Part I CONSUMER THEORY II Laura Sochat Constrained optimisation There are n goods consumed in quantities making up a bundle The agent income is M and the ID: 547668
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Slide1
Intermediate Microeconomics
Part I
CONSUMER THEORY (II)
Laura SochatSlide2
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
Slide3
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 methodObtain the F.O.C.s
Solve the system to find the values of
and , for which the Lagrangian is maximised, and the constraint holds.
Slide4
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 findingsAn alternative method could be to substitute for X, or Y, within the utility function and solve for the first order condition: , also assume that , , and Given the values of 𝑃𝑥, 𝑃𝑦, and 𝐼, It is possible to solve for the optimal bundle. Slide5
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 taxesUniform quantity taxes Slide6
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 Slide7
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
Slide8
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 priceIncreasing in IncomeHomogeneous of degree zero in price and income Slide9
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
Slide10
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:
Slide11
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 Slide12
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 Slide13
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 goodExample- Solve for the Hicksian demand in the case of a Cobb Douglas Utility function (where ) Slide14
Shepard’s Lemma
Remember Roy’s identity, obtained after solving for the utility maximisation problem, and using the Envelop Theorem
Shepard’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
Slide15
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 utilityFrom 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
Slide16
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 demandedABDemandSlide17
Using the rational choice model to derive individual
demand: A
change in the price of one good.
Price (P)Quantity demandedThe 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
7Slide18
Using the rational choice model to derive individual
demand: A
change in the price of one good.
Price (P)Quantity demanded126471522Demand curve
Price of fish
Quantity demanded
4
22
6
1512
7Slide19
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 curve609012058
12
10
1520
Income
Quantity demanded
120
12
90
8
60
5
Slide20
A change in Income: The Income-Consumption curve and the Engel curve
Income
Fish (Kg/week)
The Engel curve 60901205812
Income
Quantity demanded
12012908
60
5
Slide21
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 Slide22
Income elasticities and Income consumption curves
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.
Slide23
Difference preferences: What would the Engel curves look like?
Perfect substitutes
Perfect complements
Homothetic preferencesQuasilinear preferences Slide24
The Engel curve when one of the good is both normal and inferior
All other goods (£)
X
The income consumption curve
The Engel curve
Income
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 income consumption curve
The Engel curve
Slide25
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 Slide26
The income and substitution effects (Hicks) : A normal good
All other goods (£)
Fish (Kg/week)
All other goods (£)
Fish (Kg/week)
Substitution effect
Income 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.Slide27
The income and substitution effects (Hicks) : An inferior good
All other goods (£)
All other goods (£)
Income effect
Substitution effect
Total 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
XSlide28
The income and substitution
effect (Hicks) : A
giffen
good
Substitution effect
Income effect
Total effect
All other goods (£)
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.
Slide29
How to calculate the effects?
STEP 1
Utility maximisation
Allows us to find the initial optimising bundle of goods chosen by the consumer at initial pricesSTEP 2Expenditure minimisationAllows us to maintain the level of utility fixed at initial level, while minimising expenditure at new prices STEP 3Utility maximisationAllows us to calculate the income effect from the consumer’s maximisation problem at the new set of prices Slide30
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 effectThe own price demand curve derived before, the Marshallian demand, is the uncompensated demand curve. It accounts for both the income and the substitution effect Slide31
Compensated H
icksian
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. Slide32
The income and substitution effects (
Slutsky
) : A normal good
All other goods (£)
All other goods (£)
Substitution effect
Income effect
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
XSlide33
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 effectSlide34
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 effectSlide35
Putting the two together, we obtain the
S
lutsky
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
S
lutsky
equation Slide36
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
Slide37
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 Slide38
24
0
0
24
Work hours per day
Leisure hours per day
Time constraint
Composite good per day (£)
At point A, the consumer’s optimal choice is to consume 16 hours of leisure, and work for 8 hours.
Application: Labor-Leisure choice
The slope of the budget constraint is given by -
, the price of one extra unit of leisure is an hour of foregone earnings working.
Slide39
24
0
0
24
Work hours per day
Leisure
hours per dayComposite good per day (£)
Time constraint
Wage per hour (£)
Demand for leisure
Application: Labor-Leisure choice
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).
Slide40
Wage per hour (£)
Demand for leisure
Application: Labor-Leisure choice
Wage per hour (£)
Supply of labor
Slide41
Application: Labor-Leisure choice – Income and substitution effects
0
24
Work hours per day
Leisure
hours per day
Time constraint
Income effect
Substitution effect
Total effect
Composite good per day (£)
A
B
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?