Consumer Theory

 The study about scarcity o is the study of choices & behaviour of individual decision- making units o How we can allocate the world limited resources o How can we allocate the world’s limited resources most efficiently? o How do we allocate our limited resources?.. . Our limited time (hours in the week); . Our limited budgets; . Our limited supply of oil, gas, iron, etc. : budget constraint: o Budget constraint identifies what he can afford to buy . There are two X฀ and Y฀; . Good X฀ can be purchased at price PX and Y at price PY; . The consumer has an income of M. . A consumption bundle (x,y) ฀ tells us that the consumer gets฀x units of good X฀ and y฀ units of good Y ฀. . Bundle (x,y) is feasible if, and only if  Px+Py≤M . ฀M or m฀ stands for income; . x฀ stands for the amount of good X฀; . y stands for the amount of good Y (and similarly for other goods); . PX or px stands for the price of good X฀and PY or py stands for the price of good Y (and similarly for other goods and prices). o We will call budget line or budget constraint the set of consumption bundles that cost exactly available income. o If we assume that the consumer will spend all of his income, and rearrange, we get the budget constraint: . y=(M/Py)-(Px/Py)x o “Relative” prices are illustrated by the slope of the budget line.

 Factors that affect the budget constraint position:  The slope and position of the budget constraint are a function of two factors: income and relative prices: o Changes in income shift the budget constraint by changing the intercept; o Changes in the price of one good pivots the budget line by changing the slope. Comparative statistics overview: o If M, PX or PY change then we get a new budget constraint: . Changing M changes the intercept but not the slope. . Changing PX or PY changes the slope.  Why only two goods: o It makes our life easy without sacrificing too much. This is because we can think of one of the goods as a numeraire. o A numeraire or is a good that stands for ‘everything else’. The obvious candidate is . o If good Y is the numeraire then we set PY = 1 and the budget constraint shows how much of good X the consumer can afford as a function of M, y and Px.  Taxes: o Value tax (also known as ad valorem tax): tax on the value (price) of purchased good or service . if the price of the good is ฀, then cost for the consumer will be: (1+t)p฀ or฀p+tp฀฀  where t is the value tax o Quantity tax: per purchased unit . IF the price of the good is p then the cost for the consumer will be  P+t o Where t is the quantity tax  Subsidies o Value subsidy: government gives back a share (%) of purchasd good or service: . (1- )P  where is the value subsidy and p is the original price of the good 휎 휎

1 o Quantity subsidy: per purchased unit . p-s  where s is the quantity subsidy  Taxes subsidies and rationing:  Lump-sum tax: government takes away a fixed amount, regardless of the consumer’s behaviour: o The budget line will shift inward, because the income was reduced.  Rationing: maximum level of consumption is fixed. o Example: good x฀ is rationed so that no more than ฀ҧ can be consumed by one consumer.  Combinations of taxes, subsidies and rationing: Good x can be consumed at price p below quantity x (with dash on top) and then a tax t can be introduced so that the cost for the consumer becomes (p+t) for any additional unit  To sum up: o Microeconomics is about using simple models to get some understanding of ‘real world’ phenomena. Simplicity is gained through assumptions. o We were building a consumer choice model. Economic theory of the consumer is based on the idea that a consumer chooses: ‘the best bundle of goods that he/she can afford’.

o A consumer’s budget constraint identifies what he can afford to buy:

 + =

푝 푥 푝 푦 푚 o Slope of the budget line:푥 rate푦 of substitution of one good for the other or the :

∆ / =-Px/Py . ∆ 푦 o If m, px or py change푥 then we get a new budget constraint: changing ฀m changes the intercept but not the slope; changing px or py changes the slope. o Taxes and subsidies affect prices; taxes, subsidies and rationing change budget line and budget set.

Consumer choice: Preference: o Three typical assumptions are made on preference to give consistent preferences: . Completeness: Any two bundles can be compared, i.e. either A B, B A, or both.  Interpretation: The consumer is able to express an opinion on each and every pair of consumption bundles ≽ ≽ . Reflexivity: Any bundle is at least as good as itself, i.e. A A. . Transitivity: If bundle A is as good as B and bundle B as good as C then A should be . as good as C, i.e. if A B and B C then A C. ≽ o these assumptions or preferences properties are related to the key assumption: the representative consumer makes≽ rational ≽choices ≽ o Why require consistency? . If preferences are not consistent it is difficult for us to say anything useful about a consumer. It will be also much more difficult to build a model. o Additional assumption is often (but not always) made: nonsatiation (more is always better).  o Plots a set of bundles between which the consumer is indifferent . Indifference curve are a way to describe preferences o Two goods are perfect substitutes if the consumer is willing to substitute one good for the other at a constant rate. . The simplest example is one to one rate of substitution. . In this case indifference curves have a constant slope. o Perfect complements are goods that are consumed together in a fixed proportion:

2 . In this case indifference curves are L-shaped. o A « bad » good is a good that the consumer doesn’t like, . In other words this type of goods gives our consumer some negative or disutility; . Observe carefully the direction of utility increase on the graph. o A « neutral » is a good that the consumer doesn’t care about, . The additional consumed unit gives zero utility; . Indifference curves are vertical lines. o There is an overall best bundle for the consumer, that we can illustrate by a satiation or a bliss point. . The further away from this point the consumer is the worse off he is o Discrete goods . Sometimes it makes more sense to consider consumer’s in terms of discrete units.  Indifference curves: behaviour o Monotonicity- indifference curves have a negative slope: . Nonsatition (more is better than less) o Convexity . Averages are preferred to extremes (diversity is preferred . Foe any t between 0 and 1 we have:

( 1 + (1− ) 1, 2 + (1− ) 2 ≥( 1, 2) o Continuity: .푡푥 A small푡 change푦 푡푥 in consumption푡 푦 푥 bundle푥 results in a small change in utility  Are indifference curve well behaved in reality: o The assumption of well behaved indifference curves is much stronger than that of consistency. For example: if goods only come in whole units there cannot be continuity o Later we will examine examples where indifference curve are not well behaved  Marginal rate of substitution (MRS): o The MRS of a good y for good x is the rate wat which the consumer is willing to trade or exchange good y for good x . IF you take away one unit of good x you need to give the consumer MRS units of good y to leave her indifferent . MRS is typically a negative number o The MRS measures the slope of the indifference curve . Change of y/ change of x o MRS is the marginal willingness to pay: . As we can suppose that good 2 represents all other goods o Example: . For perfect substitutes MRS is constant, for neutrals MRS is infinite, for perfect complements MRS is zero or infinity . For strictly convex indifferent curves, MRS is diminishing (in absolute value)  The more you have of one good, the more willing you are to give up some of it in order to have the other one o Summary of MRS: . MRS is typically negative . Is about marginal amount of a good . Is about willingness to pay o Summary: . A consumers budget constraint identifies what he can afford to buy

 + =

푝 푥 푝 푦 푚 and the slope푥 of the푦 budget line is the rate of substitution of one good for the other

∆ / =-Px/Py . ∆ 푦 . If m, px, or py change푥 then we get a new budget constraint: changing m changes interception but not the slope; changing px or py changes the slope

3 . Preferences show which bundle is the best. Indifference curves were drawn and assumptions about their behaviour were made  Utility function: o Is used to describe preferences: . Assigns a number to each possible consumption bundle so that a more-preferred bundle is assigned a higher number . Note: order matters, but not the number itself o We made several assumptions about consumer preferences: . Preferences are complete and can always be ranked . More is preferred to less . Ranking is consistent (transivity), etc . They will be used for utility function characteristics o Utility: ordinal vs cardinal: . Note that we consider ordinal utility functions and so all that matters is whether U(A) is bigger or smaller than U(B)  This means the difference U(A)-U(B) is meaningless and we cannot make interpersonal comparisons . Cardinal utility attaches a significance to the magnitude of utility. The size of the utility difference between two bundles has some sort of significance  We will not use this type of utility o Neuroeconmics: . The utility we are used to: measure of how desirable is the outcome of a choice . Neuroeconomic utility: the average firing rate of population of neurons that encodes the subject value of the object  This is a real number which cant be measured  It predicts choices . Subjective value is the average firing rate of a population of neurons coding behavioural preferences . From neurobiological perspective, an object has a subjective value if it is a reward or a punishment  How to measure subjective value: o Idea: a being will work for a reward or work to avoid punishment o In other words a being will work to obtain activation of certain neurons (e.g. nucleus accumbens and orbitofrontal cortex) o A utility function assigns a number to each consumption bundle such that: . A>B implies that U(A)>U(B) and . A=B implies that U(A)=U(B)  Definition of utility function: o A numerical score representing the satisfaction that a consumer derives from a given consumption basket o Utility function: positive monotonic transformation: . Utility function can be constructed using the indifference curves  Each curve is labelled  Due to our assumption of monotonic preferences, all bundles that are on a higher indifference curve have to have nigger labels . Utility function can be represented in a general form (for two goods, x and y): U(x,y) . Indifference curve can be drawn from the utility function  All bundles that have the same utility follow U(x,y)=const  Each value of constant will give us a different indifference curve . E.G.  Lets assume a utility function of the following form: U(x,y)=xy o Each indifference curve will be constructed for a certain level o Lets call this constant k or U =K o We can solve the above for y and we have y=k/x o We can draw indifference curve for different values of k . Examples of different utility functions: algebraic expressions  For goods which are perfect substitutes the utility function will be of the following form: o U=ax+by: . Example: U(x,y)=x+3y

4  For goods which are perfect complements the utility function will be of the following form: . U(x,y)=min(ax,by) . Example: U(x,y)=min(x,3y)  If one of the considered goods (second good y) is a , the utility function will be of the following form: o U(x,y)=ax . E.g.: U(x,y)=x  If one of the considered goods (second good y) is a bad good the utility function can be of the following form: o U(x,y)=ax-by . This a particular example for a bad good and perfect substitute o

o

 Marginal utility: o Is a ratio o What additional utility is given to the consumer if we have one additional unit of good 1

o o We can calculate the change in utility due to a small change in the amount of good, if we know the marginal utility:

.  Marginal utility and MRS: o o

o

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Consumers Optimal choice:  Economic theory of the consumer is based on the idea that a consumer chooses: ‘the best bundle of goods that he/she can afford  Optimisation =: o Every consumer wishes to maximise her utility but they are bound by their income (budget constraint) . Max U(x,y) . Subject to PxX+PyY=m o The consumer chooses an optimal consumption bundle (x*,y*) . If price and/or income change the optimum consumption bundle will change  Consumer rational choice: o Rational choice of the representative consumer under constraint: . Indifference curve is tangent to the budget line . (x*,y*) is the optimal consumption bundle . This is the optimal consumption bindle given prices and income: o Px,Py,m  Consumers rational choice: tangency condition: o Rational choice of the representative consumer under constraint:

6 . Tangency condition implies that indifference curve cannot cross the budget line and does not take into account several possible cases:  Kinky taste: Shows were indifference curves does not have a tangent  Boundary solution: The optimal consumption involves summing zero units of good 2. The indifference curve is not tangent to the budget line  Multiple tangencies: more than one tangency. . Tangency conditions is a sufficient condition for interior tangency solutions  Consumer demand: o Optimum consumption bundle is the demanded bundle . IF price and or income change then the demand bundle will change as well o The demand function explains the relationship between different demanded consumption bundles and different prices and incomes o Demand functions can be denoted by x(Px,Py,m) and y(Px,Py,m)  Optimal choice/consumer demand for selected examples: o perfect substitutes . Different cases are possible, depending on price of goods: if Py>Px: budget line is flatter, corner solution (x1,0) . Boundry solution, depending on the price of goods:  If Px

between 0 and m/px can be the optimal choice . Bad good y:  All income will be ised for the good good x=m/Px . Neutral good y:  A  ll income will be used for the good good x=m/Px . Concave preferences:  Optimal choices is the boundry point (when x*=0 or y*=0) .

 Concave preferences: o Consumer prefers a variety of bundles o Here we have the boundary optimum, as the optimal choice is at Z

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 Perfect complements: o Optimal choice would be at the angle of the L shape curves (not a real tangency) o

 Discrete goods o Impossible to locate tangency solutions as there are no proper indifference curves

 Optimal choice for well-behaved preferences o Graphically: draw the consumers budget constraint and indifference curves and see which consumption bindle would maximise utility: . This is not a very precise option o Mathematically: . Using Lagrangian or . Using tangency condition

8  These methods are useful only for interior (both goods >0) tangency solution

 Px/Py> MRSyx means the consumer would increase his utility by consuming more Y  Px/Py< MRSxy means the consumer would increase his utility by consuming more X

Conclusions:  Between an income tax and a quantity tax, both of which provide the same revenue for the government, the representative consumer is better off with the income tax Limitations:  WE have considered one representative consumer: o Just one o Uniform preferences, tax and goods and consumption  WE have assumed that income does not change  WE have not studied the response of the supply side to the tax

Consumer demand:

9  Consumer demand function x1(p1;p2;m) and x2(p1;p2;m) tells us what consumers optimal choices are for a given price p1;p2 and income m Income effect:  Change in income: normal and inferior goods o , as income increase so will the demand for the good . Change in x/change in M >0 o Inferior goods, as the income rises demand for these goods falls . Change in x / change in M <0  Income expansion path o Shows all (optimal) bundles of goods demanded for different levels of income o All points on the income expansion path are solutions to the consumers optimisation problem:

. o e.g. To find the income expansion path we have to express good 2 in terms of good 1:

a 1−a consider utility function u(x1 , x2 ) = x1 x2

Optimal condition is :

o WE can express the good 2 in terms of good 1: .

Engel curve: Shows the optimal demand for good 1 for different levels of income  This curve Is true for a given set of preferences  Price of goods are held constant; if p1 or p2 change we will have a new curve  We will have another Engel curve for the second good

 For a normal good the Engel curve is upwards sloping  For the Engle curve is downward sloping  o The Income expansion path (IEP) shows the relationship between two goods o Engel curve shows the relationship between income and the demand for the project. WE ALWAYS TALIKING ABOUT OPTIAML CONSUMPTION o Because we have 1;1 compliments, the curve will be a upwards sloping and it will be at 90 (Px+Py) as goods x and goods y are consumed in equal proportions .

10 . As long as price of X is lower than price of Y, we will only consume good x

. : IF the demand of the good 1 goes up greater proportion than income increase . If the demand of good 1 goes up by a lesser proportion than the income increase this good is called a necessary good

Own-Price effect: . Anticipated reaction: If the price increases then the demand for a good decreases . Goods that follow this patter are called ordinary goods o The demand for such a good decreases when price increases and increases when price decreases

. . Goods that follow the reverse pattern are called Giffen goods: o The demand for such goods increases when price increases and decreases when price increases

. Price offer curve (POC) shows all (optimal consumption bundles of goods demanded for different level of price

. Demand Curve: o Shows the optimal demand for good one for different levels of price of good one . This curve is true given set of preferences . Price of good two and income are held constant; if p1 or m change we will have a new curve . We will have another demand curve for the second good . This is the case where as price increased demand decreases  Change in x1/change in p1 <0 . Substitutes and complements: Imperfect case o The nature of preference (the shape of the utility function) determines whether consumption of one good is increasing or decreasing in the price of the other good

11 o Substitutes (gross): if demand d1 is increasing in p2 (or if the demand of good one increases when the price of good two increases): . Change in x1/change in p2 >0  Example Audi and BMW cars o Complements (gross): IF demand d1 is decreasing in p2 (or if demand of good one decreases when the price of good two increases: . Change in x1/change in p2 <0  Example Gin and Tonic water . As long as the price fo good 1 or y is lower than good 2 or good y, we will consume zero of the other good. We represent it by the vertical line. . IF the prices are exactly the same the part of demand curve will be a horizontal line . Between zero to m/p1. After this the price of good x is cheaper than the price of good y then we know that we only consume only good x. The higher the price the less of good x we are consuming. The more the price falls the more we consume of x

Discrete goods: demand curve and reservation price: . If the price of the discrete good (good 1) is too high, consumer will consume zero units. At some point consumer, will strictly prefer to consume a unit of this discrete good . Between these two situations there is a price at which consumer will be indifferent between consuming one unit or not. This price is known as reservation price o The price at which the consumer will be exactly indifferent with consuming one or more units . The cut off evaluation of the consumer if they are consuming the additional unit of the discrete goods . The demand of this discrete good can be understood as a sequence of reservation prices (for different number of units) . You can draw a line through any two points meaning you can find such a budget constraint which will go through any two points. Meaning the consumer is very indifferent for consuming one or two units

Inverse demand function: . the inverse demand function: example and interpretation: o For Cobb-douglas preferences . Which are expressed by the utility function U(x,y)=XcYd We had the solution for optimal consumption of good x:  . WE can transform푐 푚it and then: 푥 = 푐+푑 푃푥

푚푐 . The first representation is a demand function,푃푥 = the second is the inverse demand function . Interpretation: the downward sloping inverse푥 demand(푐 + 푑) function shows also the willingness to pay  More when the amount of good x is little and less as x grows larger

12 o . Review: o Decrease in price increases demand – and decrease in price decreases demand-

Income and substitution effect: o Changes in the price of good . Lets consider the case when the price of good one decreases  Substitution effect: good one is relatively cheaper so the consumer may one for good two  Income effect: the consumer has relatively more real income so he may change (increase) consumption of both goods o Overall effect= Substitution + income effect

. Price change for different types of goods: o Normal goods: . Substitution and income effects reinforce one another  When price falls, both effects lead to a rise in quantity demanded  When price rise both effects lead to a drop in the quantity demanded o Inferior goods: . Substitutional and income effects move in opposite directions and the combined effect is indeterminate:  When price rises, substitution effect leads to a drop in quantity demanded, but the income effect is opposite  When price falls, the substitution effect leads to a rise in quantity demanded but the income effect is opposite . Slutskys Effects for normal good: o Most goods are normal (i.e. demand increases with income) o The substitution and income effects reinforce each other when a normal goods own price changes: . Since both the substitution and income effect increase demand when own-price

13 falls, normal good ordinary demand curve slopes down: . The law of downward sloping demand therefore always applies to normal goods  Before we have good x and good y.  Before we have a original budget constraint and tangency point which is X1Y1  Then there is a change in price for good X, so we have a new budget constraint and a new optimal consumption bundle at X3,Y3. The overall effect is between X1X3 o To determined the income and substitution effect we need to construct a hypothetical budget constraint o This is holding the purchasing power constant, so it will go through the original optimal consumption bundle X1Y1. The hypothetical budget constraint has to have the same slope as the final budget constraint o The substitution effect is: X2-X1 o Income effect is X3-X2 o Overall effect= the sum of two effects . Slutskys effects for income-inferior goods o Some goods are income-inferior (i.e. demand is reduced by higher income) o The substitution and income effects oppose each other when an income- inferior goods own price changes . We have old budget constraint and old optimum consumption bundle X1Y1 . Price of X decreases, so we can find the new optimum and budget constraint X3Y3 . Overall effect = X3-X1 . We have to hold the purchasing power constant and have the hypothetical budget constraint has to be parallel to the new budget constraint. WE then find the hypothetical optimum bundle, which is where the indifference curve is at tangent to the budget constraint  The substitution effect is the difference between X2 and X1, it is larger than the income effect which works in opposite direction. The income effect is the difference in consumption of X due to the parallel shift of the budget line, the difference between hypothetical budget constraint and new budget constraint (DIFFERENCE BETWEEN X2 and X3) . Slutskys effect for Giffen goods: o In the rare case of extreme income-inferiority, the income effect may be larger in size than the substitution effect causing quantity demanded to fall as own-price rises o Such goods are Giffen goods: . Slutskys break down of the effect of a price change into a pure substitution effect and income effect thus explains the Law of Downwards- Sloping demand is violated for extremely income inferior goods  Overall effect= X3-X1  Income effect= X3-X2  Substitution effect= X2-X1 . Price changes for different types of goods: normal and inferior (algebraically): o Sign of the substitution effect: . IF the price decreases, then the demand increases or the substitution effect is negative o Total change in demand:

.  Or the total change in demand is equal to the sum of the substitution and the income effect: Slutsky identity . To sum up:  Microeconomics is about using simple models to get some understanding of ‘real world’ phenomena. Simplicity is gained through assumptions.

14  We are building the consumer choice model. Economic theory of the consumer is based on the idea that a consumer chooses: ‘the best bundle of goods that he/she can afford’.  In our model, a rational representative consumer maximises his utility under budget constraint and chooses the optimum consumption bundle.  We can describe consumer demand and the changes in this demand following a change of price, income, etc.

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