x1∂u(x1, x2 ) /∂x2
= −MRUx1
MRUx2
=MRS
§ The Marginal Rate of Substitution is equal to the ratio of Marginal Utilities
§ Which makes sense: the more utility I get from a good, the more I am willing to substitute for it
§ The negative of the partial derivative with respect to x1 over the partial derivative with respect to x2
2—21
2.1.4.1 Types of Preferences • Perfect substitutes • Perfect Complements • Diminishing
3—22
3 “Doing the Best We Can” – budgets and preferences Chapters 5 and 6 of the textbook
3.1 Preferences Cont’d Preferences have the following assumptions
• Rationality o Completeness (can always choose between preferred, not
preferred and indifferent) o Transitivity (syllogistically sound)
• Other o Convexity (averages preferred) o Monotonicity (more of both is better) o Continuity (no jumps in happiness)
Utility Functions
• MRS = − MRU1
MRU2
o Or, the Marginal Rate of Substitution is equal to negative the ratio of the Marginal Rates of Utility (change in utility for a given change in x1 or x2 )
• MRS = − ∂(x1, x2 ) /∂x1∂(x1, x2 ) /∂x2
o Or, the MRS is equal to negative the partial derivative of the utility function with respect to x1over the partial derivative of the utility function with respect to x2 .
Types of Preferences • Perfect Substitutes
o u =αx1 +βx2
o MRS = − ∂(x1, x2 ) /∂x1
∂(x1, x2 ) /∂x2
= −αβ
o Thus, the MRS is constant for perfect substitutes
u=ax1+bx2
MRS = -a/b
3—23
• Perfect complements o u =min{αx1,βx2}
o
MRS = ?@pivot = undefined@vertical =∞@horizontal = 0
• Cobb-‐Douglas (most common/logical)
o u = x1αx2
β
o
MRS = − ∂(x1, x2 ) /∂x1∂(x1, x2 ) /∂x2
= −α(x1
(α−1) )(x2β )
β(x2(β−1) )(x1
α )
= −α • x1
α • x1−1 • x2
β
β • x2β • x2
−1 • x1α
= −α • xβ2 • x
−β2 • x2
β • xα1 • x−α1 • x1
= −αx2
1+β−β
βx11+α−α
= −αx1βx2
• This week – we have a new one! Quasilinear preferences
o This type of preference is like Cobb-‐Douglas, but instead of the preference depending on the quantity of both goods, it is dependent on 1.
o u(x1, x2 ) =α •v(x1)+βx2 § Or, the utility is dependent on a constant multiplied by a
function of x1à e.g.→ v(x1) plus a constant multiplied by x2
o MRS = − ∂(x1, x2 ) /∂x1
∂(x1, x2 ) /∂x2
= −α •v '(x)
β
u=min{ax1,bx2}
MRS = infinity
MRS = undefined MRS = 0
MRS = -ax2/bx1
u= x1^a*x2^b
3—24
o Eg:
§
u =α • ln(x1)+βx2
MRS = − ∂(x1, x2 ) /∂x1∂(x1, x2 ) /∂x2
= −α •
1x1
β= −
αx1β
= −αβx1
§ In this case, the gradient is only dependent on x1 : the gradient is unchanged through x2
3.1.1 Essential Goods • Essential goods are goods where you are not willing to take a package
where there are non of that good
3.2 Choice When trying to optimize choice we look to be on the furthest possible indifference curve possible for a given budget.
u= aln(x1)+bx2
MRS=-a/bx1
3—25
3.2.1 Cobb-‐Douglas functions With a Cobb-‐Douglas function, the optimum choice occurs when the budget line is tangential to the indifference curve at the point of consumption. This can be expressed as: the optimum choice occurs when the gradient of the budget line is equal to the gradient, at that point, of the indifference curve à mbudget =MRS . To find this, we simultaneously equate the MRS equation and the budget line. u(x1, x2 ) = x1
αx2β
MRS = − ∂(x1, x2 ) /∂x1∂(x1, x2 ) /∂x2
= −αx1
α−1x2β
βx2β−1x1
α
= −α(x2
β )(x21)(x2
−β )β(x1
α )(x11)(x1
−α )= −
αx2(1+β−β )
βx1(1+α−α )
m =MRS = −αx2βx1
⇒ "eq.1"
C(P1,P2, I ) = x1,C( )∈ R+2 | P1x1 +P2x2 = I{ }
I = P1x1 +P2x2
x2 =IP2−P1P2x1
m = −P1P2⇒ "eq.2"
Simultaneously equate the gradients to isolate a variable. "eq.1"⇒ into⇒ "eq.2"
−P1P2= −
αx2βx1
P1βx1 = P2αx2
x2 =P1βP2α
x1⇒ "eq.3"
Substitute this variable into the Budget Line Function to find optimum points (sub “eq.3” into the Budget Line Function). I = P1x1 +P2x2
I = P1x1 + P2P1βP2α
x1
P1x1 +P1βαx1 = I
P1x1α +P1βx1 = IαP1x1(α +β) = Iα
x1 =Iα
P1(α +β)
Substitute this variable in to find x2 The point generated (x1, x2 ) is, prima facie, the optimum consumption point.
3—26
3.2.2 Special Cases: Indifference Curves (Perfect Complements, Substitutes and Quasilinear)
• Perfect complements are straight line graphs o Thus, if their gradient is steeper than the budget line, the x
intercept is the point of optimum consumption o If it is shallower, the y intercept o If they are equal, any point
• Perfect substitutes o u(x1, x2 ) =min αx1,βx2{ } o The optimum usually occurs at the corner, so we simultaneously
equate the budget line with αx1 = βx2
§ i.e. by substituting x1 =βαx2 into I = P1x1 +P2x2
• Quasilinear o Solve as per Cobb-‐Douglas, but be aware, if the optimum
consumption point is negative, you have a corner solution
3.2.3 Special Cases: kinked lines • If kinked out, most likely optimum at the kink • If kinked in, more complex, could have a point with two tangencies.
4—27
4 Income and Substitution Effects
4.1 Extending last week’s theory Last week, we matched utility functions with budget lines to find the points of optimum consumption. Now, by solving this using “blank” formula, we can come up with demand functions – a formula expressing how much is consumed of each good, depending on the economic circumstances.
4.1.1 Cobb-‐Douglas Preferences
4.1.1.1 Utility and Budget Line u(x1, x2 ) = x1
α • x2β I = P1x1 +P2x2
4.1.1.2 Finding MRS and Op. Cost
MRS = −δ(x1, x2 )
δx1δ(x1, x2 )
δx2
= −αx1
α−1 • x2β
x1α •βx2
β−1
= −α • x1
α • x1−1 • x2
β
β • x1α • x2
−1 • x2β= −
α • x2 • x2β • x2
−β
β • x1α • x1
−α • x1
= −αx2βx1
I = P1x1 +P2x2P2x2 = I −P1x1
x2 = −P1P2x1 +
IP2
y =mx + b
m = op.Cost = − P1P2
4.1.1.3 At optimum consumption, MRS=Op.Cost – isolate a variable
op.Cost =MRS
−P1P2= −
αx2βx1
βx1P1 =αx2P2
x1 =
αP2βP1
x2
x2 =βP1αP2
x1
4.1.1.4 Find the Demand Function, mother schlucker
x1 =αP2βP1
x2 → I = P1x1 +P2x2
I = P1αP2β P1
x2 +P2x2 =αβP2x2 +P2x2
I = P2x2αβ+1
"
#$
%
&'= P2x2
α +ββ
"
#$
%
&'
x2 =β
α +β( )•IP2
x2 =βP1αP2
x1→ I = P1x1 +P2x2
I = P1x1 + P2βP1α P2
x1 =βαP1x1 +P1x1
I = P1x1 1+βα
"
#$
%
&'= P1x1
α +βα
"
#$
%
&'
x1 =α
α +β( )•IP1
4—28
4.1.2 Quasilinear preferences
4.1.2.1 Utility and Budget Line u(x1, x2 ) =α • ln x1 +βx2 I = P1x1 +P2x2
4.1.2.2 Finding MRS and Op. Cost
MRS = −δ(x1, x2 )
δx1δ(x1, x2 )
δx2
MRS = −αx1β
= −αβx1
I = P1x1 +P2x2
x2 = −P1P2x1 +
IP2
m = op.Cost = − P1P2
4.1.2.3 At optimum consumption, MRS=Op.Cost – isolate a variable MRS = op.Cost
−αβx1
= −P1P2
P1βx1 =αP2
x1 =αP2βP1
4.1.2.4 Find the Demand Function, mother schlucker
x1 =αP2βP1
→ I = P1x1 +P2x2
I = P1αP2β P1
+P2x2
I = αP2β
+P2x2
P2x2 = I −αP2β
x2 =IP2−αP2βP2
=Iβ −αP2βP2
x2 =Iβ −αP2βP2
x1 =αP2βP1
4—29
4.2 Creating the model – Changing Income
4.2.1 Changes in Consumption associated with a change in Income (Normal, neutral and inferior goods)
For a change in income, what is the effect on consumption? • Normal good
o Increase in Income is echoes by an increase in consumption o Thus, there is a positive income effect
• Neutral good (quasilinear) o Increase in Income leads to a borderline result
• Inferior good o Increase in Income leads to a decrease in consumption
So, what are we actually finding?
• δx1(P1,P2,Y )δI
: the change in x1 related to a change in Income
• If this derivative is greater than 0, it is a normal good; if it is equal to 0, it is a quasilinear good; and, if it is less than 0, it is an inferior good.
Mathematically
• Cobb-‐Douglas: U = x1α • xβ2
o If the demand function is: x1 =α
α +β( )•IP1
o Then δx1(P1,P2,Y )δI
(the derivative with respect to I)
o x1 =
αα +β( )
•IP1
δx1(P1,P2,Y )δI
=α
(α +β)P1> 0
o Since all the variables here are positive, Cobb-‐Douglas functions are necessarily “normal” goods
x1
x2
Y1 Y2
x2a
x2b
x1a x1b
4—30
o Here an increase in Y from Y1 to Y2 leads to an increase in x1 (from x1a to x1b) and an increase in x2 (from x2a to x2b) – Cobb-‐Douglas functions behave as “normal” goods
• Quasilinear goods: u(x1, x2 ) =α • ln x1 +βx2
o Regarding x1 § The demand function of a quasilinear good in x1 is:
x1 =αP2βP1
§ Then δx1(P1,P2,Y )δI
(the derivative with respect to I)
§ x1 =
αP2βP1
δx1(P1,P2,Y )δI
= 0
o Regarding x2 § The demand function of a quasilinear good in x2 is:
x2 =Iβ −αP2βP2
§ Then δx1(P1,P2,Y )δI
(the derivative with respect to I)
§ x2 =
Iβ −αP2βP2
=IββP2
−αP2βP2
δx2 (P1,P2,Y )δI
=ββP2
> 0
§ This is necessarily positive, and thus “normal” o Thus, quasilinear goods have a neutral income effect in the
variable which is a function, and are normal in the other variable
o Here, an increase in Y from Y1 to Y2 leads to an increase in x2
(from x2a to x2b) but no increase in x1a – Quasilinear preferences may lead to one good being “neutral” and the other being normal
x1
x2
Y1 Y2
x2a
x2b
x1a
4—31
• Inferior goods are difficult to model mathematically, but here is a graphical example
o Here, an increase in Y from Y1 to Y2 leads to an increase in x2 (from x2a to x2b) but a decrease in x1 (from x1a to x1b) – here, we have a normal good in x2, and an inferior good in x1
x1
x2
Y1 Y2
x2a
x2b
x1ax1b
4—32
4.2.2 Relative consumption – luxuries and necessities Now we look to the scale of the increase in consumption associated with the increase in income (as apposed to the “sign” of the change, as above).
• Where the percentage change in consumption is greater than the change in income, there is a luxury good.
o x1(P1,P2, t( )• I )> tx1(P1,P2, I ) |∀t >1 o i.e. for an increase of income by a multiple of t, there is an increase
of consumption of x1 of more than t • Where the percentage change in consumption is equal to the change in
income, there is a borderline (homothetic) good. o x1(P1,P2, t( )• I ) = tx1(P1,P2, I ) |∀t >1
• Where the percentage change in consumption is less than the change in income, the good is a necessity
o x1(P1,P2, t( )• I )< tx1(P1,P2, I ) |∀t >1 Examples
• Quasilinear Preferences as luxury goods
o
u x1, x2( ) = βx1 +α ln x2( )
x1(P1,P2, I ) =βI −P1αβP1
I = 6 | P1 =1| P2 =1|α = 2 | β =1
x1(P1,P2, I ) =(1)(6)−1(2)(1)(1)
= 4
I = 6→ I =12 | "t"= 2[ ]
x1(P1,P2, (2)I ) =(1)(12)−1(2)
(1)(1)=10 = (2.5)x1(P1,P2, I )
∴x1(P1,P2, (t)I )> (t)x1(P1,P2, I )
o Increasing Income by a multiple of 2 lead to an increase of consumption of x1 by a multiple of 2.5. Thus, there is a greater increase in consumption than the increase in income – this is a luxury good
• Cobb-‐Douglas Preferences as borderline goods
o
u(x1, x2 ) = x1α • x2
β
x1(P1,P2, I ) =α
α +βIP1
I = 6 | P1 =1| P2 =1|α = 2 | β =1
x1(P1,P2, I ) =(2)
(2)+ (1)(6)(1)
= 6
I = 6→ I =12 "t"= 2[ ]
x1(P1,P2, (2)I ) =(2)
(2)+ (1)(12)(1)
=12 = 2x1(P1,P2, I )
∴x1(P1,P2, tI ) = tx1(P1,P2, I )
o Increasing Income by a multiple of 2 lead to an increase of consumption of x1 by a multiple of 2. Thus, there is an equal
4—33
increase in consumption than the increase in income – this is a borderline good
• Quasilinear preferences as necessities
o
u(x1, x2 ) =α ln(x1)+βx2
x1(P1,P2, I ) =P2αP1β
I = 6 | P1 =1| P2 =1|α = 2 | β =1
x1(P1,P2, I ) =(1)(2)(1)(1)
= 2
I = 6→ I =12 |"t"= 2
x1(P1,P2, (2)I ) =(1)(2)(1)(1)
= 2
∴x1(P1,P2, (t)I )< tx1(P1,P2, I )
o Here, a doubling of income lead to no increase in consumption of x1 (a multiple of 1 instead of 2), therefore this is a necessity.
4—34
4.3 Creating the model – Changing prices This section looks at changing the opportunity cost of good, and seeing the effects on consumption and the requisite compensation required to maintain utility.
4.3.1 Example: A carbon tax (let’s talk it through) The standard example shows that, given a tax, and a proportionate redistribution of the taxed revenue so that utility is unchanged, consumption choices may change. A brief explanation of this example – the carbon tax -‐ is below: The government sees a problem in that a carbon externality is not being factored into production decisions, causing over-‐emission of carbon. The government, therefore taxes carbon emissions. This causes emissions in carbon to fall, because emitting carbon becomes more expensive. However, it also leads to substantial hardship in lower income households. Thus, the government reimburses the tax to lower income households, either through direct contributions, or through tax credits. This is worked out to equal out, in utility terms, the “hardship” created by the carbon tax. Detractors say that, since the tax is offset by an equal payout, the effect of the tax is reduced. However, this is a simplistic analysis. Whilst the “utility” has stayed equal, the change in relative prices has made carbon intensive goods more expensive than non-‐carbon intensive goods. Thus, there is a shift in consumption – even though the same amount (in utility terms) is consumed, the consumption shifts to the relatively cheaper (i.e. non-‐carbon intensive) good. There is a shift along the indifference curve.
4—35
4.3.2 Example – A Carbon Tax (in pictures!) First, the government sees that consumption of carbon intensive goods is too high (e.g. – the starting point).
So, the government taxes carbon intensive goods, making them less affordable.
Non-Carbon Intensive Good
Car
bon
Inte
nsiv
e G
ood
C1
Non-Carbon Intensive Good
Car
bon
Inte
nsiv
e G
ood
C1
Ctax
4—36
After public outcry, the government raises income so that utility was as per the beginning.
This shows the point of the example. Whilst Ctax had the lowest consumption of carbon intensive goods, it also had a lower utility (as shown by the lower indifference curve that it sits on). When redistribution occurred, so that consumption was now at Cfinal, we have the same utility as in the first example (both consumption points lie on the same indifference curve). However, because of the change of slope of the budget line (change in relative prices of carbon intensive and non-‐carbon intensive goods), the consumption is on a different point on the indifference curve where there is the same utility, but lower consumption of carbon intensive goods.
Non-Carbon Intensive Good
Car
bon
Inte
nsiv
e G
ood
C1
Ctax
Cfinal
4—37
4.3.3 Separating the Income and Substitution Effects The income and substitute effects are separated thusly:
Verbally
• The movement from CA to CB is due to the change in opportunity cost – this is the substitution effect.
• The movement from CB to CC is due to the change in income – the income effect.
• These together, or, the movement from CA to CC, is the “total effect” of the price change.
Non-Carbon Intensive Good
Car
bon
Inte
nsiv
e G
ood
C(B)
C(C)
C(A)
4—38
4.3.3.1 The Slutzky Equation with Hicksian substitution • This has been formalized in the “Slutzky Equation”
o
x1A → x1
C{ }= x1A → x1
B{ }− x1B → x1
C{ }δx1(P1,P2, I )
δP1
#$%
&'(=
δx1δP1
| u#$%
&'(−δx1δI
x1#$%
&'(
total.effect{ }= sub.effect{ } Y.effect{ }>=<{ }= <={ } >=<{ }
o Simply, there are 3 propositions § The move from Xa to Xc = move from Xa to Xb minus the
move from Xb to Xc § Or, the change in X1 for a given change in P1 (derivative of
demand function with respect to P1) (can be greater, equal to or less than 0)
• Is equal to the derivative of the demand function with respect to P1, holding utility equal (must be equal to or greater than 0)
• Minus the derivative of x1 with respect to income, multiplied by x1 (can be equal to, less than or greater than 0
o The total effect is equal to the substitution effect minus the income effect
4.3.3.2 The Law of Demand For a normal good, there is a decrease in demand for a relative increase in the price of that good. δx1(P1,P2, I )
δP1> 0
4.3.3.3 Giffen Good Where an increase in the relative price of a good leads to an increase in demand for the same good. δx1(P1,P2, I )
δP1< 0
4—39
4.3.4 The substitution effect in more detail The “size” of the substitution effect depends on the substitutability between goods – the more substitutable goods are, the greater the substitution effect. What we know from the graphical analysis:
1. The Consumption point B satisfies the following a. MRS=new op.Cost b. Utility is unchanged
2. Thus, we have a system of two equations in two unknowns
4—40
4.3.4.1 Four Steps 1. Set MRS(x1, x2 ) = −P1
new P2 , and solve for x2 a. Find MRS:
i.
u = x1αx2
β
MRS = − ux2 'ux1 '
MRS = −αx1α−1x2
β
βx2β−1x1
α= −
αx2βx1
b. Set equality
i.
−αx2βx1
= −P1new
P2αx2P2 = P1
newβx1
x2 =P1newβP2α
x1
2. Substitute this expression for x2 into the utility function
a. x2 = P1
newβ P2α • x1→ u = x1αx2
β
u(a) = x1α P1
newβP2α
x1"
#$
%
&'
β
3. Solve for optimal x1new*
a.
u(a) = x1α P1
newβP2α
x1!
"#
$
%&
β
u(a) = x1αx1
β P1newβP2α
!
"#
$
%&
β
= x1α+β P1
newβP2α
!
"#
$
%&
β
x1α+β = u(a)[ ] P1
newβP2α
!
"#
$
%&
β
= u(a)[ ] P1newβP2α
!
"#
$
%&
−β
x1*new = u(a)[ ]
1α+β
P1newβP2α
!
"#
$
%&
−β
α+β
4. Substitute x1
*new to find x2
x1*new = u(a)[ ]
1α+β
P1newβP2α
!
"#
$
%&
−β
α+β
→ x2 =P1newβP2α
x1
x2 = u(a)[ ]1
α+βP1newβP2α
!
"#
$
%&
−β
α+β P1newβP2α
= u(a)[ ]1
α+βP1newβP2α
!
"#
$
%&
1− βα+β
= u(a)[ ]1
α+βP1newβP2α
!
"#
$
%&
α+β −βα+β
x2*new = u(a)[ ]
1α+β
P1newβP2α
!
"#
$
%&
αα+β
4—41
4.4 The Wealth effect The wealth effect changes income from being exogenous to income coming from the sale of assets. Thus, where before we had only the purchasing decisions change, now we also have a change in the amount of income at our disposal. Where before we decomposed into the income and the substitution effect, now we decompose to the wealth and substitution effect
4.4.1 What’s the deal with Income? In this model, Y is 0, and you have an endowment. Thus, for an increase in prices, we have an increase in the endowment: an increase in prices gives us more choice. Watch out for changes in Y, and then it is the same.
x1A → x1
C{ }= x1A → x1
B{ }− x1B → x1
C{ }δx1(P1,P2, I )
δP1
#$%
&'(=
δx1δP1
| u#$%
&'(−δx1δI
x1#$%
&'(
total.effect{ }= sub.effect{ } Wealth.effect{ }>=<{ }= <={ } >=<{ }
100 1 2 3 4 5 6 7 8 9
10
0
1
2
3
4
5
6
7
8
9
X Axis
Y A
xis
A
B
C
5—42
5 Demand for Goods, Supply of Labor and Capital
5.1 Last week • What were the changes in consumption for changes in income and prices? • We devolved it into:
o Y effect (change in demand for a change in Y (op.Cost unchanged)) o Substitution effect (change in demand for a change in op.Cost) o Wealth effect (change in demand for a change in wealth (op.Cost
constant)) • This allowed us to compensate consumers for changing prices • And illustrated different responses to changes in price (did the consumer
own any of the good?)
5.1.1 How? • We solved:
o min x1,x2 Exp = P1newx1 +P2x2 | u
o i.e. what is the minimum budget line (with the new opportunity cost) that would satisfy the condition that utility is unchanged.
• This took us from the original demand functions:
o x1old. price =
αα +β
IP1
x2old. price =
ββ +α
IP2
• To the compensated demand functions
o
x1new = [u(a)]
1α+β •
P1newβP2α
!
"#
$
%&
−βα+β
x2new = [u(a)]
1α+β •
P1newβP2α
!
"#
$
%&
αα+β
o Which give us the new optimal point of consumption, at the new prices, such that utility is unchanged from the original price
5.1.2 Goal for this week: • Find the
o Goods demand curve (x1, x2) o Labor supply curve (c, l) o Capital supply curve (c1, c2) (consumption, saving)
• Define Marginal and Total willingness to pay o To measure consumer surplus
• Find the compensated demand curve
5—43
5.2 Demand and Supply Curves There are three types of demand curve:
• Income demand curves (relationship between Y and Demand) • Own-‐price demand curves (relationship between Price and Demand) • Cross-‐price demand curves (relationship between Price of other goods
and Demand)
5.2.1 Demand for Goods Take the following setting:
• I = 6P1 =1;P2 =1u(x1, x2 ) = x1
αx2β = x1
2x21 →Cobb.Douglas
• Allows us to derive:
o MRS = −ux1 'ux2 '
= −2 x1x2x12 = −
2x2x1
• And uncompensated demand functions
o x1 =α
α +βIP1
o x2 =β
β +αIP2
5—44
• What happens when we vary P1? o P1 =1→ P1
+ = 2→ P1− = 0.5
§ x1*P1 =
22+1
•61= 4; x1
*P1+
=22+1
•62= 2; x1
*P1−
=22+1
•60.5
= 8
§ x2*P2 =
11+ 2
61= 2 , since x2
* does not depend on the P1 ,
changing P1will not change it.
• So, we are looking at the change in x1 for a given change in Price, sounds
like a derivative!
o
x1 =α
α +βIP1=
αIα +β
•P1−1
δx1δP1
=αIα +β
•(−1)•P1−2
= −α
α +βIP12 < 0
§ So, demand is decreasing in price
o δ 2x1δP1
2 =α
α +βIP13 > 0
§ So, it is decreasing at a decreasing rate
0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
6
7
8
x1
x2
P1
P1-
P1+
5—45
• Graphically
• Changing income will shift the own price demand function
o Same demand function, with different levels of Income
x1
x2P1
P1-P1+
P1
6
2 4 8
.5
1
2
42 8
Optimum CChanging P1
Own-Price DemandFor x1
0 1 2 3 4 5 6 7 8 9 10
0.5
1
1.5
2
2.5
3
3.5
4
X1
P1
5—46
5.2.2 Supply of Labor We use the old Consumption, Labor model from week 3.
• First, we determine the supply of leisure l = (w,L) • Then, we subtract that from hours available to find the supplied labor
labour(w,L) = L − l(w,L) Given quasilinear preferences in leisure, and a usual budget
• u(l,c) = γ ln(l)+θc • wl + c = wL
We get the following demand functions • With respect to labor
o u(l,c) = γ ln(l)+θc
MRS = − u 'lu 'c
= −γl/θ = − γ
lθ
o wl + c = wLc = −wl +wLop.Cost = −w
o
MRS = op.Cost
−γlθ= −w
wlθ = γ
l = γwθ
• With respect to consumption
o
l = γwθ
→wl + c = wL
w γwθ
+ c = wL
c = wL − γθ
5—47
Now, let us model this in the following circumstance
o L = 60;w = Pl = 20;Pc =1;γ = 400;θ =1u = γ ln(l)+θcwl + c = wL
o At this circumstance, we have the following optimum consumption:
o l*[w=20] = 400
(20)(1)= 20
c*[w=20] = (20)(60)− 4001
= 800
o What happens if we change the prices? (w=10, 40)
o l*[w=40] = 400
(40)(1)=10
c*[w=20] = (40)(60)− 4001
=1600
o l*[w=10] = 400
(10)(1)= 40
c*[w=10] = (10)(60)− 4001
= 400
o We can graph this, as above o With the “x1” being replaced by leisure o We flip this around to find the labor curve
§ Just puzzle that shit o What does math tell us about the demand for leisure?
o l = γ
wθ=γθw−1
δlδw
=γθw−2 = −
γw2θ
< 0
§ demand for leisure is decreasing with increasing wages
o
δlδw
= −γw2θ
= −γθw−2
δ 2lδw2 =
2γw3θ
> 0
§ it is decreasing at a decreasing rate
5—48
5.2.3 Supply of Capital Same idea as above: lets have a look at the shit we need to do! We use the intertemporal model (c1,c2)
• To find the demand for current consumption c1(r,e1,e2 ) • Capital supply is then s1(r,e1,e2 ) = e1 − c1(r,e1,e2 ) • If savings are negative, we are borrowing
A Cobb-‐Douglas example: here are our circumstances
• Utility function:u = c1αc2
β • Budget: (1+ r)c1 + c2 = (1+ r)e1 + e2
Let’s work out those demand functions! • Try the easy way:
o C1* =
αα +β
IP1;P1 = (1+ r); I = e1(1+ r)+ e2
C1* =
αα +β
e1(1+ r)+ e2(1+ r)
o C2* =
ββ +α
IP2;P2 = (1+ r); I = e1(1+ r)+ e2
C2* =
ββ +α
e1(1+ r)+ e2(1+ r)
• The Easy Way WORKS! • So, how do we find savings? Savings is e1 −C1
o S1* = e1 −
αα +β
e1(1+ r)+ e2(1+ r)
So … now let’s have a look at what the capital supply curve looks like:
• S1* = e1 −
αα +β
e1(1+ r)+ e2(1+ r)
δSδr
= +ve;δ2Sδr2
= −ve
• just look to the “r” at the bottom; it flips the sign with each derivative • Thus, savings increases with the interest rates, at a decreasing rate
Savings function, plot Savings against IR
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5.3 Consumer Welfare To work out consumer welfare, we need to work out the following:
1. The Marginal Willingness to Pay for each unit of x1 2. The total willingness to pay (TWTP) for x1 (the sum of the MWTP) 3. The consumer surplus is the difference between the amount paid and the
TWTP
5.3.1 Method 1: step by step
5.3.1.1 Step 1: MWTP For each unit, the MWTP is equal to the MRS at that point.
• u = x1
αx2β
MRS = −ux1,
ux2, = −
αx2βx1
• at (x1i, x2
j )→MWTP =MRS(x1
i ,x2j )= −
αx2j
βx1i
We can plot these to make the MWTP curve.
5.3.1.2 Step 2: TWTP The TWTP is the area below the MWTP curve.
5.3.1.3 Step 3: Consumer Surplus Consumer surplus is the area below the MWTP curve, above the price paid.
5.3.2 Method 2: MWTP and the Compensated Demand Function The Compensated Demand Function is equal to the MWTP
5.3.3 My idea: I think that the MWTP=MRS=Slope … so we can just derive the demand function, and find the area underneath that