EC201 LT Flashcards
Marshal’s consumer surplus
The most you would be willing to pay for good x if is no good x is the value of good x
Consumer surplus is value - cost
Consumer surplus is the difference between what consumer would have been prepared to pay compared to what they actually payed
Completeness
Completeness - If a consumer is choosing between two bundles A and B one of the following possibilities holds: she prefers A to B or B to A or is indifferent
Can rank any two possibilities
U(X1A, X2A) > U(X1B, X2B) - prefers bundle A
U(X1A, X2A) < U(X1B, X2B) - prefers bundle B
U(X1A, X2A) = U(X1B, X2B) - indifferent (A and B equally attractive)
Transitivity
If A is preferred to B and B to C then A is preferred to C
U(X1A, X2A) > U(X1B, X2B) and,
U(X1B, X2B) > U(X1C, X2C) then,
U(X1A, X2A) > U(X1C, X2C)
Continuity
If A is preferred to B and C is close to B, then A is preferred to C
Nonsatiation
More is better, Increasing consumption of goods increases utility -> implies that indifference curves slope downwards, points above indifference curve give higher utility than points on or below
If du/dx1 > 0 and du/dx2 > 0 then increasing x1 and or x2 increases utility. The indifference curve slopes downwards, the preferred set is above the indifference curve and nonsatiation is satisfied
Nonsatiation requires if X1B > X1A then U(X1B, X2A) is preferred to U(X1A, X2A)
Nonsatiation implies that points above an indifference curve are preferred to points on the indifference curve. These points are the preferred set
Convexity
Any straight line joining two points lies inside the set/lies entirely on or above the graph, increasing first derivatives with positive second derivatives
Convex functions are functions with increasing first derivatives and positive second derivatives
The assumption of diminishing MRS is equivalent to the assumption that all combinations of x and y a preferred or indifferent to a particular combination of x,y for a convex set
If indifference curves are convex (if they obey the assumption of diminishing MRS), then the line joining any two points that are indifferent will contain point preferred to either of the initial combinations. Intuitively, balanced bundle of goods are preferred to extreme bundles
Mathematically a set is convex if any straight line joining two points in the set lies in the set
Can indifference curves cross?
No because if u(a) = u(b) and u(b) = u(c), transitivity implies that a and c are indifferent but nonsatiation implies that a had higher utility than c
MRS
Amount of good 1 given up, amount of extra good 2 needed to get back to the same u
Rate at which the consumer would be willing to give up one good for the other while maintaining the same level of utility
Ordinal Utility
The order of number attached to indifference curves does not change
Relative utility ranking/utility of two bible
Any transformation that preserves the ordering will give the same ordering of utility
(can multiple by a positive number, take it at a power of a positive number, take logs)
What is a utility function?
A utility function assigns a number to bundles then, to compares to bundles, it suffices to check which gives high utility
Definition of uncompensated demand
The consumer’s demand functions x1(p1,p2,m) maximise utility u(x1,x2) subject to the budget constraint p1x1 + p2x2 <= m and non negativity constraints x1>=0, x2>=0
To get uncompensated demand, fix income and prices which fixes the budget line. Get onto highest possible indifference curve
Uncompensated demand is a function of prices and income
Homogeneity of uncompensated demand
All uncompensated demand functions are homogeneous of degree 0 in prices and income
E.g. what happens to uncompensated demand when prices and income are all multiplied by 2? Demand does not change, the values of x1 and x2 do not change when p1,p2 and m are all multiplied by t>0 (draw diagram and show budget line/budget set doesn’t change)
Demand curves: if p1 increases, there is movement up the demand curve, if income m increases the demand curve shifts outwards and quantity demanded increases
Own price elasticity
%change in quantity/%change in own price
Income elasticity of demand
%change in quantity/%change in income
Normal vs Inferior goods
A good is normal if consumption increases when income increases, positive income elasticity if x1 is a normal good
A good is inferior if consumption decreases when income increases, negative income elasticity if x1 is an inferior good
Cobb Douglas Utility
With Cobb-Douglas utility, the consumer with income m>0 will never choose to be at a corner where the budget line meets one of the axes (Because Cobb-Douglas curves never touch the axes!)
Income has an effect in demand for good 1, income elasticity of demand for good 1 = 1
Quasilinear utility
Quasilinear utility: u(x1,x2) = V(x1) + x2
V(x1) is increasing in x1 and concave
Demand for good 1 depends on prices and not income, demand for good two depends on prices and income (one good will not be dependent on income so there will be no income effect)
Income elasticity on a good with no income effect is equal to 1??
When m satisfies the inequality (=/>) -> income has no effect on demand for good 1, income elasticity of demand for good 1 = 0
If m doesn’t satisfy the inequality ( demand for good 1 depends on prices and income, there is an income effect on demand for good 1
Substitute
If demand for good 1 increases when the price of good 2 increases
Complements
If demand for good 1 decreases when the price of good 2 increases
Cross Price Elasticity
measures responsiveness of demands for x1 following the change in price of good x2
Perfect Complements
utility maximisation -> implies that (x1, x2) lies at the kink of the indifference curves
With a perfect complements utility function the ratio of consumptions of the goods does not vary with price or income
Price of good 1 is dependent on price of good 2
Demand curves: increase in p1 results in movement along demand curve, increase in p2 results in shift down the demand curve, increase in m results in shift up demand curve
Perfect Substitutes
MRS is a/b which does not depend on x1 and x2, there is a jump in the demand curve, may be a point where no price equals demand and supply
Price Consumption Curve
Indicates the various amounts of a commodity bought by a consumer when its price changes
A line drawn through all the equilibrium points between indifference curves and expenditure function/budget sets
Income Consumption Curve
The locus of points showing the consumption bundles chosen at each of the various levels of income
How the consumer’s optimal bundle of purchases vary with corresponding income
Compensated Demand
Minimises the cost of obtaining utility u at prices p1 and p2 and is function of obtaining utility u at prices p1 and p2 and is a function of utility u, p1, p2, notation h1(p1,p2,u), h2(p1,p2,u)
Compensated demand h1(p1,p2,u), h2(p1,p2,u) is the cheapest way of getting utility u at prices p1 and p2
To get uncompensated demand fix income and prices which fixes the budget line. Get onto the highest possible indifference curve
To get compensated demand fix utility and prices which fixes the indifference curve and gradient of the budget line. Get onto the lowest possible budget line
A demand curve composed solely of substitution effects
The Expenditure Function
E(p1,p2,u) is the minimum amount of money you have to spend to get utility u with prices p1 and p2. It is function of p1,p2 and u
The cost of the cheapest way of getting utility u at prices p1 and p2
The amount of goods which minimises the cost of getting utility u is compensated demand h1(p1,p2,u), h2(p1,p2,u) so E(p1,p2,u) = p1h1(p1,p2,u) + p2h2(p1,p2,u)
Homogeneity of Compensated Demand and Expenditure Function
Compensated demand is homogeneous of degree 0 in prices
The expenditure function is homogeneous of degree 1 in prices
If all prices are multiplied by t, the cheapest way of getting utility u doesn’t change (degree 0?)
The cost of the cheapest way of utility does change (degree 1?)
Properties of the Expenditure Function
1) Increasing in utility
2) The expenditure function increases or does not change when a price increases
3) Homogeneous of degree 1 in prices
4) Shepherd’ s lemma - derivative of expenditure = compensated demand for good 1
5) Concave in prices
The Slutsky Equation
It is way of knowing how big income and substitution effects are. Combined with elasticity estimates it tells us that income effects are too small to bother with except for goods that are a large proportion of the budget
For a normal good, income and sub effects work in the same direction
Income effects are small if the income elasticity or the budget share are small
The Income Effect
The change in uncompensated demand due to an income change holding prices constant
The Substitution Effect
is the change in compensated demand due to a price change holding utility constant
Quasilinear Utility (C.D.)
Quasilinear utility -> uncompensated demand for good 1 does not depend on income except at low levels of income and is equal to compensated demand
Quasilinear utility: at a tangency compensated demand for good 1 depends only on prices and uncompensated demand for good one depends only on prices
There is no income effect on demand for good 1 with quasilinear utility
Perfect Complements (C.D.)
Perfect complements utility -> Compensated demand does not depend on prices. No substitution effect
With perfect complements, compensated demand does not depend on prices (perfectly inelastic) -> No substitution effect
Compensated demand is completely inelastic and demand curve is vertical
Price Changes for Normal Goods
If a good is Normal, substitution and income effects work in the same direction
When price falls, both effects lead to a rise in quantity demanded
When price rises, both effects lead to a drop in quantity demanded
Utility maximisation implies that a rise in price leads to a decline in quantity demanded
Sub effect causes less to be purchased as the individual moves along an indifference curve
Income effect causes less to be purchased because the resulting drop in purchasing power moves the individual to a lower indifference curve
Price Changes for Inferior Goods
If a good is inferior, substitution and income effects move in opposite directions
Combined effect is indeterminate
When price rises, sub effect leads to a drop in q demanded, I.E is opposite
When price falls, sub effect leads to a rise in q demanded, I.E is opposite
Utility maximisation implies that no definite prediction can be made for changes in price
The sub an income effect move in opposite directions
Income effect outweighs sub effect then we have Giffen’s paradox
The Base Weighted Price Index
The base weighted price index: measures the proportional increase in the cost of (x1A, x2A) which has utility uA
Initially prices at p1A, p2A, consumption (x1A,x2A) is optimal, income is p1Ax1A + P2Ax2A
Prices change to p1B, p2B if income changes at the same rate as the base weighted price index it changes to p1Bx1A+p2Bx2A
is a weighted average of proportionate price increases where weight for good I is the proportion of expenditure spent on good I at date A
Base weighted price index: proportional increase in income needed to continue to buy (x1a, x2a) after the price change
Expenditure Function Price Index
Expenditure function price index: initially prices p1A, p2A, utility uA, income E(p1A,p2A,uA). Price changes to p1B,p2B, income changes at the same rate as the expenditure function price increase so changes to E(p1B,p2B,uA)
Expenditure function price index: proportional increase in income needed to continue to have utility uA after the price change
The expenditure function prices index is lower than or equal
Fall in Consumer Surplus
Fall in consumer surplus: measures the reduction of the gain from trade (total value - total cost) in money
The fall in consumer surplus is the amount of extra money you would have to be given to get back to the same utility as before the price increase -> This is the compensating variation?
If demand is elastic/substitute available, fall in consumer surplus is smaller
Change in consumer surplus is the area bounded by the uncompensated demand curve
Compensating Variation
Compensating variation for a price increase p1A to p2A is the amount of extra money the consumer needs to get back to the same level of utility as before the price change
Compensating variation is the area between the compensated demand curve and the vertical axis (why not uncompensated?) - the amount of extra money you need after the price increase to make you as well of as you were before the price increase
Compensating variation is the area bounded by the compensated demand curve with utility uA
Equivalent Variation
Equivalent variation: EV is the amount of money that taken away from the consumer without changing prices has the same effect on utility as the price change
E(p1B,p2,uB) - E(p1A,p2,uB)
EV if it is what is the monetary equivalent of the price change
Assume the price of good 2 is 1. The equivalent variation is a line on the indifference curve diagram and an area of the demand curve diagram
EV vs CS vs CV
For a price rise of a normal good: EV < change in consumer surplus < CV because EV is measured at a lower level of utility
When there are no income effects, e.g. with quasilinear utility at a tangency, uncompensated and compensated demand are the same so the loss in consumer surplus due to an increase in p1 is the same as the CV & EV
There is no difference between compensating variation, equivalent variation and change in consumer surplus with quasilinear utility as there is no income effect
• EV = tax revenue if no sub effect
• EV = CV if no income effect
• CV = CS in quasilinear utility (no income effect)
The difference between CV and EV and the change in consumer surplus is due to income effect
When there is no income effect (quasilinear), dealing only with sub effects and so compensated demand meaning that CV = change in CS, because income elasticity = 0, demand for good 1 is inelastic meaning same quantity of good 1 is consumed each time which means utility for good 1 remains constant there is no difference between CV and EV
Income effects are small when either or both of the income elasticity of uncompensated demand and the budgets share are small
If income effects are small the change in consumer surplus is a good approximation to the compensating variation
The difference between CV, EV and CS is small when the income effect is small and/or when the share of the budget spent on a good is small
There is no measurement consumer surplus in compensated demand because the demand function corresponds to changes in utility and utility expenditures and not income
Excess Burden
The excess burden of an excise tax is equivalent variation - tax revenue = loss to consumer - tax revenue (With an excise tax there is an excess burden)
Lump Sum Tax
A tax is lump sump if the amount paid does not depend on anything the consumer can change e.g. does not depends on income, wealth, spending and saving etc..
Suppose the government wants to raise revenue R from a household: a lump sum tax that reduces income by R that does not depend on anything consumer does reduces utility by less than a non lump tax which raises the same revenue
A poll tax is a lump sum tax which is the same for everyone
Does compensating a consumer for a price increase imply that the price increase has no effect on demand?
A price increase reduces demand even if the consumer is compensated
If income effects are small compensation has little effect on the demand for the good
Value Judgement
Adding up consumer surplus geometrically implies a value judgement that giving £1 to one consumer has the same social benefit as giving £1 to any other consumer
You want to evaluate the losses to each group, and then consider how to use them as input into a decision
Budget Constraint and Utility Assumptions
Budget constraint and utility assumptions:
Utility u(c,n) depends on consumption c and time outside paid employment (leisure) n, households can choose how many days of hours to work (e.g. gig economy)
Nonsatiation: utility is increasing in both consumption and leisure (also completeness, transitivity, continuity and convexity)
Spending on conusumption <= total earnings
Nonsatiation implies that that budget constraint is satisfied as an equality c + wn =wT
MRS = w = real wage
Substitution and Income effect (Labour supply)
Sub effects dominates -> labour supply increases when wage rises (workers prefer to work as opportunity cost of leisure increases)
Income effect dominates -> labour supply decreases when wage rises (more wealth gained per unit of time work means workers can increase leisure time)
The labour supply curve is backward-bending when the substitution effect dominates the income effect for wages below a certain w, and the income effect dominates the substitution effect above w
Cut in marginal tax rate for incomes:
Substitution effect increases labour supply
Income effect decreases labour supply
Income v Consumption tax
Income tax is a proportion of total earnings
Consumption tax is a proportion of total spending
In general a Proportional income tax at rate tm and a promotional consumption tax at rate tc raise the same revenue and have the same effect on the budget constraint if (1-tm)(1+tc) = 1
Equivalent Variation for a Price Change
Definition for Equivalent Variation for a price change: taking away EV without changing p1 from p1A has the same effect on utility as increasing p1 from p1A to p1B without changing income
Equivalent Variation for a Tax
Definition for Equivalent Variation for a tax: Taking away the equivalent variation without changing the price of leisure has the same effect on utility as imposing the tax
Lump Sum Tax
Budget constraint with lump sum tax raising same revenue as income tax. This gives higher utility than the income tax
A lump sum tax that reduces income by a fixed amount that does not depend on anything the consumer does reduces utility by less than a tax raising the same amount of revenue where the revenue can be changed by changing consumption, work or saving
The only feasible lump sum tax is a poll tax where everyone pays the same amount
Marginal Income Tax Rate
Marginal income tax rate is the number of extra pennies tax you pay on £1 extra earnings
Average Income Tax Rate
Average income tax rate = total income tax/total income
The Benefit Withdrawal Rate
The benefit withdrawal rate: this is the amount by which the benefit is withdrawn if someone earns £1 more
Upward Sloping Indifference Curves
Pre-tax and post-tax income: Indifference curves are upward sloping (less pre-tax income is equivalent to more leisure, more post-tax income is equivalent to more consumption)
Hours worked and post-tax income: Indifference curves are upward sloping (less hours worked is equivalent to more leisure, more post-tax income is equivalent to more consumption)
Indifference curves upward sloping for the withdrawal rate (More income before benefits implies more work, less leisure, More income after benefits implies more consumption)
Reducing the withdrawal rate from 100% to 50% improves work incentive for people earning less than £200 but worsens work incentives for people earning between £200 and £400
More generally lower withdrawal rates improve work incentives for low earners and worsen work incentives for moderate earners
Lower withdrawal rates result in more benefits being paid so are more expensive
Universal Credit
Intends to integrate part of the tax and benefit system to make it easier to understand and improve work incentives
Present Discounted Value
The present discounted value at date 0 of income stream y0,y1,y2…yt,yT paying yt at date t, discounted at an interest rate r…..
The pdv of y1,y2…… is the maximum amount of debt at date 0 which you could repay using the entire income stream
If you get the income stream y0,y1…… at interest rate r, start with no saving and no debt at date 0, you can follow any consumption path and leave no debt or savings at date T whose pdv is equal to the pdv of the income stream
The consumer consumes his/her endowment bundle if savings are exactly zero (s=0) and no borrowing takes place
Perfect Capital Markets
1) No uncertainty
2) You can borrow and lend at the same interest rate r
3) The only constraint on borrowing is that you must have enough income too repay your debt
4) There is a perfect and costless mechanism which ensures that no one takes on loans which they cannot repay and that all debts are paid
No Arbitrage Argument
The market can only clear if current price = pdv -> no arbitrage argument
Change in Asset Prices
If an increase in current asset prices leads people to expect that future asset prices will be even higher, an increase in asset prices can increase demand and further increase prices
If a fall in current asset prices leads to people to expect that futureP asset prices will be even lower, a decrease in asset prices can decrease demand and further decrease prices
Interest Rate Increase and Income and Substitution Effects
Interest increase: never makes a saver worse off, usually makes a borrower worse off
Borrowing is more expensive so the sub effect decreases borrowing
Borrowing is more expensive so the household is poorer. If current consumption is a normal good this reduces current consumption so decreases borrowing
Income and substitution effects on borrowing work in the same direction. Both decrease current consumption and borrowing
Saving is more rewarding so the substitution effect increases saving
The higher interest rate makes the household richer. If current consumption is a normal good this increases current consumption so decreases saving
Income and sub effects work in opposite directions: If the SE dominates current consumption falls and saving increases, if the IE dominates current consumption increases & saving decreases
In reality the interest rate at which you borrow is higher than the rate at which you can save
If there are different interest rates for borrowing and lending or credit limits the choice between income streams depends on preferences
Compensated Demand curves cannot slope upwards
The substitution effect of an increase in the price of a good decreases or leaves unchanged the demand for the good
Compensated demand is by definition the cheapest way of getting utility u at prices (p1,p2)
At prices (p1,p2) any other way of getting utility u must cost the same or more than (h1.h2) so must lie on or above the budget line through (h1,h2) withnslope -p1/p2
Any other (x1,x2) with Utility u cannot lie in the shaded area below the budget line through (h1,h2) with slope -p1/p2
H1a, h2a is the cheapest way of getting utility u at prices p1A, p2
H1b, h2b is another way of getting utility u, therefore h1b, h2b cannot cost less than h1a, h2a at prices p1A, p2
H1b, h2b, cannot cost less than h1a, h2a at prices p1A, p2 so (H1b, h2b) cannot lie in shaded area below budget line
Saving and Borrowing Decisions
Assume preferences satisfy the standard assumptions of completeness, transitivity, continuity, nonsatiation and convexity so can be represented by a utility function u(c0,c1)
The consumer maximises u(c0,c1) subject to the budget & non-negativity constraints where c0+c1/(1+r) <= y0 + y1/(1+r), c0 >= 0, c1 >= 0
If the household can continue to do after the rate change what it did before the rate change, the rate change can’t make it worse off and usually makes it better off
Nonsatiation Implies that any consumer would the income stream with higher present value
In reality the interest rate at which you borrow is higher than the rate at which you can save
If there are different interest rates for borrowing and lending or credit limits the choice between income streams depends on preferences
It is optimal to choose the income stream with the highest pdv regardless of preferences
