1.2 Utility Maximisation Flashcards
What is a utility function
basically assign a number to every possible consumption bundle, more preferred get assigned higher numbers (ordinal not cardinal)
What are preferences?
what the individual wants to do, given choice
Five properties: completeness, transitivity, continuity, non-satiation & convexity
How do Quasi linear functions look like again?
e.g. u(x1,x2) = x1^1/2 +x2
How do Non-convex functions look like?
What is the mathematical representation of a budget line?
Where p1 = price of good 1
Where p2 = price of good 2
M = income the consumer has.
What is a budget set? represent budget line and set on diagram and what are the axes of the line?
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.
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.
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)
The next question we ask is where will the consumer consume? what are we here referring to?
Uncompensated demand
What is uncompensated demand?
When finding uncompensated demand what are we pinning down?
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.
Is this graph accurate and where is the point that maximises the consumer utlitiy?
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.
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
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.
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?
the slopes must be the same so MRS = P1/P2. ( slope of indifference curve = slope of budget line)
So how do we solve for uncompensated demand?
We have 2 equations to solve unknowns
1) first equation is the tangency condition
2) is the budget line
3) we solve simultaneously.
Intrepreting the tangency condition MRS = P1/P2?
This tells us an individual will reach optimal consumption bundle when deviating away from that consumption bundle will not lead to gaining.
The tangency condition can only hold if what?
If non-saitation and convexity hold.
Here what assumptions fail and is the tangency the maximising thing to consume?
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.
How do you determine preferred set again?
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?
What is this a failure of?
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.
Here what assumption fails?
Convexity, extremes are preferred to averages.
What are the 3 steps in calculating the uncompensated demand mathematically?
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.
What is step 1 of this problem?
Sketch the cobb douglas diagram?
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.
What is step 2 of the problem?
Check if non satiation and convexity are satisfied ( they always are with cobb-douglas
What is step 3?
Step 3: Solve the tangency and budget line conditions and express the solution
as a function of prices and income; check non-negativity constraints.
Lets think about the result, what can we infer 2 things?
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!
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?
What is the general forumla for uncompensated demands
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?
All uncompensated demand functions are homogeneous of degree 0
in prices and income.
The t’s cancel out
Now what do we want to do with the indifference curve map, where do we want to move to?
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)
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?
What is special about Cobb douglas ( Linked to Xped)?
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!
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?
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)
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?
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)
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?
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.
How would income expansion path look like on the indifference curve map, is this realistic, compare to quasi linear
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.).
How do we show income expansion path on a demand curve for cobb douglas preferences?
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.
Now we will look at elasticities, what elasticities will we look at ( First of all for Cobb douglas)
Own PeD
Income elasticity
Cross price elasticity
How do you calculate Own Ped
Calculate Ped of good 1 and what does it tell us about the cobb douglas function?
For income elasticity what is the formula and what does it mean when E1m >0 and E1m<0?
- E1m > 0: normal good ( buy more when richer)
* Elm < 0: inferior good ( buy more when prices go down)
Calculate the income elasticity of
m.x1 is the reporcial of 2m/5p1
What is the forumla for Xped and what does it mean if its >0, <0 = 0
p2/x1 is the reciprocal of the x1 ( of cobb douglas uncompensated demand)
Work out Xped for good 1 ?
The answer should be 0. Remember thats always the case fo Cobb douglas.
With Quasi linear preferences (ONLY whats a cool thing you can do that makes solving for x1 and x2 compensated demands easy?
We can use the MRS = P1/P2 and solve for x1 or x2, then sub that in the budget constraint.
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?
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.
Are non negativity constraints satisfied?
There are 2 cases when
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?
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.
How does Case 1 look on an indifference map? Does MRS depend on good 2.
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
How does Case 2 look on Indifference map?
If m is p2^2/4p1 or lower all income is spent on good 1.
What is the income expansion path for quasi linear preferences?
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.
What shape would the income expansion path have if u(x1,x2) = x1 + v(x2)
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.
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)
So for Quasi linear preferences ( protising x1) what is the income elasticity and Xped compared to CB?
The income elasticity for x1 is 0 ( not the same for CB)
Also the Xped is not 0( it was for CB)
Now lets look at Perfect complements, as they are not differentiable, what can we infer already?
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.
How do you say this and show on diagram?
You need 2 wheels for each frame
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.
1) plug x2 into budget constraint and simplfy to find x1 then plug into x2 so
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?
Rearrange the uncompensated demand in terms of p1
How would the Income and Price expansion path look like for PC?
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
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.
2 and 6
What are normal, inferior and giffen goods?
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)
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?
1) If first derartive with respect to x1 is positive, it means its upward sloping, if it is negative it means its downward sloping.
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?
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.
Find the uncompensated demand for the function
What does the price expansion and income expansion path look like?
They are the same
What are perfect susbitutes ?
Goods that can be substituted for each other at a constant rate, i.e. have a constant MRS,
are called perfect substitu
Find the uncompensated demand for perfect substuties? u(x1,x2)= 3x1+2x2.
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
How would the price expansion and income expansion path look like for each case?
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.
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.
Sketch the income expansion path for Demand foe good 1 and provide initution on shape
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..
Find the first and second deravitive?
foc = -1/2(U-X1^2)^1/2 X -2X1 = -X1(U-X1^2)^1/2
SECOND DERARTIVE : Using the product rule
PROBLEM SET COMPLETE THE WHOLE OF QUESTION 1?
PROBLEM SET QUESTION 2
Does a tangency exist with L shaped preferences( always mention this in perfect complements questions?
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.
Why is utility maximised when 1/2x1 = x2?
What is the utility function here?
Whenever we have x1 and x2 and greater than one x1 and x2, divide do the reciprocal then find x2
Finish the sentence, because nonsaitation and convexity are satisfied ( this is for uncompensated and compensated demand?
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?
