5- choice and demand

Question 1

what is the utility maximisation problem?

Answer 1

a rational consumer maximises her utility u(x1,x2) given her budget constraint p1x1+ p2x2=M

Question 2

what is the optimal consumption position on the indifference curve and budget line?

Answer 2

the optimal consumption position is when the indifference curve is tangent to the budget line ie when the MRS = - p1/p2

Question 3

how do you find the maximimal utility for a cobb douglous utility function?

Answer 3

you use a lagrangian function where L = x1^(a)x2^b + h(M-p1x1-p2x2) . you then do partial differentiation for x1, x2 and h and then you equate them all to 0

Question 4

what does the cobb douglas utility function at the optimum satisfy?

Answer 4

the following property is satisfied : MRS=-p1/p2

Question 5

does the tangency condition for a cobb douglous utility function always hold?

Answer 5

no as an interior solution doesnt always exist so sometimes we need to look for a corner solution. if an interior solution exists but a utility function is not differentiable everywhere, the condition can be violated

Question 6

what is a boundary optimum?

Answer 6

in the situation where the slope of the budget constraint is always smaller (bigger) than the slope of the indifference curve so the equality MRS = -p1/p2 is never satisified therefore we should have a corner solution

Question 7

what are the important implications of the MRS condition?

Answer 7

in well organised markets, it is typical that everyone roughly faces the same prices for goods. therefore everyone who is consuming the two goods must have the same marginal rate of substitution

Question 8

what is the optimal choice of goods 1 and 2 at some set of prices and income called?

Answer 8

the demand bundle?

Question 9

what is the demand bundle?

Answer 9

the optimal choice of goods 1 and 2 at some set of prices and income

Question 10

what occurs to the consumers optimal choice and demand when the prices and income changes?

Answer 10

the consumers optimal choice changes

Question 11

what is the demand function?

Answer 11

the demand function is the function that relates the quantities demanded to the different values of prices and incomes

Question 12

what is the equation of the demand function?

Answer 12

the demand fucntion is xi(pi,pj,M) where pi is the own price, pj is the price of the other good and M is income

Question 13

what does the demand curve depict??

Answer 13

it depicts the optimal choice of good 1 as a function of its price

Question 14

what does the price offer curve depict

Answer 14

the price offer curve depicts the optimal choices as the price of good 1 changes

Question 15

what does the indirect utility fucntion show?

Answer 15

the indirect utility function shows the maximum level of utility which can be achieved at given prices and income so v(px, py,M) = Max(x,y), xPx +yPy=M
in other words, its utility as a function of prices and income at the optimal consumption point

Question 16

what are the properties of the indirect utility function?

Answer 16

v(px,py,M) is non increasing in prices and non decreasing in income. it is also homogenous of degree 0

Question 17

what is the marshallian demand for good x with a utility function u=xy?

Answer 17

x*=M/(2Px)

Question 18

what is the marshallian demand for good y with a utility function u=xy?

Answer 18

Y*=M/(2Py)

Question 19

what is the indirect utility curve for a utility function u=xy?

Answer 19

v=M^2/(4PxPy)

Question 20

what is the marshallian demand for good x for the utility function u=min{x,y}?

Answer 20

x*=M/(p1+p2)

Question 21

what is the marshallian demand for good y for the utility function u=min{x,y}?

Answer 21

y*=M/(p1+p2)

Question 22

what is the indirect utility curve for the utility function u=min{x,y}?

Answer 22

v=M/(p1+p2)

Question 23

what is the marshallian demand for good x when the utility fucntion u= x+y?

Answer 23

X=M/p1 if p1<p2, X= exponential when p1=p2 and 0 when P1>p2

Question 24

what is the marshallian demand for good y when the utility fucntion u= x+y?

Answer 24

Y=M/p1 if p1<p2, Y= exponential when p1=p2 and Y*=0 when P1>p2

Question 25

what is the indirect utility curve for the utility function u=x+y?

Answer 25

v=M/P1 if p1<p2, v=M/p1 if p1=p2 and v=M/p2 if p1>p2

Question 26

why do we need indirect utility?

Answer 26

it is convient to use indirect utility for welfare analysis for example if we want to understand how consumer welfare changes if price px increases to price p’x, we may use indirect analysis to answer this

Question 27

what does it mean to be an ordinary good?

Answer 27

ordinarily, when the price of a good increases, the demand for that good will decrease ie the good has a negative PED

Question 28

what is a giffen good?

Answer 28

if for some values of price, the quantity demanded of a good rises as its own price increases then the good is a giffen good. ie the PED is positive. usually an essential low income good with a lack of close substitutes

Question 29

what is the own price elasticity of demand?

Answer 29

own price elasticity of demand measures the change in consumption of good 1 in relation to a change in its price:
𝜺_𝟏𝟏=(𝝏𝒙_𝟏)/(𝝏𝒑_𝟏 ) 𝒑_𝟏/𝒙_𝟏

Question 30

what is the elasticity of demand when the good is elastic and what does it mean?

Answer 30

if the own elasticity of demand is elastic then if the price rises the expenditure falls. the PED is greater than 1

Question 31

what is the elasticity of demand when the good is inelastic and what does this mean?

Answer 31

the price elasticity of demand is less than 1 and it means if you increase the price the expenditure will increase

Question 32

what is unit elasticity and what does this mean for expenditure?

Answer 32

it has a ped of 1 and expenditure does not change when the price increases

Question 33

Why not just use the slope of a demand curve to measure the sensitivity of quantity demanded to a change in a commodity’s own price?

Answer 33

Because the value of sensitivity then depends upon the (arbitrary) units of measurement used for quantity demanded.

Question 34

what are the units of measurement for elasticity?

Answer 34

However, the elasticity measures the percentage change, and therefore it is independent of units of measurement

Question 35

what is the cross price elasticity of demadn?

Answer 35

Cross price elasticity of demand measures the change in consumption of good 1 in relation to a change in the price of good 2

Question 36

what is the formula for cross price elasticity of demand?

Answer 36

𝜺_𝟏𝟐=(𝝏𝒙_𝟏)/(𝝏𝒑_𝟐 ) * 𝒑_𝟐/𝒙_𝟏

Question 37

what is the cross price elasticity for a substitute and what would be the effect of a price increase in good 2?

Answer 37

Substitutes: 𝜀_12>0, price 𝑝_2↑ ⇒ buy more of good 1

Question 38

what is the cross price elastiticy of demand for a complement and what is the effect of a price rise in good 2?

Answer 38

Complements: 𝜀_12<0, price 𝑝_2↑ ⇒ buy less of good 1

Question 39

what is the cross price elastiticy of demand for an independent good and what is the effect of a price rise in good 2?

Answer 39

Independent” goods: 𝜀_12=0, consumption of good 1 doesn’t depend on the price of good 2.

Question 40

is it possible for 𝑥_1 to be a substitute for 𝑥_2 and at the same time for 𝑥_2 to be a complement of 𝑥_1?

Answer 40

It is possible, by the definitions, for 𝑥_1 to be a substitute for 𝑥_2 and at the same time for 𝑥_2 to be a complement of 𝑥_1.

Question 41

what is the income offer curve?

Answer 41

Income offer curve define as the curve which depicts the optimal choice of two goods at different levels of income at constant price.

Question 42

what is the engel curve?

Answer 42

The Engel curve describes how the spending on a certain good varies with household income by either proportion or absolute dollar amount.

Question 43

how do you produce the engel curve from a cobb douglas function?

Answer 43

you find the optimal solution for each good according to the cobb douglas preferences and then you rearrange to isolate M

Question 44

are all engel curves straight lines?

Answer 44

no this is not true in general

Question 45

what type of preferences are needed in order for the engel curves to be straight lines?

Answer 45

the preferences need to be homothetic

Question 46

what does it mean for preferences to be homothetic?

Answer 46

A consumer’s preferences are homothetic if and only if
(𝑥_1,𝑥_2 )≽(𝑦_1,𝑦_2 )⇔(𝑘𝑥_1,𝑘𝑥_2 )≽(𝑘𝑦_1,𝑘𝑦_2 ), ∀𝑘>0
Homothetic preferences can be represented by the utility function that has the following property:
𝑢(〖𝑘𝑥〗_1,𝑘𝑥_2 )=𝑘𝑢(𝑥_1,𝑥_2 ),∀𝑘>0

Question 47

what are common examples of homothetic preferences?

Answer 47

Examples of homothetic preferences:
Cobb-Douglas
Perfect substitutes
Perfect complements
Some separable (𝑢=√(𝑥_1 )+√(𝑥_2 ))

Question 48

what is an important property of the homothetic preferences?

Answer 48

MRS is a function of only the 𝑥2/𝑥1 ratio, but not of 𝑥2 and 𝑥1 individually.
Therefore, when income increases consumption of both goods increases proportionally.

Question 49

what is the slope of the engel curve for a normal good?

Answer 49

normal goods quantity rises with income therefore a normal goods engel curve is positively sloped

Question 50

what is the slope of the engel curve for an inferior good?

Answer 50

a good for which quantity demanded falls as income increases is called income inferior and therefore an income inferior goods engel curve is backwards falling

Question 51

what is income elasticity?

Answer 51

Income elasticity shows how sensitive the quantity demanded for a good to an income change

Question 52

what is the equation for income elasticity?

Answer 52

𝜀_1𝑚=(𝜕𝑥_1)/𝜕𝑀 𝑀/𝑥_1

Question 53

what is the income elasticity of a luxury good?

Answer 53

Income elasticity 𝜀_1𝑚 > 1: Luxury good

Question 54

what is the income elasticity of a necessary good?

Answer 54

Income elasticity 𝜀_1𝑚 < 1: Necessary good

Question 55

what is the income elasticity of a normal good?

Answer 55

Income elasticity 𝜀_1𝑚>0: Normal good

Question 56

what is the income elasticity of an inferior good?

Answer 56

Income elasticity 𝜀_1𝑚<0: Inferior good

Question 57

what does the income offer curve look like for a graph where x2 is a luxury and x1 is a necessity?

Answer 57

the x2/x1 ratio is increasing with M and so the curve is positive and getting steeper

Question 58

what does the engel curve look like for a luxury good?

Answer 58

the slope is positive and getting flatter as when income increases, more luxury goods are purchased

Question 59

what does the engel curve look like for a necessity good?

Answer 59

it is a postive slope which is getting steeper as when income increases a lower proportion of the income is spent on necessities

Question 60

are income elasticities of different goods linked?

Answer 60

the income elasticities of different goods are linked

Question 61

what are the equations that link together elasticities for a two good economy assuming all income is spent?

Answer 61

The weighted average of the income elasticities (weights being the proportion of income spent on goods)equals 1:
𝒔_𝟏 𝜺_𝟏𝑴+𝒔_𝟐 𝜺_𝟐𝑴=𝟏
where s is the budget share of each good and 𝜺 is the income elasticities

Question 62

what are the conclusions that arise from the equation that links together the elasticities.

Answer 62

We can see that
If 𝜀_1𝑀>1, then 𝜀_2𝑀<1⇒ if good 1 is luxury, then good 2 is necessary;
If 𝜀_1𝑀<1, then 𝜀_2𝑀>1⇒ if good 1 is necessary, then good 2 is luxury;
Both goods can’t be luxury or necessary at the same time;
If 𝜀_1𝑀=𝜀_2𝑀⇒ they must be equal to 1, and therefore both goods are neither luxury nor necessary;

Question 63

for homothetic preferences, what is feature of budget shares?

Answer 63

the budget shares are constant meaning that the elasticities are equal to 1