Indifference Curves and Consumer Optimal Choice PDF

Preview

Summary

INDIFFERENCE CURVES UNI OF LIVERPOOL ECON 121...

Description

Faye Henaghan

Indifference Curves and Consumer’s Optimal Choice Slope of indifference curve  Slope of IC: Marginal Rate of Substitution (MRS) 

MRS: How much food the consumer is willing to give up in exchange to an additional unit of shelter, such that his/her utility stays the same.



MRS vs price ratio: o MRS is the rate at which the consumer changes one product for another.

o Price ratio is the rate at which the market changes one product for another.  Slope of Budget Line: Rate at which we substitute good x for good y, without changing total expenditure.

 Slope of Indifference Curve: Rate at which a consumer substitutes good x for good y, without changing total satisfaction. 



Convexity  Preference for variety, rather than for the extremes. Technically, decreasing MRS means IC are convex.  We like to consume a little bit of everything instead of a lot of just one thing. (Example: fish and chips, food and shelter, etc.)  There are some exceptions where some goods are not good to be consumed together. (Example: drink or drive).

Faye Henaghan

Faye Henaghan Different Consumer Preferences  Consumers have different preferences Tex prefers potatoes, Mohan prefers rice.  Tex’s MRS of potatoes for rice is smaller than Mohan’s. Tex is willing to sacrifice fewer potatoes for a unit of Rice.  Smaller MRS: Indifference Curve is flatter.

The Best Affordable Bundle 1. We know how to model budget constraints. 2. We know how to model preferences.  Combine the two to determine Optimal Choice.  Decision Problem: Maximize satisfaction (2) conditional on the limited amount of Income (1). Continue with the Food & Shelter Example:  Income: £ 100 per week,  PF =£10kg,  PS =£5sq.m,  Combine the Budget Line & Indifference Curves.

Faye Henaghan 

Point of Tangency between IC and Budget Line. Slope of IC and BL must be equal at the best affordable bundle.



P5 We know that the slope of the Budget Line is:



The slope of the IC must also be: MRS=

PS 5 = P F 10

5 . 10

 If the slopes are not equal, the consumer can always choose a better bundle.  At tangency: MRS (slope of IC) = 0.5.  0.5kg of food compensates exactly for 1 sqm of shelter  this is equal to the price ratio.

 0.5kg of good = £5, 1 sqm of shelter = £5.  MRS = Price Ratio. No Tangency  A tangency does not always exist (e.g. corner solution and perfect complements).  The MRS may always be above/below the slope of the budget line.  Corner solution  a solution where only one good is consumed.  Perfect complement  no corner solution, no tangency.

Common Case  In practice, we don’t consume some of every product- we don’t consider every item in the supermarket, every book at the library  we just don’t value some products at all.  Therefore, corner solutions are not rare, but we often focus on ‘interior’ solutions. Indifference Curves with Multiple Goods  Same as budget line.  Use one specific good (x) and a composite good.  IC: rate at which the consumer will exchange composite good for good x. Utility Function, Budget Line and Optimal Choice

Faye Henaghan     

We have used ICs to illustrate preferences. ‘Satisfaction’ is constant along each IC. A higher IC = higher satisfaction. We want to say more than satisfaction, we want to say how much one bundle is preferred to another. Utility- a measure of satisfaction  measured in ‘Utils.

Utility Function  The utility function from consuming the quantity x and y of two goods is defined as U (x, y). Examples:  U (x, y) = x + y o Perfect substitutes 1:1 (rise vs pasta, blue pen vs black pen) o Constant slope and equal to -1. o Consume only the cheapest good.  U (x, y) = xy o Preference for variety. o Slope changes with consumption. U (x, y) = 2x + y o Perfect substitutes 2:1 (2 cups of tea vs 1 cup coffee) o Constant slope and equal to -1/2 (-2 depending on the axes label). o Consume only the relatively cheapest good.  Higher utility = higher IC  How do we construct an IC? Find all combinations of consumed quantities of the two goods such that utility is the same.

 Food and Shelter goods.: o Suppose U (S, F) = F * S. o Suppose we want to construct the indifference curve for U (S, F) = 1. o This curve is such that F * S = 1. o To construct the indifference curve, we connect all the points:

o For U=2, we have: (we then continue with U=3, U-4, etc).

Faye Henaghan





In theory, we cannot construct the indifference curves  the set of indifference curves is compact. o There is an infinite number of indifference curves in the plane. o There is an infinite number of bundles in each indifference curve. We know where the ICs are, we also know where the budget constraint is. Why do we need both? To derive the optimal consumption of the two goods.

Conditions for Optimal Consumption  Sufficient condition: the slope of the indifference curve must be equal to the slope of the budget line.  MRS = PRICE RATIO.  This equivalence isn’t necessary for optimality though (e.g. corner solution).  There are cases where consumption is optimal, but the slope of the two curves is different (corner solution) Assume a bundle of food and shelter. What happens to the utility if we change (a little bit) the consumed quantities by ∆F and ∆S? Answer: ∆U = ∆F * MUF +∆S * MUS where MUF and MUS is the change of the utility as a result of a marginal (i.e. very small) change in F and S, respectively. Now, we want the utility to be the same, after the change, i.e. ∆U = 0. If ∆U = 0, then from the above equation, we get ∆F * MUF +∆S * MUS =0 (1) Note: the marginal utilities are always strictly positive, i.e. MUF > 0 and MUS >0. Therefore, for the above equation to be satisfied, ∆F and ∆S need to have opposite signs. The intuition for this is straightforward: if I increase consumption of F, then consumption of S has to decrease to keep the total utility constant. ΔF MUS = From (1), we get ΔS MUF which is true only for ∆S and ∆F small enough (note that MU refers to small changes). M US ΔF (and therefore ) approaches to the slope of For ∆S and ∆F small enough, M UF ΔS the indifference curve.

Faye Henaghan We know that the marginal rate of substitution of shelter for food, MRSSF , is the slope of the indifference curve. Therefore, MU S MRS SF= MU F Suppose that MUS = 2 and MUF = 1. If the consumer receives one additional unit of food, utility increases by 1 utils. It increases by 2 if the consumer receives one additional unit of food. Therefore, in utility terms, 2 units of food bring the same utils of 1 unit of shelter. Therefore, the marginal rate of substitution of shelter for food, MRSSF, is 2.

Optimal Bundle  The optimal bundle is such that: P MRS SF= S PF PS , where is the slope of the budget line. PF  When the condition is rewritten using marginal utilities: MU S PS = MU F P F  Rearranging, we get: MU F MU S = PF PS

 The ratio of marginal utility to price must be the same for all goods. The intuition is straightforward. If one good has a higher MU/P ratio, then it has to be relatively more valuable for the consumer, and vice versa.

 Suppose MUF =6, MUS =4, PF =1, and PS =2. MU F 6 4 MU S = > = PF PS 1 2 Then the choice is not optimal: An additional unit of Food generates higher util-per-£ than Shelter. If you reduce shelter by 1 unit, you save £ 2 and lose 4 utils. If you spend the £ 2 buying food, you get 2 more units of food and enjoy 12 more utils. Overall you gained 8 utils.

Then

Faye Henaghan

Uniqueness of optimal bundle How many optimal bundles do exist? Answer: if indifference curves are strictly convex, the optimal bundle is unique. Two counterexamples:  Concave indifference curve.  Perfect substitutes

Concave indifference curve

Faye Henaghan

Perfect substitutes  MRS = price ratio...