1.2 Utility Maximisation

Question 1

What is a utility function

Answer 1

basically assign a number to every possible consumption bundle, more preferred get assigned higher numbers (ordinal not cardinal)

Question 2

What are preferences?

Answer 2

what the individual wants to do, given choice

Five properties: completeness, transitivity, continuity, non-satiation & convexity

Question 3

How do Quasi linear functions look like again?

Answer 3

e.g. u(x1,x2) = x1^1/2 +x2

Question 4

How do Non-convex functions look like?

Answer 4

Question 5

What is the mathematical representation of a budget line?

Answer 5

Where p1 = price of good 1
Where p2 = price of good 2
M = income the consumer has.

Question 6

What is a budget set? represent budget line and set on diagram and what are the axes of the line?

Answer 6

1) rearrange the budget line in terms of x2
the Y axis is m/p2 andn the x axis is m/p1.
Remember x2 and x1 are quantities brought.

Question 7

Why is btw p1x1+p2x2=m and why does the y axis m/p2 and the x axis m/p1 and what is gradient of budget line.

Answer 7

1) because of non-saitiation you want to spend every penny to get as many foods as possible, in real life you save.
2) x1 intercept = m/p1 ( ifyou spend all your money on good 1 that means tou buy 0 of good 2, hence you rearrange and get in terms of x2 same thing for x2 intercept)
3) gradient = -p1/p2 ( Mux1/Mux2)

Question 8

The next question we ask is where will the consumer consume? what are we here referring to?

Answer 8

Uncompensated demand

Question 9

What is uncompensated demand?

Answer 9

Question 10

When finding uncompensated demand what are we pinning down?

Answer 10

Assume fixed income and price;
this pins down the budget line
• The consumer aims to get onto the
highest possible indifference curve, given prices and income.

Question 11

Is this graph accurate and where is the point that maximises the consumer utlitiy?

Answer 11

The graph is accurate if our assumptions of non-saitation and convexity hold, if not it wouldnt be a good fit.
The tangency point of IC and bugdet, gives us coordinates to find us the uncompensated demand.

Question 12

What is the slope of budget line again, what happens if there is a change in one and what happens if there is a change in both

Answer 12

It is the relative price of good 1 over good 2, so if one changes the slope changes, if they both change by same amount the slope is the same.

Question 13

If we want to find Uncompensated demand, what must be true for the slope of the indifference curve and the slope of the budget line?

Answer 13

the slopes must be the same so MRS = P1/P2. ( slope of indifference curve = slope of budget line)

Question 14

So how do we solve for uncompensated demand?

Answer 14

We have 2 equations to solve unknowns

1) first equation is the tangency condition
2) is the budget line
3) we solve simultaneously.

Question 15

Intrepreting the tangency condition MRS = P1/P2?

Answer 15

This tells us an individual will reach optimal consumption bundle when deviating away from that consumption bundle will not lead to gaining.

Question 16

The tangency condition can only hold if what?

Answer 16

If non-saitation and convexity hold.

Question 17

Here what assumptions fail and is the tangency the maximising thing to consume?

Answer 17

This is a failure of non-satiation
MRS = P1/P2 HOLDS AT TANGENCY A,
But A doesn’t maxmise utlitiy as the preferred set is below budget line, as you can see b is breffered to A.

Question 18

How do you determine preferred set again?

Answer 18

1) you take Utility function, test for non saiation and convexity, and it those assumptions hold you know where the preferred set is.
2) if the two assumptions fail, you have to think what preferences mean and try to sketch IC and figure it out?

Question 19

What is this a failure of?

Answer 19

It is a failure of convexity. there are 2 points where MRS = P1/P2
A is not maximising as B is better ( its in the preferred set.

Question 20

Here what assumption fails?

Answer 20

Convexity, extremes are preferred to averages.

Question 21

What are the 3 steps in calculating the uncompensated demand mathematically?

Answer 21

1)Identify the maximisation problem to be solved and what the solution is a function of: draw an indifference curve. ( maxmise the utlitiy function subject to budget constraint and non-negativitiy constraints)
2) Check for non-saitation and convexity using calculus
3) If Non-saitation and convexity satisfied then solve the tangency and budget line conditions and express the solution as a function of prices and income. ( If non-negativity constraints x1 ≥ 0 and x2 ≥ 0 are satisfied, then this solution
is uncompensated demand for goods 1 and 2
If violated look for corner solution.

Question 22

What is step 1 of this problem?

Answer 22

Question 23

Sketch the cobb douglas diagram?

Answer 23

The axes are the indifference curve for u= 0
• u > 0 requires both x1 > 0 and x2 > 0
• If u > 0 the indifference curve does not meet
the axes.

Question 24

What is step 2 of the problem?

Answer 24

Check if non satiation and convexity are satisfied ( they always are with cobb-douglas

Question 25

What is step 3?

Answer 25

Step 3: Solve the tangency and budget line conditions and express the solution as a function of prices and income; check non-negativity constraints.

Question 26

Lets think about the result, what can we infer 2 things?

Answer 26

1) Demand for each good does not depend on the price of the other ( Cross price elasticity = 0, thus subsitution effect doesn't hold here), only price of its on good. 2) The consumer spends a fixed proportion of income on each good!

Question 27

What is the general rule about cobb douglas if we have power of x1 alpha and x2 1- alpha? what must be the uncompensated demands?

Answer 27

Question 28

What is the general forumla for uncompensated demands

Answer 28

Question 29

What degree of homogeneity are uncompensated demand functions in prices and income? Illustrate it with the previous cobb douglas uncompensated demands adding a factor t prices and income?

Answer 29

All uncompensated demand functions are homogeneous of degree 0 in prices and income. The t's cancel out

Question 30

Now what do we want to do with the indifference curve map, where do we want to move to?

Answer 30

We now want to go from the indifference curve map to the demand curve diagram with quantity and price (x1 and p1) on the axes • And to see how changes in income m and p2 affect the demand curve for good 1 (and how m and p1 affect the demand curve for good 2)

Question 31

How can we translate our Cobb douglas uncompensated demands into a demand curve, whats the first step ( HINT REARRANGING uncompensated demands, to get p1 and p2 as functions and what do these functions say?

Answer 31

Question 32

What is special about Cobb douglas ( Linked to Xped)?

Answer 32

In this special case income is spend in fixed proportions so a price change of one good do not affect quantity demanded of the other!

Question 33

Now lets do some diagramtation, Lets say price of good 1 increases from P1a to P1b ( the budget line pivots inwards, whilst the y axis stays the same, and the gradient gets steeper, hence uncompensated demand for good 1 has fallen, but not for good 2, show this on a demand curve?

Answer 33

Remember the demand for good 1 is not affected by price of good 2 ( its only affected by its own prices and income as a whole) vice versa)

Question 34

What happens when prices of both goods for a cobb douglas example are kept constant and income increases and what happens if there is an increase in price of good 2?

Answer 34

There would be a shift outwards, bundle increases. d curve.  Increase in p2 results in no change ( as remember also the demand function isn't a function for p1)

Question 35

What is the price expansion path of when p1 changes? and what would be the be the price expansion path when x2 changes for cobb douglas preferences?

Answer 35

It is a horizontal line from each tangent to the budget line which pivots according to what has happened to price of good 1 It would be vertical when p2 changes with cobb douglas preferences.

Question 36

How would income expansion path look like on the indifference curve map, is this realistic, compare to quasi linear

Answer 36

Income increases, meaning there is a parallel shift outwards, it looks like a wray from origin( this is not realistic tho, e.g. quasi linear, you don't scale up everything when income increases, no income effect for good 1.).

Question 37

How do we show income expansion path on a demand curve for cobb douglas preferences?

Answer 37

So changes in price led to a movement along the demand curve and changes in income shows a shift in demand curve, keeping prices constant.

Question 38

Now we will look at elasticities, what elasticities will we look at ( First of all for Cobb douglas)

Answer 38

Own PeD Income elasticity Cross price elasticity

Question 39

How do you calculate Own Ped

Answer 39

Question 40

Calculate Ped of good 1 and what does it tell us about the cobb douglas function?

Answer 40

Question 41

For income elasticity what is the formula and what does it mean when E1m >0 and E1m<0?

Answer 41

* E1m > 0: normal good ( buy more when richer) | * Elm < 0: inferior good ( buy more when prices go down)

Question 42

Calculate the income elasticity of

Answer 42

m.x1 is the reporcial of 2m/5p1

Question 43

What is the forumla for Xped and what does it mean if its >0, <0 = 0

Answer 43

p2/x1 is the reciprocal of the x1 ( of cobb douglas uncompensated demand)

Question 44

Work out Xped for good 1 ?

Answer 44

The answer should be 0. Remember thats always the case fo Cobb douglas.

Question 45

With Quasi linear preferences (ONLY whats a cool thing you can do that makes solving for x1 and x2 compensated demands easy?

Answer 45

We can use the MRS = P1/P2 and solve for x1 or x2, then sub that in the budget constraint.

Question 46

Now lets look at Uncompensated demand for a quasi linear function? Find uncompensated demand of this quasi linear function. Remember outline the steps, what does this say?

Answer 46

Work out x1 and x2 on paper When you don't have enough income you consume only good x1 but when you you have enough income you consume the uncompensated demand function you found.

Question 47

Are non negativity constraints satisfied?

Answer 47

There are 2 cases when

Question 48

Why is it the case that people want to spend a fixed amount on heating, then when they have enough income, they don't consume any more of the good being proritised?

Answer 48

The Mu for heating x1 is falling. When your house is cold and you put the heating on the Mu is high, but some point house is warm enough that you switch to the other goods which has a constant MU. ( Thus there is a diminishing MU for heating the house.

Question 49

How does Case 1 look on an indifference map? Does MRS depend on good 2.

Answer 49

If don't have a lot of money but i get richer and richer, i increase consumption of good x2 but not x1. Remember the MRS doesn't depend on good 2

Question 50

How does Case 2 look on Indifference map?

Answer 50

If m is p2^2/4p1 or lower all income is spent on good 1.

Question 51

What is the income expansion path for quasi linear preferences?

Answer 51

So when income is smaller than p2^2/4p1 you only consume x1 till income is grreater than p2^2/4p1 you stop increasing consumption of good 1 then increase consumption of good 2.

Question 52

What shape would the income expansion path have if u(x1,x2) = x1 + v(x2)

Answer 52

Question 53

How does a demand curve look like for Quasi linear preferences and what happens when there increase in p1. Increase in p2 and increase in m compare to CB.

Answer 53

1) Rearrange x1 to find p1 and you find the demand curve is the same as with CB but is not a function of income 2) Increase in p1 results in a movement along the demand curve. 3) Increase in p2 results in a shift out of the demand curve. ( with CB an increase in price of p2 will lead to no change in demand) 4) Increase in m results in no change in demand.( whereas for CB it did)

Question 54

So for Quasi linear preferences ( protising x1) what is the income elasticity and Xped compared to CB?

Answer 54

The income elasticity for x1 is 0 ( not the same for CB) | Also the Xped is not 0( it was for CB)

Question 55

Now lets look at Perfect complements, as they are not differentiable, what can we infer already?

Answer 55

The IC's are L-shaped and as averages are preferred to extremes( fixed proportions though) the function is convex as it when you draw 2 lines from IC and the line is in the preferred set. Non saitation holds if both x1 and x2 are increased.

Question 56

How do you say this and show on diagram?

Answer 56

You need 2 wheels for each frame

Question 57

As we know for PC there is an interior solution as there is no substation effect, how do we find uncompensated demand for the example. We know that the budget constraint satisfies p1x1 + p2x2 = m and we know x2=1/2x1, how can we solve for x1 and x2?( solution is at kink) Also tell me about the function.

Answer 57

1) plug x2 into budget constraint and simplfy to find x1 then plug into x2 so

Question 58

How do we find the demand curve for the perfect complements example and illustrate it?What happens when there is an increase in p1 p2 and m?

Answer 58

Rearrange the uncompensated demand in terms of p1

Question 59

How would the Income and Price expansion path look like for PC?

Answer 59

1) Income expansion ( an increase in income means you consume more of both proportionality) 2) Price expansion ( E.g. lets say there is a high price for good 2, then you reduce consumption of both goods ( remember no income effect) Hence Price expansion and income expansion are the same

Question 60

``` Quiz question cobb douglas? p1 increases, p2 and m do not change. What happens to demand for goods 1 and 2? 1. Demand for good 1 increases. 2. Demand for good 1 decreases. 3. Demand for good 1 does not change. 4. Demand for good 2 increases. 5. Demand for good 2 decreases. 6. Demand for good 2 does not change. ```

Answer 60

2 and 6

Question 61

What are normal, inferior and giffen goods?

Answer 61

A good is normal if consumption increases when income increases.( income elasticity of demand is positive) A good is inferior if consumption decreases when income increases ( income elasticity of demand is negative) Giffen goods are goods whose demand increases with the increase in its price and vice versa ( has nothing to do with income)

Question 62

KEY: When testing for convexity we have to rearrange the function in terms of x2 as a function of x1 and U. What does the first derative tell us and the second derative?

Answer 62

1) If first derartive with respect to x1 is positive, it means its upward sloping, if it is negative it means its downward sloping.

Question 63

KEY: When testing for convexity we have to rearrange the function in terms of x2 as a function of x1 and U. What does the second derative tell us?

Answer 63

If the 2nd derative is positive then we can say We can say that the function is increasing (or decreasing) at an increasing rate. if the second derative is negative this tells us that the function is increasing (or decreasing) at a decreasing rate.

Question 64

Find the uncompensated demand for the function

Answer 64

Question 65

What does the price expansion and income expansion path look like?

Answer 65

They are the same

Question 66

What are perfect susbitutes ?

Answer 66

Goods that can be substituted for each other at a constant rate, i.e. have a constant MRS, are called perfect substitu

Question 67

Find the uncompensated demand for perfect substuties? u(x1,x2)= 3x1+2x2.

Answer 67

For convexity the second derative 0. Case 1 X1( p1,p2,m) = m/p1 and x2(p1,p2,m) = 0 Case 2 X1 (p1,p2,m) = 0 and x2(p1,p2,m) = m/p2 Case 3 solution at any x1 x2 satisfying x1 ≥ 0 x2 ≥ 0 and budget constraint

Question 68

How would the price expansion and income expansion path look like for each case?

Answer 68

Case 1 ) Price and Income would be along the x axis Case 2 ) Price and income would be along the y axis Case 3 )if income changes, then the budget line shifts out but MRS is still equal to p1/p2. The implication is there is no path as such...the entire positive quadrant is the income expansion 'path'....as income expands you just move upwards anywhere in positive (x1, x2) space. With price expanding, when MRS = p1/p2, the entire budget set is optimal. So it zigzags...one axis, the budget line and then the other axis.

Question 69

Draw the demand curve for the function and show when show 2 diagrams, showing when P1 is greater than p2, when equal and when p1 is less than p2 ( hint remember we can rearrange to find p1 as a function of p2. Also show what happens when there is an increase in p1, increase in p2 and increase in m.

Answer 69

Question 70

Answer 70

Question 71

Sketch the income expansion path for Demand foe good 1 and provide initution on shape

Answer 71

Quasi linear preferences in good 1 means that the consumer will opt for good 1 that doesn't vary with income, provided income is enough to cover this expenditure. Increasing income above this point does increase consumption of good 1 but increases consumption of good 2 ( talk about MU) ( there is no income effect for good 1, interior solution). If income were to fall below p1^-3p2^4, there would be an income effect for good 1 in the corner solution..

Question 72

Find the first and second deravitive?

Answer 72

foc = -1/2(U-X1^2)^1/2 X -2X1 = -X1(U-X1^2)^1/2 | SECOND DERARTIVE : Using the product rule

Question 73

PROBLEM SET COMPLETE THE WHOLE OF QUESTION 1?

Answer 73

Question 74

PROBLEM SET QUESTION 2

Answer 74

Question 75

Does a tangency exist with L shaped preferences( always mention this in perfect complements questions?

Answer 75

NO!!, with L shaped preferences, utility is maximised at kink, it is optimised at kink because because if not it would be possible to reduce the quantity of one good and increase the quantity of the other without changing expenditure but increasing utility.

Question 76

Why is utility maximised when 1/2x1 = x2?

Answer 76

Question 77

What is the utility function here?

Answer 77

Whenever we have x1 and x2 and greater than one x1 and x2, divide do the reciprocal then find x2

Question 78

Finish the sentence, because nonsaitation and convexity are satisfied ( this is for uncompensated and compensated demand?

Answer 78

Any point on the indifference curve at which the tangency condition MRS = price ratio, the utility condition/budget constraint and non negativity constraint are satisfied solves the minimization/maximisation problem?