Microeconomics I * Term 1, Part 2

Intermediate Microeconomics

Part ICONSUMER THEORY (II)

Laura Sochat

Constrained optimisation

There are n goods consumed in quantities , …, making up a bundle , …, The agent income is M, and the given market prices of each good are , …, .Agent’s 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:

, or

Optimisation with two goods

We have seen before the graphical method of solving for the optimising bundle of goods, using the tangency condition:

Let’s now look at the Lagrangean method

1. Obtain the F.O.C.s

Solve the system to find the values of and , for which the Lagrangian is maximised, and the constraint holds.

Examples, and alternative method

Suppose that represents the consumer’s income, the price of good X, and the price of good Y. Assume that the consumer’s preferences for goods X and Y, are defined by the following utility function:

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:

a) , also assume that , , and Given the values of , , and , It is possible to solve for the optimal bundle.𝑃𝑥 𝑃𝑦 𝐼

Choosing between two types of taxes, using consumer theory

Assuming the original budget constraint is given by

Suppose the government wishes to raise tax revenue. Should they raise quantity tax (on good 1)? Or income tax?– Show the results in a graphThink about the limitations of this example in terms of:– 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:

– …

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:

That is we can say that a dollar of extra income should increase the consumer’s utility by

Roy’s identity

Substituting for the Marshallian demands into the original utility function, we obtain an expression for the actual level of utility obtained:

This is called the indirect utility function, and has the following properties:– It is non-increasing in every price, decreasing in at least one price– Increasing in Income– Homogeneous of degree zero in price and income

Roy’s identity- The envelop theorem

Consider the case of two goods. Taking the total derivative of the Indirect utility function we get that (1):

Next we need to make use of two results found before:– The first order conditions tell us the value of the marginal utilities, which we can use in (1)

, – Taking the total derivative of the budget constraint and substituting to obtain the following

This gives us an important result, Roy’s Identity

The Envelop Theorem

More generally, the result above can be assumed, by using the envelop theorem. Consider the following maximisation problem:

The constant is given exogenously. We can solve the problem as usual:

F.O.Cs are given by

Substituting for and into the objective function, we obtain the value function:

The value function is the maximised value of our objective function. Taking the total derivative of the value function with respect to :

Differentiating the constraint, with respect to

The Envelop Theorem

Expenditure minimisation

We can find optimal decisions of our consumer using a different approach.We can minimise the consumer’s expenditure subject to a given level of utility that the consumer must obtain – 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 , and his preferences are rationalThe consumer is choosing to solve the following problem

Expenditure minimisation solution

Solution to the expenditure minimisation problem are called Hicksian demand functions and take the form

• 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 )

Shepard’s Lemma

Remember Roy’s identity, obtained after solving for the utility maximisation problem, and using the Envelop TheoremShepard’s 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

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:

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 xs…

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)

Quantity demanded

A

B

Demand


Price (P)

Quantity demanded

The price consumption curve

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

Price of fish Quantity demanded

4 22

6 15

12 7


Price (P)

Quantity demanded

12

6

4

7 15 22

Demand curve

Price of fish Quantity demanded

4 22

6 15

12 7

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.

A change in Income: The Income-Consumption curve and the Engel curve

All other goods (£)

Fish (Kg/week)

The income consumption curve

60

90

120

5 8 1210 15 20

Income Quantity demanded

120 12

90 8

60 5

𝐴𝐵𝐶

A change in Income: The Income-Consumption curve and the Engel curve

Income

Fish (Kg/week)

The Engel curve

60

90

120

5 8 12

Income Quantity demanded

120 12

90 8

60 5𝐴

𝐵

𝐶

Different types of goods

The income elasticity tells us how quantity demanded responds to a change in income. It is given by:

– As income increases by 1%, quantity demanded increases by ξ%.

A good is said to be normal, if ξ>0, the quantity demanded of a normal good increases (decreases) as income increases decreases) A good is said to be inferior, if ξ<0, the quantity demanded of an inferior good decreases (increases) as income increases (decreases)A good is said to be a luxury good if ξ>1A good is said to be a necessary good if ξ<1

Income elasticities and Income consumption curves

𝑋 1

𝐼𝐶𝐶1

𝑋 2

𝐼𝐶𝐶2

𝐼𝐶𝐶3

Assume income increases; The budget constraint shifts to the right.

: Both goods are normal, quantity demanded of both goods has increased following the increase in income : is a normal good, while is inferior. Quantity demanded of good 2 has fallen following the increase in income : Good 2 in normal, while good 1 is inferior.

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


X


𝐼 1

𝐼 2

𝐼 3

𝐼𝐶1

𝐼𝐶2

𝐼𝐶3

The Engel curve

Income

𝑋

𝐼 1

𝐼 2

𝐼 3

𝑋 1 𝑋 2𝑋 3 𝑋 1 𝑋 2𝑋 3

From to , the increase in income lead the consumer to demand more of X.From to , however, the increase in income lead the consumer to demand less of X.


The Engel curve

𝐴𝐵

𝐶 𝐵

𝐴

𝐶

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. – 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 ratioThe 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. – 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 (£)

Fish (Kg/week)𝑋 1𝑋 3


Fish (Kg/week)𝑋 1𝑋 3𝑋 2

𝐼𝐶1

𝐼𝐶2

𝐼𝐶2

𝐼𝐶1𝐴

𝐵

𝐶𝐴𝐶

Substitution effectIncome effect

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.

The income and substitution effects (Hicks) : An inferior good


𝑋 1𝑋 3

𝐼𝐶2

𝐼𝐶1

𝐴𝐶

𝐴𝐵

𝐶


𝑋 1𝑋 2𝑋 3Income effect

Substitution effectTotal effect

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.

X X

The income and substitution effect (Hicks) : A giffen good


Total effect


𝑋 1

𝐵

𝐴

𝐶

Suppose the price of falls, leading to a new (rotated) BL. 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.

In this situation, the substitution effect is completely offset by the income effect.

𝐼𝐶1

𝐼𝐶2

How to calculate the effects?

STEP 1Utility maximisation– Allows us to find the initial optimising bundle of goods chosen by the consumer at initial prices

STEP 2Expenditure minimisation– Allows us to maintain the level of utility fixed at initial level, while minimising expenditure at

new prices

STEP 3Utility maximisation– Allows us to calculate the income effect from the consumer’s 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

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 (£)

𝑋 1𝑋 2


𝑋 1𝑋 2𝑋 3

𝐼𝐶1

𝐼𝐶2

𝐼𝐶2

𝐼𝐶1𝐴

𝐵

𝐶𝐴𝐶


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.

X X

As seen in the graph above, the ‘pivoted’ 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 changes from to How can we calculate the amount of money income needed to keep that initial bundle affordable?

The substitution effect is the change in demand for a good when its price changes, and at the same time, money income is compensated.

An algebraic interpretation: The substitution effect

Consider still a change is the price of good , from to

The income effect will be the change in the demand for the good, when we change income from to , while holding the price of the good at the new level. That is:

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 Income effect

Putting the two together, we obtain the Slutsky identity:

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:

An algebraic interpretation: The Slutsky equation

Marshallian demand elasticities

The price elasticity of demand measures the proportionate change in quantity demanded in response to a proportionate change in a good’s own price. Apart from the exception of a Giffen good, the own price elasticity of demand is always negative.

Cross price elasticity of demand, , measures the proportionate change in quantity demanded in response to a proportionate change in the price of some other good

Application: Labor-Leisure choice

Consider a consumer choosing how to spend his time. He has a choice between working, and consuming leisure (N).

The consumer spends his total income on a variety of goods (a composite good) which costs £1 per unit. – How many goods the consumer buys depends on how much he earns; so does the cost the leisure.

When the consumer isn’t working, he is losing earnings.

The consumer’s utility depends on how many goods he buys, and how many hours he spends not working (consuming leisure)

The consumer’s total income is given by :

Where represents hourly wage

24 00 24

Work hours per day Leisure hours per day

Time constraintComposite good per day (£)

𝑌 1

𝐼𝐶1

At point A, the consumer’s optimal choice is to consume 16 hours of leisure, and work for 8 hours.

𝐴


The slope of the budget constraint is given by -, the price of one extra unit of leisure is an hour of foregone earnings working.

24 00 24

Work hours per dayLeisure hours per day

Composite good per day (£)

𝑌 1

𝑌 2

𝐴

𝐵

𝐻 𝐴=8𝐻 𝐵=12𝑁𝐵=12 𝑁 𝐴=16

Time constraint Wage per hour (£)

Demand for leisure𝐴

𝑁 𝐴=16𝑁𝐵=12


𝐵

𝑤1

𝑤2

We can now derive a demand curve for leisure. Increasing the wage from to , we obtain a new rotated budget constraint and a new optimal bundle of work and leisure (12, 12).

𝐵

Wage per hour (£)

Demand for leisure

𝐴

𝑁 𝐴=16𝑁𝐵=12

𝑤1

𝑤2


Wage per hour (£)

𝑤1

𝑤2

Supply of labor

𝐻 𝐵=12𝐻 𝐴=8

𝐵

𝐴

Application: Labor-Leisure choice – Income and substitution effects

024

Work hours per dayLeisure hours per day

𝑌 1

Time constraint

Income effectSubstitution effect Total effect

Composite good per day (£)

AB

C

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?

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Microeconomics I * Term 1, Part 2