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Intermediate Microeconomics Part I CONSUMER THEORY (II) Laura Sochat Constrained optimisation There are n goods consumed in quantities π₯1 , π₯2 β¦, π₯π making up a bundle (π₯1 , π₯2 β¦, π₯π ) β π The agent income is M, and the given market prices of each good are π1 , π2 β¦, ππ . Agentβs preferences are represented by a utility function U (i.e the agent has rational preferences). Preferences are monotonic and (generally) convex. We have seen the graphical representation of optimisation. The utility function is our objective function, and the budget set gives us the constraint, and is given by: πΌ π2 ππ π₯1 π1 + β― + π₯π ππ β€ πΌ, or π₯1 = β π₯2 β β― β π₯π π π π 1 1 1 Optimisation with two goods We have seen before the graphical method of solving for the optimising bundle of goods, using the tangency condition: ππ1 = ππ π ππ2 Letβs now look at the Lagrangean method πππ₯ππππ π π(π₯1 , π₯2 ) π π’πππππ‘ π‘π πΌ = π1 π₯1 + π2 π₯2 max πΏ = π π₯1 , π₯2 β Ξ»( π1 π₯1 + π2 π₯2 β I) π,π,Ξ» 1. Obtain the F.O.C.s ππΏ ππ₯1 = ππ ππ₯1 β Ξ»π1 = 0 ππΏ ππ₯2 = ππ ππ₯2 β Ξ»π2 = 0 ππΏ πΞ» = πΌ β π1 π₯1 β π2 π₯2 = 0 Solve the system to find the values of π₯1 and π₯2 , for which the Lagrangian is maximised, and the constraint holds. Examples, and alternative method Suppose that πΌ represents the consumerβs income, ππ₯ the price of good X, and ππ , the price of good Y. Assume that the consumerβs preferences for goods X and Y, are defined by the following utility function: a) π(π, π) = π π π π Solve for the optimal bundle of X and Y, and comment on the findings An alternative method could be to substitute for X, or Y, within the utility function and solve for the first order condition: max π π, π π π’πππππ‘ π‘π πΌ = ππ₯ π + ππ π π,π a) π(π, π) = π 1/3 π 2/3 , also assume that πΌ = £24, ππ₯ = £4, and ππ = £2 Given the values of ππ₯, ππ¦, and πΌ, It is possible to solve for the optimal bundle. Choosing between two types of taxes, using consumer theory Assuming the original budget constraint is given by π1 π₯1 + π2 π₯2 = I Suppose the government wishes to raise tax revenue. Should they raise quantity tax (on good 1)? Or income tax? β Show the results in a graph Think about the limitations of this example in terms of: β β Uniform income taxes Uniform quantity taxes Marshallian demand functions Solutions to the earlier maximisation problem, the optimal values for the quantity of goods consumed by the consumer can be expressed as a function of prices and income. These are demand functions such as: β β β β π₯1β = π·1 π1 , π2 , β¦ , ππ , πΌ π₯2β = π·2 π1 , π2 , β¦ , ππ , πΌ β¦ π₯πβ = π·π (π1 , π2 , β¦ , ππ , πΌ) Prices and income, has before, are exogenous and the consumer has no power over their values. Marshallian demand functions are homogeneous of degree zero Interpretation of the Lagrange Multiplier It is interpreted as the marginal utility of an extra dollar of consumption expenditure, that is the marginal utility of income: ππ’ Ξ»= ππ₯1 = π1 ππ’ ππ₯2 =β―= π2 ππ’ ππ₯π ππ That is we can say that a dollar of extra income should increase the consumerβs utility by Ξ». ππ = Ξ»β ππΌ Royβs identity Substituting for the Marshallian demands into the original utility function, we obtain an expression for the actual level of utility obtained: π£ π1 , π2 , β¦ , ππ , πΌ = π(π₯1β ( π1 , π2 , β¦ , ππ , πΌ , π₯2β ( π1 , π2 , β¦ , ππ , πΌ , β¦ , π₯πβ ( π1 , π2 , β¦ , ππ , πΌ ) This is called the indirect utility function, and has the following properties: β β β It is non-increasing in every price, decreasing in at least one price Increasing in Income Homogeneous of degree zero in price and income Royβs identity- The envelop theorem Consider the case of two goods. Taking the total derivative of the Indirect utility function we get that (1): ππ ππ ππ₯1β ππ ππ₯2β = + ππ1 ππ₯1 ππ1 ππ₯2 ππ2 Next we need to make use of two results found before: β The first order conditions tell us the value of the marginal utilities, which we can use in (1) ππ ππ₯1 β ππ = Ξ»π1 , ππ₯ = Ξ»π2 2 Taking the total derivative of the budget constraint and substituting to obtain the following This gives us an important result, Royβs Identity ππ ππ ππ1 β = βΞ»π₯1 β β = π₯1β ππ ππ1 ππΌ The Envelop Theorem More generally, the result above can be assumed, by using the envelop theorem. Consider the following maximisation problem: max π(π₯, π¦, π½) π . π‘. π(π₯, π¦, π½) β€ π π₯,π¦ The constant π½ is given exogenously. We can solve the problem as usual: πΏ π₯, π¦, π, π½ = π π₯, π¦, π½ + π(π β π(π₯, π¦, π½)) F.O.Cs are given by ππΏ ππ(π₯ β , π¦ β , π½) ππ = βπ ππ₯ ππ₯ ππΏ ππ(π₯ β , π¦ β , π½) ππ = βπ ππ¦ ππ¦ π₯β, π¦β, π½ =0 ππ₯ π₯β, π¦β, π½ =0 ππ¦ Substituting for π₯ β and π¦ β into the objective function, we obtain the value function: πΉ π½ = π(π₯ β (π½), π¦ β (π½), π½) The Envelop Theorem The value function is the maximised value of our objective function. Taking the total derivative of the value function with respect to π½: ππΉ(π½) ππ ππ₯ β ππ ππ¦ β ππ = + + ππ½ ππ₯ ππ½ ππ¦ ππ½ ππ½ β β ππΉ(π½) ππ ππ₯ ππ ππ¦ ππ = πβ + πβ + ππ½ ππ₯ ππ½ ππ¦ ππ½ ππ½ ππΉ(π½) ππ ππ₯ β ππ ππ¦ β ππ β =π + + ππ½ ππ₯ ππ½ ππ¦ ππ½ ππ½ Differentiating the constraint, with respect to π½ ππ ππ₯ β ππ ππ¦ β ππ ππ ππ₯ β ππ ππ¦ β ππ + + =0β + =β ππ₯ ππ½ ππ¦ ππ½ ππ½ ππ₯ ππ½ ππ¦ ππ½ ππ½ ππΉ(π½) ππ ππ ππΏ = β πβ = ππ½ ππ½ ππ½ ππ½ Expenditure minimisation We can find optimal decisions of our consumer using a different approach. We can minimise the consumerβs expenditure subject to a given level of utility that the consumer must obtain β The goal and the constraint have been reversed. This will be important to separate income and substitution effects. The basic set up consists again of n goods making up a bundle, and each good has a specific price. The consumer has a utility target, say π£, and his preferences are rational The consumer is choosing π₯1 , π₯2 , β¦ , π₯π to solve the following problem πππ π1 π₯1 , π2 π₯2 β¦ ππ π₯π π . π‘. π(π₯1 , π₯1 , β¦ , π₯1 ) β₯ π£ Expenditure minimisation solution Solution to the expenditure minimisation problem are called Hicksian demand functions and take the form π»1β (π1, π2 , π£) β’ They are also called compensated demand, and represent the cost minimising value of each good Example- Solve for the Hicksian demand in the case of a Cobb Douglas Utility function (where Ξ± + π½ = 1) π π₯, π¦ = π₯ πΌ π¦ π½ Shepardβs Lemma Remember Royβs identity, obtained after solving for the utility maximisation problem, and using the Envelop Theorem Shepardβs Lemma is obtained the same way, and will recover the following result. If the price of a good changes by a small amount, then demand (compensated), will also change by a small amount, therefore the increased cost of consumption will be equal to the compensated demand. Using the expenditure function (minimised objective function), we get the following: E π1 , π2 , β¦ , ππ , π£ = π1 π₯1β ( π1 , π2 , β¦ , ππ , π£ , π2 π₯2β π1 , π2 , β¦ , ππ , π£ , β¦ , ππ π₯πβ π1 , π2 , β¦ , ππ , π£ ππΈ = π»1β (π1, π2 , π£) ππ1 Connecting the two results- Two sides of the same coin From utility maximisation, we obtained the Marshallian demands, from which we can obtain the indirect utility function: π£ ππ , ππ , β¦ , ππ , πΌ = π’(π₯1β π1 , π2 , β¦ , ππ , πΌ , π₯2β π1 , π2 , β¦ , ππ , πΌ , β¦ , π₯πβ π1 , π2 , β¦ , ππ , πΌ ) The indirect utility function tells us that utility indirectly depends on prices and Income. It maps prices and income into maximum utility From expenditure minimisation, we obtain the expenditure function, using Hicksian demands: In both cases, prices and income are given, and you choose the xsβ¦ πΈ ππ , ππ , β¦ , ππ , π£ = π1 π»1β π1 , π2 , β¦ , ππ , πΌ , π2 π»2β π1 , π2 , β¦ , ππ , πΌ , β¦ , ππ π»πβ π1 , π2 , β¦ , ππ , πΌ The constraint in the primal becomes the objective in the dual Using the rational choice model to derive individual demand: A change in the price of one good. Remember the demand curve we have seen before, giving us relationship between the price of a good and the quantity demanded of that good. Price (P) A B Demand Quantity demanded Using the rational choice model to derive individual demand: A change in the price of one good. Price (P) Changing the price of good X, we obtain different budget lines-Using rational consumer theory, we can find the optimal bundles corresponding to the different budget line and obtain the price-consumption curve by linking them The price consumption curve Price of fish Quantity demanded 4 22 6 15 12 7 Quantity demanded Using the rational choice model to derive individual demand: A change in the price of one good. Price (P) 12 Price of fish Quantity demanded 4 22 6 15 12 7 6 4 Demand curve 7 15 22 Quantity demanded A change in Income: The Income-Consumption curve and the Engel curve Recall the effect of a change in income on the budget constraint: It leads to a shift in the budget constraint, and therefore to an increase in the feasible set. All other goods (£) 120 Income Quantity demanded 120 12 90 8 60 5 The income consumption curve 90 πΆ 60 π΅ π΄ 5 8 10 12 15 20 Fish (Kg/week) A change in Income: The Income-Consumption curve and the Engel curve Income The Engel curve πΆ 120 π΅ 90 π΄ Income Quantity demanded 120 12 90 8 60 5 60 5 8 12 Fish (Kg/week) Different types of goods The income elasticity tells us how quantity demanded responds to a change in income. It is given by: π= β Ξπ₯/π₯ ΞπΌ/πΌ = Ξπ₯ I ΞπΌ π₯ = ππ¦ π₯ ππ₯ πΌ As income increases by 1%, quantity demanded increases by ΞΎ%. A good is said to be normal, if ΞΎ>0, the quantity demanded of a normal good increases (decreases) as income increases decreases) A good is said to be inferior, if ΞΎ<0, the quantity demanded of an inferior good decreases (increases) as income increases (decreases) A good is said to be a luxury good if ΞΎ>1 A good is said to be a necessary good if ΞΎ<1 Income elasticities and Income consumption curves Assume income increases; The budget constraint shifts to the right. π2 πΌπΆπΆ3 πΌπΆπΆ1 : Both goods are normal, quantity demanded of both goods has increased following the increase in income πΌπΆπΆ2 : π1 is a normal good, while π2 is inferior. Quantity demanded of good 2 has fallen following the increase in income πΌπΆπΆ3 : Good 2 in normal, while good 1 is inferior. πΌπΆπΆ1 πΌπΆπΆ2 π1 Difference preferences: What would the Engel curves look like? Perfect substitutes Perfect complements Homothetic preferences Quasilinear preferences The Engel curve when one of the good is both normal and inferior All other goods (£) πΌ3 Income πΌ3 The income consumption curve πΌ2 The Engel curve πΆ πΌ2 πΆ π΅ πΌπΆ3 πΌ1 πΌ1 From πΌ1 to πΌ2 , the increase in income lead the consumer to demand more of X. From πΌ2 to πΌ3 , however, the increase in income lead the consumer to demand less of X. π΄ π΅ π΄ πΌπΆ2 πΌπΆ1 π1 π3 π2 The income consumption curve X π1 π3 π2 The Engel curve π The effect of a change in the prices of goods: The income and substitution effects From the law of demand, we know that an increase (decrease) in the price a good leads to an decrease (increase) in the quantity demanded of that good. We can divide the total effect of a price change into two effects: The substitution effect refers to the change in the relative price of the good. As the price of a good rises (falls), other goods become relatively cheaper (more expensive), making them more (less) attractive to the consumer. β Even if the consumer was to stay on the same indifference curve, optimisation will lead to the consumer having to equate the marginal rate of substitution to the new price ratio The income effect refers to the change in real income from a rise (fall) in the price of one good. The consumer is now poorer (richer), leading to a change in quantity demanded. β The individual cannot stay on the same indifference curve and will have to move to a new one The income and substitution effects (Hicks) : A normal good All other goods (£) All other goods (£) Assume that we compensate the consumer, by providing him with enough money to achieve the same level of utility than before the price of fish increased. We draw an imaginary budget constraint tangent to the old IC. π΅ πΆ πΆ π΄ π΄ πΌπΆ1 πΌπΆ1 πΌπΆ2 πΌπΆ2 π3 π1 Fish (Kg/week) π3 π2 π1 Income effect Substitution effect Fish (Kg/week) The income and substitution effects (Hicks) : An inferior good The income elasticity of an inferior good being negative, the income effect from a price increase will be positive, while the substitution effect is still negative. All other goods (£) All other goods (£) π΅ π΄ π΄ πΆ πΌπΆ1 πΆ πΌπΆ2 π3 π1 π2 π3 X Income effect X π1 Total effect Substitution effect The income and substitution effect (Hicks) : A giffen good All other goods (£) Suppose the price of π1 falls, leading to a new (rotated) BL. π1 being an inferior good, the substitution effect will lead to the consumer consuming more of good 1, while the income effect will lead the consumer to consume less of the good. πΆ πΌπΆ2 π΄ In this situation, the substitution effect is completely offset by the income effect. π΅ πΌπΆ1 Total effect Substitution effect Income effect π1 How to calculate the effects? STEP 1 Utility maximisation β Allows us to find the initial optimising bundle of goods chosen by the consumer at initial prices STEP 2 Expenditure minimisation β Allows us to maintain the level of utility fixed at initial level, while minimising expenditure at new prices STEP 3 Utility maximisation β Allows us to calculate the income effect from the consumerβs maximisation problem at the new set of prices Compensated Hicksian demand The compensated demand is the solution obtained from the expenditure minimisation problem (subject to a fixed level of utility). It gives us the smallest possible expenditure at the old level utility- It is often called the compensated demand, as it accounts only for the substitution effect π₯ β = π»(ππ₯ , ππ¦ , π’) The own price demand curve derived before, the Marshallian demand, is the uncompensated demand curve. It accounts for both the income and the substitution effect Compensated Hicksian demand The compensated Hicksian demand can be derived as shown on the graph to the left. The effect of the price change are compensated so as to force the individual to remain on the same indifference curve. The income and substitution effects (Slutsky) : A normal good All other goods (£) All other goods (£) Assume that we compensate the consumer, by providing him with enough money to achieve the same purchasing power than before the price of fish increased. We draw an imaginary budget constraint tangent to go through the original optimal bundle. π΅ πΆ πΆ π΄ π΄ πΌπΆ1 πΌπΆ1 πΌπΆ2 πΌπΆ2 π2 π1 X π2 π3 π1 Income effect Substitution effect X An algebraic interpretation: The substitution effect As seen in the graph above, the βpivotedβ budget line represents a situation where the consumer has been compensated to ensure its purchasing power remained unchanged (at the new set of prices, the consumer can still consume the initial optimal bundle) Consider the general situation the price of good π₯1 changes from π1 to π1β² How can we calculate the amount of money income needed to keep that initial bundle affordable? πΌβ² = π1β² π₯1 + π2 π₯2 πΌ = π1 π₯1 + π2 π₯2 πΌβ² β πΌ = π₯1 (π1β² β π1 ) The substitution effect is the change in demand for a good when its price changes, and at the same time, money income is compensated. βπ₯1π = π₯1 π1β² , πβ² β π₯1 (π1 , π) An algebraic interpretation: The Income effect Consider still a change is the price of good π₯1 , from π1 to π1β² The income effect will be the change in the demand for the good, when we change income from πβ² to π, while holding the price of the good at the new level. That is: βπ₯1π = π₯1 π1β² , π β π₯1 π1β² , πβ² What can you say about the direction of the income effect, based on the type of good, good 1 is? What about the sign of the substitution effect? An algebraic interpretation: The Slutsky equation Putting the two together, we obtain the Slutsky identity: βπ₯1 = π₯1 π1β² , πβ² β π₯1 (π1 , π) + (π₯1 π1β² , π β π₯1 π1β² , πβ² βπ₯1 = π₯1 π1β² , π β βπ₯1 π1 , π = βπ₯1π + βπ₯1π While we often see the Slutsky identity in terms of absolute changes, it is often useful to look at it in terms of rate of change: βπ₯1 βπ₯1π βπ₯1π = β π₯1 βπ1 βπ1 βπ Marshallian demand elasticities The price elasticity of demand ππ₯,ππ₯ measures the proportionate change in quantity demanded in response to a proportionate change in a goodβs own price. Apart from the exception of a Giffen good, the own price elasticity of demand is always negative. Ξπ₯/π₯ Ξπ₯ ππ₯ ππ₯ ππ₯ ππ₯,ππ₯ = = = Ξππ₯ /ππ₯ Ξππ₯ π₯ πππ₯ π₯ Cross price elasticity of demand, ππ₯,π¦ , measures the proportionate change in quantity demanded in response to a proportionate change in the price of some other good Ξπ₯/π₯ Ξπ₯ ππ¦ ππ₯ ππ¦ ππ₯,ππ¦ = = = Ξππ¦ /ππ¦ Ξππ¦ π₯ πππ¦ π₯ Application: Labor-Leisure choice Consider a consumer choosing how to spend his time. He has a choice between working, and consuming leisure (N). π» = 24 β π The consumer spends his total income on a variety of goods (a composite good) which costs £1 per unit. β How many goods the consumer buys depends on how much he earns; so does the cost the leisure. When the consumer isnβt working, he is losing earnings. The consumerβs utility depends on how many goods he buys, and how many hours he spends not working (consuming leisure) π = π(π, π) The consumerβs total income is given by : π = π€π» = π€(24 β π) Where π€ represents hourly wage Application: Labor-Leisure choice The slope of the budget constraint is given by -π€1 , the price of one extra unit of leisure is an hour of foregone earnings working. Composite good per day (£) Time constraint At point A, the consumerβs optimal choice is to consume 16 hours of leisure, and work for 8 hours. πΌπΆ1 π΄ π1 0 24 24 0 Leisure hours per day Work hours per day Application: Labor-Leisure choice We can now derive a demand curve for leisure. Increasing the wage from π€1 to π€2 , we obtain a new rotated budget constraint and a new optimal bundle of work and leisure (12, 12). Composite good per day (£) Time constraint π€2 π΅ π2 Wage per hour (£) π΅ π΄ π€1 Demand for leisure π΄ π1 0 24 ππ΅ = 12 π»π΅ = 12 ππ΄ = 16 π»π΄ = 8 24 0 Leisure hours per day Work hours per day ππ΅ = 12 ππ΄ = 16 Application: Labor-Leisure choice Wage per hour (£) Wage per hour (£) Supply of labor π€2 π΅ π΅ π€2 π΄ π΄ π€1 π€1 Demand for leisure ππ΅ = 12 ππ΄ = 16 π»π΄ = 8 π»π΅ = 12 Application: Labor-Leisure choice β Income and substitution effects Composite good per day (£) Time constraint C B A π1 Substitution effect Total effect 24 Income effect 0 Leisure hours per day Work hours per day From A to B is the substitution effect: At the higher wage, leisure is now more expensive. The consumer will substitute leisure for work. From B to C is the income effect, with the now higher wage, the consumer consumes more leisure. What does this tell you about leisure? What would happen if leisure becomes an inferior good after the wage increases above a certain threshold?