Consumption- Preferences Goods and bundles Consumers make choices over bundles of goods. Consumer theory models the way in which these choices are made uch as apples or bananas. A good may be specified in terms of time- such as ples today or apples tomorrow, or place -such as apples in Oxford or apples in London. a bundle of goods is a collection of goods- such as(3 apples, 2 bananas). The bundle can contain as many goods as necessary. For much of economics, only 2 goods are required. This allows a graphical representation Number of Apples(= r1) Write a bundle of goods as r=(r1, T2)where rn is the amount of good I and I2 the amount of good 2 in the bundle. How do consumers choose between bundles such as r and different bundles like y=(yI, yz)and z=(21, 22)? Consumption-Preferenees Preferences The basic premise of consumer theory is that consumers have preferences over bundles of goods. Write r >y if and only if bundle r is strictly preferred to bundle y. Write r w y if bundle r is not preferred to bundle y and bundle y is not preferred to bundle r. The cor Is indifferent between the two bundles. Write r> y if the consumer strictly prefers bundle r to bundle y or the consumer is indifferent. Equivalently, r is weakly preferred to Armed with such preferences the consumer can compare different bundles of goods. However, preferences are difficult things to work with is there a simple alternative?
Consumption — Preferences 1 Goods and Bundles • Consumers make choices over bundles of goods. Consumer theory models the way in which these choices are made. • A good is simply a product — such as apples or bananas. A good may be specified in terms of time — such as apples today or apples tomorrow, or place — such as apples in Oxford or apples in London. • A bundle of goods is a collection of goods — such as (3 apples, 2 bananas). The bundle can contain as many goods as necessary. For much of economics, only 2 goods are required. This allows a graphical representation. Number of Bananas (= x2) Number of Apples (= x1) ·x · y · z 0 1 2 3 4 5 0 1 2 3 4 5 • Write a bundle of goods as x = (x1, x2) where x1 is the amount of good 1 and x2 the amount of good 2 in the bundle. How do consumers choose between bundles such as x and different bundles like y = (y1, y2) and z = (z1, z2)? Consumption — Preferences 2 Preferences • The basic premise of consumer theory is that consumers have preferences over bundles of goods. • Write x  y if and only if bundle x is strictly preferred to bundle y. • Write x ∼ y if bundle x is not preferred to bundle y and bundle y is not preferred to bundle x. The consumer is indifferent between the two bundles. • Write x º y if the consumer strictly prefers bundle x to bundle y or the consumer is indifferent. Equivalently, x is weakly preferred to y. • Armed with such preferences the consumer can compare different bundles of goods. • However, preferences are difficult things to work with — is there a simple alternative?
Consumption- Preferences Representatio Economists make three assumptions about preferences. These are 1. Completeness: Either r y or y > r for all r and y. In other words, all the different bundles can be compared 2. Reflexivity: r>r for all r. In other words, all the bundles are at least as good as themselves. 3. Transitivity: If z>y and y >2 then r> z for all r, y and z. In other words. if a consumer prefers bundle r to bundle y and bundle y to bundle z then the consumer also prefers bundle r to bundle z. At a first glance these do not seem unreasonable. Nevertheless they are assumptions and not facts. Theorem: Any preferences that satisfy the above three assumptions can be represented by a utility functio The utility function is u(). It takes each bundle of goods and a number to it. The theorem states that there is a utility function u()such that r >y if and only if u(r)>u(y). That is, a consumer prefers bundle r to bundle y if and only if they get bigger" utility"from bundle r. The ordering induced by a consumer's preferences be very simply represented by numbers. The numbers u() themselves might have no meaning it is the relative size that matters. Utility functions allow the construction of indiffe Consumption-Preferenees Indifference Curves 1 As the theorem says,“ utility numbers”can to each bundle of goods. This can be graphically represented. Suppose all the bundles with a utility number greater than 5 are found and a line is drawn to separate them from the bundles with a utility number less than 5. This line is called an indifference cu = In the picture above u(r)>5 and u(y)y. Also z> z but more information is required to rank y and z. More indifference curves need to be drawn. Clearly they can be drawn for any utility number With the current assumptions indifference curves can be of various shapes. However, they cannot cross. Why not?
Consumption — Preferences 3 Representation • Economists make three assumptions about preferences. These are: 1. Completeness: Either x º y or y º x for all x and y. In other words, all the different bundles can be compared. 2. Reflexivity: x º x for all x. In other words, all the bundles are at least as good as themselves. 3. Transitivity: If x º y and y º z then x º z for all x, y and z. In other words, if a consumer prefers bundle x to bundle y and bundle y to bundle z then the consumer also prefers bundle x to bundle z. • At a first glance these do not seem unreasonable. Nevertheless they are assumptions and not facts. • Theorem: Any preferences that satisfy the above three assumptions can be represented by a utility function. • The utility function is written as u(·). It takes each bundle of goods and assigns a number to it. The theorem states that there is a utility function u(·) such that x º y if and only if u(x) ≥ u(y). That is, a consumer prefers bundle x to bundle y if and only if they get bigger “utility” from bundle x. The ordering induced by a consumer’s preferences can be very simply represented by numbers. • The numbers u(·) themselves might have no meaning — it is the relative size that matters. Utility functions allow the construction of indifference curves. Consumption — Preferences 4 Indifference Curves 1 • As the theorem says, “utility numbers” can be assigned to each bundle of goods. This can be graphically represented. Suppose all the bundles with a utility number greater than 5 are found and a line is drawn to separate them from the bundles with a utility number less than 5. This line is called an indifference curve. ................................................................................................................................................................................................................................................................................ 0 x2 x1 ·x · y · z u = 5 u > 5 u 5 and u(y) < 5 therefore x º y. Also x º z but more information is required to rank y and z. More indifference curves need to be drawn. Clearly they can be drawn for any utility number. • With the current assumptions indifference curves can be of various shapes. However, they cannot cross. Why not?
Consumption- Preferences Indifference Curves 2 Suppose that two indifference curves were to cross, as below. 2 The consumer is indifferent between r and y, and between r and 2. z is strictly preferred to y. But by transitivity and A contradiction Even with the current"minimal"assumptions two things can be said. First, indifference curves can be drawn and second they do not cross. What do they look like? Consumption-Preferenees Examples of Indifference Curves Perfect Substitutes 12 0 工1 "Neutral The arrows are in the direction of increasing utility. A great variety of other shapes are possible-economists restrict the sorts of indifference curves possible by making further technical assumptions on preferences
Consumption — Preferences 5 Indifference Curves 2 • Suppose that two indifference curves were to cross, as below. ................................................................................................................................................................................................................................................................................ 0 x2 x1 x y z u = 5 u = 6 . ................................................................................................................. . .................................................................................................................................... • The consumer is indifferent between x and y, and between x and z. z is strictly preferred to y. But by transitivity x ∼ y and x ∼ z implies z ∼ y. A contradiction. • Even with the current “minimal” assumptions two things can be said. First, indifference curves can be drawn and second they do not cross. What do they look like? Consumption — Preferences 6 Some Examples of Indifference Curves . x2 x1 . . . ................................................................................................................................................................................................................................................................................ Perfect Substitutes x2 x1 .............................................................................................................................................................................................. ........................................................................................................................................ .................................................................................. .................................................................................................................................................................................................................................................................................. Perfect Complements 0 0 . ......... ........ .... . ........ ......... .... . x2 x1 . . ................................................................................................................................................................................................................................................................................ A “Bad” — x2 x2 x1 .................................................................................................................................................................................................................................................................................. A “Neutral” — x2 0 0 . ......... ........ . . .......................................................... ...... ...... ... ...... ...... ...... ... • The arrows are in the direction of increasing utility. A great variety of other shapes are possible — economists restrict the sorts of indifference curves possible by making further technical assumptions on preferences
Consumption- Preferences Well-Behaved Preferences Two further assumptions are made in order to obtain well-behaved preferences. monotonicity: If ri y and r2 y2 or if ri yI and rz y2 then r>y. In other words, more is better. 2. Convexity: If r w y then Ar+(1-A)y>r. In other words, averages are better than extremes. It is easiest to see what these assumptions entail by looking at the diagrams below. The second diagram is ruled out by the convexity assumption. It is these two assumptions which give indifference curves their nice regular shape that will become so familiar during the course. Monotonicity makes the curves downward sloping and convexity makes thembowed Consumption-Preferenees The Marginal Rate of Substitution The marginal rate of substitution or MRS is the rate at which the Ist willing to give up a small of good 1 in order to gain a small amount of good 2. It is the private rate of erchange In the diagrams this is represented by the slope of the indifference Slope=MRs=△x2/△x Arn and AIz are both taken to be very small numbers- hence the word marginal More mathematically derivatives, or difference ratios involving A(capital delta) The second diagram shows how the MRS changes along the indifference curve. Notice it is decreasing(in absolute value)as Il increases. This is called a diminishing marginal rate of substitution. It is a direct consequence of the convexity assumption. The consumer is willing to give up of good 1 in exchange for good 2 as the amount of
Consumption — Preferences 7 Well-Behaved Preferences • Two further assumptions are made in order to obtain well-behaved preferences. 1. Monotonicity: If x1 ≥ y1 and x2 > y2 or if x1 > y1 and x2 ≥ y2 then x  y. In other words, more is better. 2. Convexity: If x ∼ y then λx + (1 − λ)y º x. In other words, averages are better than extremes. • It is easiest to see what these assumptions entail by looking at the diagrams below. ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. x2 x1 Well-Behaved x2 x1 Non-Convexity 0 0 . ................................................................................................................... .................................................................................................. . ......... ........ .... • The second diagram is ruled out by the convexity assumption. It is these two assumptions which give indifference curves their nice regular shape that will become so familiar during the course. Monotonicity makes the curves downward sloping and convexity makes them “bowed”. Consumption — Preferences 8 The Marginal Rate of Substitution • The marginal rate of substitution or MRS is the rate at which the consumer is just willing to give up a small amount of good 1 in order to gain a small amount of good 2. It is the private rate of exchange. • In the diagrams this is represented by the slope of the indifference curve. ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. x2 x1 The MRS x2 x1 Diminishing MRS 0 0 . . ......................................... .................................................................................................................................................................................................................................................................................................... ......................................... . .............................................. ∆x2 -∆x1 Slope = MRS = ∆x2/∆x1 ............ ............ ............ ............ ............ ............. ..................... . ............................................................................................................... . ............................................................................................................... • ∆x1 and ∆x2 are both taken to be very small numbers — hence the word marginal. More mathematically ∆x2/∆x1 ≈ dx2/dx1, the derivative. The concept of marginal is central to economics and is always associated with derivatives, or difference ratios involving ∆ (capital delta). • The second diagram shows how the MRS changes along the indifference curve. Notice it is decreasing (in absolute value) as x1 increases. This is called a diminishing marginal rate of substitution. It is a direct consequence of the convexity assumption. The consumer is willing to give up more of good 1 in exchange for good 2 as the amount of good 1 increases
Consumption- Preferences Ut A utility function assigns a number to each bundle of goods, r Early economists thought of utility as being a measure of happiness or satisfaction. In this case, the numbers attached to various bundles of goods matter. It means something for a bundle to have twice the utility of another. most economists think of utility as being ordinal. The numbers themselves only matter insofar as the rank the different bundles. If u(r)>u(y) then r >y. Therefore preferences can be represented by many different utility functions. All of the below utility functions are equivalent u2 298 mic transformation of a utility function yields another utility function which represents the preferences. A monotonic transformation is simply one which leaves order unchanged- like multiplying by 10. Utility functions can be constructed from indifference curves and vice-versa. In fact, indifference curves are like cross sections of the utility function. Some examples will clarify this point. Consumption-Preferenees Some Example 1. Perfect Substitutes: Consider u(I1, I2)=I1+I2. To draw indifference curves from such a function set utility equal to a constant, c. It must be the case that the consumer is indifferent between any combination of goods which sults in the same level of utility. c. I1+I2= c defines a straight line(I2 intercept of this line is at c. The below graph can be drawn. Perfect Substitutes ts:The same operation can be done for u(r1, I2)= min(I1, I21. See abowe. 3. Quasilinear Preferences: u(rn, I2)=v(z1)+I2 where u(- )is an increasing function. In this case each identical shape, simply 1.Cobb-Douglas Preferences: u(r1, r2)=r r2. Another useful way to write this is to take logarithms(a monotonic transformation) yielding the equivalent v(rn, I2)=aln r+8lnz2
Consumption — Preferences 9 Utility Functions • A utility function assigns a number to each bundle of goods, x. • Early economists thought of utility as being a measure of happiness or satisfaction. In this case, the numbers attached to various bundles of goods matter. It means something for a bundle to have twice the utility of another. This is called cardinal utility. • Nowadays most economists think of utility as being ordinal. The numbers themselves only matter insofar as they rank the different bundles. If u(x) > u(y) then x  y. Therefore preferences can be represented by many different utility functions. All of the below utility functions are equivalent. Bundle u1 (·) u2 (·) u3 (·) x 1 0.01 -10 y 2 98 0 z 3 101 65 • Any monotonic transformation of a utility function yields another utility function which represents the same preferences. A monotonic transformation is simply one which leaves order unchanged — like multiplying by 10. • Utility functions can be constructed from indifference curves and vice-versa. In fact, indifference curves are like cross sections of the utility function. Some examples will clarify this point. Consumption — Preferences 10 Some Examples 1. Perfect Substitutes: Consider u(x1, x2) = x1 + x2. To draw indifference curves from such a function set utility equal to a constant, c. It must be the case that the consumer is indifferent between any combination of goods which results in the same level of utility, c. x1 + x2 = c defines a straight line (x2 = c − x1) with a slope of −1. The intercept of this line is at c. The below graph can be drawn. . x2 x1 . . . ................................................................................................................................................................................................................................................................................ Perfect Substitutes u = 2 u = 3 u = 4 x2 x1 u = 1 u = 2 u = 3 .............................................................................................................................................................................................. ........................................................................................................................................ .................................................................................. .................................................................................................................................................................................................................................................................................. Perfect Complements 0 0 2. Perfect Complements: The same operation can be done for u(x1, x2) = min{x1, x2}. See above. 3. Quasilinear Preferences: u(x1, x2) = v(x1) + x2 where v(·) is an increasing function. In this case each indifference curve is an identical shape, simply shifted upward. 4. Cobb-Douglas Preferences: u(x1, x2) = x1 αx2 β . Another useful way to write this is to take logarithms (a monotonic transformation) yielding the equivalent v(x1, x2) = α ln x1 + β ln x2
Consumption- Preferences Marginal Utilit How much utility would er obtain if given a little ood 1? The er is found by calculating the marginal utility. This is the change in utility due to a small change in r M1=A=叫(x+Ax:2)-x、a、 Notice Iz is fixed in this calculation. To find how much extra "utility"is generated when a consumer obtains a little more of good I the change in utility is required, Au=MU1Ar1 se. This time zi is fixed △au(x1,z2+△x2)-u(x1,x2)O Again it follows that△u=MU2△r2 Ordinal marginal utility has no behavioural content however. How could it be calculated from observed behaviour? A choice might reveal a ranking over bundles but never marginal utility. However, marginal utility can be used to calculate something that does have behavioural content Consumption-Preferenees Marginal Utility and the MrS Marginal utility can be used to calculate the marginal rate of substitution. Recall that the MRS is the slope of the indifference curve. An indifference curve. by definition, maps the set bundles of goods for which utility is equal. Hence, when moving along an indifference curve by increasing consumption of good 1 and decreasing consumption of good 2 utility does not change From the last slide the change in utility from a small change in the amount of good I is MU,AI1 and from good 2 is MU2Arz. The total change is therefore: MU1Ar1+MU2△r2=△u=0 fact: MRS otice MRS is negative, a consumer is willing to give up one good in order to gain another. Economists often refer to MRS by its absolute value. MRS can be deduced from actual behaviour. By offering a consumer different ates of erchange between the two goods an economist can work out a consumers MRS. How? The above material is more simply understood with the aid of calculus. Differences become derivatives and the mathematics becomes trivial. The appendix to Varian(2002). Chapter 4 contains a useful summary
Consumption — Preferences 11 Marginal Utility • How much extra “utility” would a consumer obtain if given a little more of good 1? The answer is found by calculating the marginal utility. This is the change in utility due to a small change in x1: MU1 = ∆u ∆x1 = u(x1 + ∆x1, x2) − u(x1, x2) ∆x1 ≈ ∂u ∂x1 • Notice x2 is fixed in this calculation. To find how much extra “utility” is generated when a consumer obtains a little more of good 1 the change in utility is required, ∆u = MU1∆x1. • The same can be done for good 2 of course. This time x1 is fixed. MU2 = ∆u ∆x2 = u(x1, x2 + ∆x2) − u(x1, x2) ∆x2 ≈ ∂u ∂x2 • Again it follows that ∆u = MU2∆x2. • Ordinal marginal utility has no behavioural content however. How could it be calculated from observed behaviour? A choice might reveal a ranking over bundles but never marginal utility. However, marginal utility can be used to calculate something that does have behavioural content. Consumption — Preferences 12 Marginal Utility and the MRS • Marginal utility can be used to calculate the marginal rate of substitution. • Recall that the MRS is the slope of the indifference curve. An indifference curve, by definition, maps the set of bundles of goods for which utility is equal. Hence, when moving along an indifference curve by increasing consumption of good 1 and decreasing consumption of good 2 utility does not change. • From the last slide the change in utility from a small change in the amount of good 1 is MU1∆x1 and from good 2 is MU2∆x2. The total change is therefore: MU1∆x1 + MU2∆x2 = ∆u = 0 • Rearranging this equation yields an important fact: MRS = ∆x2 ∆x1 = − MU1 MU2 • Notice MRS is negative, a consumer is willing to give up one good in order to gain another. Economists often refer to MRS by its absolute value. MRS can be deduced from actual behaviour. By offering a consumer different rates of exchange between the two goods an economist can work out a consumer’s MRS. How? • The above material is more simply understood with the aid of calculus. Differences become derivatives and the mathematics becomes trivial. The appendix to Varian (2002), Chapter 4 contains a useful summary
