Cost-Benefit Analysis
Preference Theory and Derivation of Demand CONSUMER CHOICE THEORY (DEMAND) BUDGET CONTRAINTS: -
Suppose that the consumer (household) consumes only two goods (X and Y). Given the Prices of the two goods (PX, PY) and the consumer’s income (m), the possible quantities purchasable by the consumer (Xo, Yo) are constrained by PXXo + PYYo ≤ m
e.g., Suppose that a student has a budget for coffee and milk of $100/month. If the price of Coffee (X say) is $1/cup and the price of Milk (Y) is $2/litre, the budget constraint is $1*QC + $2*QM ≤ $100 Note: We can define one of the goods (Y say) as a composite commodity, representing all goods other than X. Budget Line: Definition: The maximum combination of two commodities purchasable by a consumer given the prices of the two commodities and the consumer’s money income Rearranging the budget equation for the assumption that all income is spent gives the Budget line Yo = m/PY – PX/PY *X0 e.g. The budget line for Coffee and Milk with Milk as the Y commodity is → QM = $100/$2 – $1/$2*QC = 50 – 0.5QC Since the Opportunity Cost of X = -dY/dX (Recall: Opportunity Cost = the best foregone option) → Opportunity Cost of X = -dY/dX = PX/PY (= -slope of the budget line) (=> dY = -PX/PY *dX)
-1-
Preference Theory and Derivation of Demand e.g. The opportunity cost of a unit of Coffee = -(-0.5) = 0.5 Milk i.e., one unit of Coffee costs ½ litre of Milk
QY
Budget Line Constraint
m/PY
slope = -PX1/PY slope = -PXo/PY
m/PXo -
m/PX1'
QX
The diagram above shows the budget line for m, PXo, PY, and the budget line that ensues given a fall in the price of X to PX1. A change in one of the prices causes a rotation of the budget line around the intercept of the commodity whose price is unchanged
Note: A change in Income causes a parallel shift in the budget line: an increase in income shifts it out and an increase in income shifts it in. PREFERENCES AND INDIFFERENCES CURVES For any two consumption bundles (X0, Y0) and (X1, Y1), define an individual’s preference ranking of each of the two bundles by 1. (X1, Y1) > (X0, Y0) means the individual prefers (X1, Y1) to (X0, Y0) -2-
Preference Theory and Derivation of Demand 2. (X1, Y1) ~ (X0, Y0) means the individual is indifferent between (X1, Y1) and (X0, Y0) Preference Assumptions 1.
Completeness For every pair of consumption bundles (X1, Y1) and (X0, Y0), (X1, Y1) > (X0, Y0) or (X0, Y0) > (X1, Y1) or (X1, Y1) ~ (X0, Y0)
i.e. The consumer prefers one or other bundle or is indifferent between them. In short the consumer does not make contradictory choices. 2.
Transitivity (X1, Y1) ≥ (X0, Y0) and (X0, Y0) ≥ (X2, Y2) => (X1, Y1) ≥ (X2, Y2) (where ≥ means either preferred to or indifferent to)
3.
Non-Satiation (More is always preferred to Less) If both X1 or Y1 are at least equal to X0 or Y0 respectively and at least one of X1 or Y1 is a greater amount than X1 or Y1 respectively, then (X1, Y1) ≥ (X0, Y0) This assumption is sometimes called the monotonicity of preferences
4.
Convex The less one has of a good, the more one requires of the other good in exchange to remain indifferent.
[Assumption 1 follows from the assumption that economic agents are rational Assumption 2 is actually an assumption since transitivity need not necessarily hold. Indeed, transitivity for groups of individuals often does not hold. Assumption 3 and 4 are actually assumptions about well-behaved indifference curves.
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Preference Theory and Derivation of Demand Assumption 3 may also not hold particularly beyond certain amounts and Assumption 4 is likely but not obvious.] Indifference Curves Definition: Combinations of two commodities between which the consumer is indifferent (or that give the same level of utility) QY
Indifference Curves
QX
Marginal Rate of Substitution (MRS) of Y for X Definition: The amount of Y that the individual will give up for an increase in X while remaining indifferent to the combinations of X and Y →
-∆Y/∆X (or –dY/dX in calculus notation) where Y and X are quantities
→
the negative of the slope of an indifference curve (for Y on the vertical axis)
(Since substitution implies giving up one of the commodities, MRS is a positive number) The Preference Assumptions => 1.
Indifference Curves are negatively sloped (Non-satiation) i.e., an increase in one commodity => a decrease in the other for possible indifference
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Preference Theory and Derivation of Demand => dQY/dQX < 0 or the MRS > 0 -> Indifference curves are not positively sloped (i.e., they don’t curl back) 2.
Indifference Curves do not intersect - Intersection would imply violation of transitivity
3.
Indifference Curves are convex => dQ2Y/dQX2 > 0 => dMRS/dQX < 0 (since MRS = - dQY/dQX) => diminishing Marginal Rate of Substitution (less Y is given for more X with increasing quantities of X)
4.
A strong preference for Y over X in general gives relatively flat indifference curves A strong preference for X over Y in general gives relatively steep indifference curves
Preference Map -
Indifference Curves are dense in commodity space, i.e., every combination of X and Y is on an Indifference Curve.
-
By the principles of Non-Satiation and Transitivity, all combinations above Indifference Curves are preferred to points on the Indifference curve and all combinations on Indifference Curves are preferred to combinations below the Indifference Curve.
-
Since Indifference Curves are derived from ordinal utility, i.e., preference ranking of combinations rather than actual measurement of levels of utility, there is no cardinal measurement (assignment of a particular number) of comparative preferences. We can say that combinations above (for example) an indifference curve are preferred to combinations on the indifference curve but we cannot measure the amount of the preference.
Consumer Equilibrium (Optimal Preference or Maximum Utility)
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Preference Theory and Derivation of Demand -
Since we assume that Consumers’ attempt to maximize their preference (utility), Consumer Equilibrium implies that the consumer’s choice of the combination of X and Y is optimal, i.e., more preferable for the individual than any other choice, given the constraints.
-
Consumer Equilibrium implies therefore that the Consumer attain the highest preference (indifference curve or utility) given the budgetary constraints of the Price of the goods and the Consumer Income. Consumer Equilibrium
QY
m/PY
QYo
slope = -PX/PY QXo
m/PX
QX
→ Maximization of Utility under Budget Constraints (a constrained optimization problem) => Tangency of the highest Possible Indifference Curve and the Budget Line (i.e., the exchange of Y for X for the Individual must equal the exchange of Y for X at the Market Price) => Consumer Equilibrium: MRS = PX/PY Note: This indifference curve can be steeper or flatter depending on the preference of the individual so that the tangent could occur virtually anywhere along the budget line. Derivation of a Consumer’s Demand
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Preference Theory and Derivation of Demand Assume: 1.
Y represents all commodities other than X
2.
Income and the Price of Y are fixed → satisfies the ceteris paribus conditions of fixed income and fixed prices of all other goods
3.
Preferences are given → Satisfies the ceteris paribus assumption of given preferences
Given these assumptions, a change in the price of X => a change in the quantity demanded of X → one point on the demand function for X We can therefore derive the demand functions by changing the price of X First we draw Qy/Qx axes and below them the P/Qx axes to show the Demand curve. Initially, we know only Income ‘m’ and Py (the price of all other goods) but this is sufficient to give us the Y-intercept ‘m/Py’ of the budget line. We then pick a price PXo in the Demand diagram; this not only gives us the X-intercept but the whole budget line. Consumer equilibrium occurs at Xo (Yo) where the budget line is tangent to the highest indifference curve. Since all relevant ceteris paribus variables are fixed, this quantity of X due the price of X ‘PXo’ is a quantity demanded of X and we have the first point on the demand function. We get the other points on the demand function by varying the price of X.
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Preference Theory and Derivation of Demand
Qy (all other goods)
m/Py
Px
m/PX2
m/PXo
m/PX1 Qx
PX2 PXo PX1 D Qx This gives the derivation of an individual’s demand. Market Demand Definition: The quantities demanded by all individuals in the market at each price → Sum of individual quantities demanded at each price e.g. Suppose that there are 10 individuals with a market demand described by P = 100 – 5q. What is market Demand? We can simply multiply the equation by 10 because that would be a multiplication of Price -8-
Preference Theory and Derivation of Demand by 10. We could invert the equation to express is as q = f(P) → q = 20 – 0.2P Since Q = 10q (Q is market quantity demanded and q is individual quantity demanded) = 10 (20 – 0.2P) = 200 – 2P Inverting this to express as P = f(Q) gives
P = 100 – 0.5Q
However, we have a much simpler method to arrive at market demand Since Q = 10q, q = Q/10. We simply substitute Q/10 for q in P = 100 – 5q = 100 – 5(Q/10) = 100 – 0.5Q
-9-
Suppose that the consumer (household) consumes only two goods (X and Y). Given the Prices of the two goods (PX, PY) and the consumer’s income (m), the possible quantities purchasable by the consumer (Xo, Yo) are constrained by PXXo + PYYo ≤ m
e.g., Suppose that a student has a budget for coffee and milk of $100/month. If the price of Coffee (X say) is $1/cup and the price of Milk (Y) is $2/litre, the budget constraint is $1*QC + $2*QM ≤ $100 Note: We can define one of the goods (Y say) as a composite commodity, representing all goods other than X. Budget Line: Definition: The maximum combination of two commodities purchasable by a consumer given the prices of the two commodities and the consumer’s money income Rearranging the budget equation for the assumption that all income is spent gives the Budget line Yo = m/PY – PX/PY *X0 e.g. The budget line for Coffee and Milk with Milk as the Y commodity is → QM = $100/$2 – $1/$2*QC = 50 – 0.5QC Since the Opportunity Cost of X = -dY/dX (Recall: Opportunity Cost = the best foregone option) → Opportunity Cost of X = -dY/dX = PX/PY (= -slope of the budget line) (=> dY = -PX/PY *dX)
-1-
Preference Theory and Derivation of Demand e.g. The opportunity cost of a unit of Coffee = -(-0.5) = 0.5 Milk i.e., one unit of Coffee costs ½ litre of Milk
QY
Budget Line Constraint
m/PY
slope = -PX1/PY slope = -PXo/PY
m/PXo -
m/PX1'
QX
The diagram above shows the budget line for m, PXo, PY, and the budget line that ensues given a fall in the price of X to PX1. A change in one of the prices causes a rotation of the budget line around the intercept of the commodity whose price is unchanged
Note: A change in Income causes a parallel shift in the budget line: an increase in income shifts it out and an increase in income shifts it in. PREFERENCES AND INDIFFERENCES CURVES For any two consumption bundles (X0, Y0) and (X1, Y1), define an individual’s preference ranking of each of the two bundles by 1. (X1, Y1) > (X0, Y0) means the individual prefers (X1, Y1) to (X0, Y0) -2-
Preference Theory and Derivation of Demand 2. (X1, Y1) ~ (X0, Y0) means the individual is indifferent between (X1, Y1) and (X0, Y0) Preference Assumptions 1.
Completeness For every pair of consumption bundles (X1, Y1) and (X0, Y0), (X1, Y1) > (X0, Y0) or (X0, Y0) > (X1, Y1) or (X1, Y1) ~ (X0, Y0)
i.e. The consumer prefers one or other bundle or is indifferent between them. In short the consumer does not make contradictory choices. 2.
Transitivity (X1, Y1) ≥ (X0, Y0) and (X0, Y0) ≥ (X2, Y2) => (X1, Y1) ≥ (X2, Y2) (where ≥ means either preferred to or indifferent to)
3.
Non-Satiation (More is always preferred to Less) If both X1 or Y1 are at least equal to X0 or Y0 respectively and at least one of X1 or Y1 is a greater amount than X1 or Y1 respectively, then (X1, Y1) ≥ (X0, Y0) This assumption is sometimes called the monotonicity of preferences
4.
Convex The less one has of a good, the more one requires of the other good in exchange to remain indifferent.
[Assumption 1 follows from the assumption that economic agents are rational Assumption 2 is actually an assumption since transitivity need not necessarily hold. Indeed, transitivity for groups of individuals often does not hold. Assumption 3 and 4 are actually assumptions about well-behaved indifference curves.
-3-
Preference Theory and Derivation of Demand Assumption 3 may also not hold particularly beyond certain amounts and Assumption 4 is likely but not obvious.] Indifference Curves Definition: Combinations of two commodities between which the consumer is indifferent (or that give the same level of utility) QY
Indifference Curves
QX
Marginal Rate of Substitution (MRS) of Y for X Definition: The amount of Y that the individual will give up for an increase in X while remaining indifferent to the combinations of X and Y →
-∆Y/∆X (or –dY/dX in calculus notation) where Y and X are quantities
→
the negative of the slope of an indifference curve (for Y on the vertical axis)
(Since substitution implies giving up one of the commodities, MRS is a positive number) The Preference Assumptions => 1.
Indifference Curves are negatively sloped (Non-satiation) i.e., an increase in one commodity => a decrease in the other for possible indifference
-4-
Preference Theory and Derivation of Demand => dQY/dQX < 0 or the MRS > 0 -> Indifference curves are not positively sloped (i.e., they don’t curl back) 2.
Indifference Curves do not intersect - Intersection would imply violation of transitivity
3.
Indifference Curves are convex => dQ2Y/dQX2 > 0 => dMRS/dQX < 0 (since MRS = - dQY/dQX) => diminishing Marginal Rate of Substitution (less Y is given for more X with increasing quantities of X)
4.
A strong preference for Y over X in general gives relatively flat indifference curves A strong preference for X over Y in general gives relatively steep indifference curves
Preference Map -
Indifference Curves are dense in commodity space, i.e., every combination of X and Y is on an Indifference Curve.
-
By the principles of Non-Satiation and Transitivity, all combinations above Indifference Curves are preferred to points on the Indifference curve and all combinations on Indifference Curves are preferred to combinations below the Indifference Curve.
-
Since Indifference Curves are derived from ordinal utility, i.e., preference ranking of combinations rather than actual measurement of levels of utility, there is no cardinal measurement (assignment of a particular number) of comparative preferences. We can say that combinations above (for example) an indifference curve are preferred to combinations on the indifference curve but we cannot measure the amount of the preference.
Consumer Equilibrium (Optimal Preference or Maximum Utility)
-5-
Preference Theory and Derivation of Demand -
Since we assume that Consumers’ attempt to maximize their preference (utility), Consumer Equilibrium implies that the consumer’s choice of the combination of X and Y is optimal, i.e., more preferable for the individual than any other choice, given the constraints.
-
Consumer Equilibrium implies therefore that the Consumer attain the highest preference (indifference curve or utility) given the budgetary constraints of the Price of the goods and the Consumer Income. Consumer Equilibrium
QY
m/PY
QYo
slope = -PX/PY QXo
m/PX
QX
→ Maximization of Utility under Budget Constraints (a constrained optimization problem) => Tangency of the highest Possible Indifference Curve and the Budget Line (i.e., the exchange of Y for X for the Individual must equal the exchange of Y for X at the Market Price) => Consumer Equilibrium: MRS = PX/PY Note: This indifference curve can be steeper or flatter depending on the preference of the individual so that the tangent could occur virtually anywhere along the budget line. Derivation of a Consumer’s Demand
-6-
Preference Theory and Derivation of Demand Assume: 1.
Y represents all commodities other than X
2.
Income and the Price of Y are fixed → satisfies the ceteris paribus conditions of fixed income and fixed prices of all other goods
3.
Preferences are given → Satisfies the ceteris paribus assumption of given preferences
Given these assumptions, a change in the price of X => a change in the quantity demanded of X → one point on the demand function for X We can therefore derive the demand functions by changing the price of X First we draw Qy/Qx axes and below them the P/Qx axes to show the Demand curve. Initially, we know only Income ‘m’ and Py (the price of all other goods) but this is sufficient to give us the Y-intercept ‘m/Py’ of the budget line. We then pick a price PXo in the Demand diagram; this not only gives us the X-intercept but the whole budget line. Consumer equilibrium occurs at Xo (Yo) where the budget line is tangent to the highest indifference curve. Since all relevant ceteris paribus variables are fixed, this quantity of X due the price of X ‘PXo’ is a quantity demanded of X and we have the first point on the demand function. We get the other points on the demand function by varying the price of X.
-7-
Preference Theory and Derivation of Demand
Qy (all other goods)
m/Py
Px
m/PX2
m/PXo
m/PX1 Qx
PX2 PXo PX1 D Qx This gives the derivation of an individual’s demand. Market Demand Definition: The quantities demanded by all individuals in the market at each price → Sum of individual quantities demanded at each price e.g. Suppose that there are 10 individuals with a market demand described by P = 100 – 5q. What is market Demand? We can simply multiply the equation by 10 because that would be a multiplication of Price -8-
Preference Theory and Derivation of Demand by 10. We could invert the equation to express is as q = f(P) → q = 20 – 0.2P Since Q = 10q (Q is market quantity demanded and q is individual quantity demanded) = 10 (20 – 0.2P) = 200 – 2P Inverting this to express as P = f(Q) gives
P = 100 – 0.5Q
However, we have a much simpler method to arrive at market demand Since Q = 10q, q = Q/10. We simply substitute Q/10 for q in P = 100 – 5q = 100 – 5(Q/10) = 100 – 0.5Q
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