Term 2 Lecture 5 Flashcards
(18 cards)
Summary: the view of consumers as choosing in their own best interests suggests
Natural ways to measure the effects of prices on consumer welfare
By Roy’s identity, the effect of a small change in price pi on utility is proportional to
The quantity consumed
- Dv/Dpi = -(Dv/Dy).fi(y,p)
- change in utility due to price changes = -(marginal utility of income) x (demand functions for good i at budget y and p)
Interpreting marginal welfare cost
- assume the marginal utility of income, Dv/Dy to be roughly constant
- means that the quantity demanded can be used to estimate the marginal welfare cost of price changes
- when price increases from piA to piB, can interpret the area under the demand curve as a measure of the welfare loss due to higher prices
- welfare loss can be represented by A+B, where A is the increase in price for the units still consumed after the price change, B is the excess of the consumer’s willingness to pay for the quantity that they no longer consume.
Consumer optimisation sets marginal utilities proportional to
- prices, Du/Dqi = kpi
- the price which the consumer is willing to pay is therefore an indicator of the marginal welfare cost of reducing quantity
- if the factor of proportionality is constant, then the effect of decreasing the quantity to a point where none of the good is demanded is roughly a trapezoidal area underneath a demand curve.
- if we subtract from thbis area the cost of buying the good piqi, we get consumer surplus
The assumption that Dv/Dy or lambda remain constant is reasonable if
If preferences are quasilinear with respect to another good
- in general, consumer surplus is only an approximation of welfare, as it assumes MU of income remains constant, but if preferences are quasilinear, this assumption holds EXACTLY, making consumer surplus a precise measure of welfare
Instead if using the uncompensated demand curve, we can use
A compensated demand curve, where utility is held constant as price changes
- method helps measure how much it costs the consumer to maintain the same level of satisfaction when prices change
- welfare loss calculated as: c(v,pB) - c(v,pA) = integrating the compensated demand function with limits of prices = integrating the derivative of the cost function with respect to dpi with same limits
Compensating and Equivalent Variation
- compensating variation is how much money a consumer would need after a price change to return to their orignal utility level
- equivalent variation is how much money a consumer would have been willing to give up before a price change to avoid it
- both are accurate welfare measures without requiring the assumption of constant MU
The role of quasilinear preferences:
- compensated demand = uncompensated demand, because income effects are removed
- consumer surplus, CV and EV all align perfectly
- makes consumer surplus a precise welfare measure, not just an approximation
- CV and EV coincide iff preferences are QL with respect to the good which did not change in price so there is no income effect on the demand for the good which did change in price
Cost of living index
Helps us measure how the cost of maintaining a certain standard of living changes over time
- expenditure function tells us the minimum cost needed to achieve a given utility under different price conditions
Konus cost of living index
Compares the minimum cost of achieving the same level of utility under two different price levels, p0 and p1
T(v,p1,p0) = (c(v,p1)/c(v,p0)), where c is the expenditure function
2 special cases where the Konus cost of living index loses dependence on V
- if price vectors in the two periods are proportional, so p1 = kp0, then the index is equal to k for all v
- if preferences are homothetic, so consumption pattern remains the same at all income levels, spending shares across goods don’t change as income increases, since budget shares remain the same at all V, cost of living index does not depend on v, they cancel out in the summation formula
Laspeyres index
- measures the cost of purchasing base bundle q0 using both old and new prices
L(q0,p1,p0) = p1q0/p0q0 = SUMwi0(pi1/pi0) - if its less than 1, that means the cost of purchasing the same bundle q0 at new prices p1 is cheaper than before, so if consumer expenditure is unchanged, they cant be worse off as they can still afford the previous
Why is L>T
We know that p0q0 = c(v0,p0), where v0 is the base period utility, c is the minimum cost to reach that utility under the base period p0
- however at the new prices p1, cost of q0 is not necessarily the cheapest way to achieve v0, consumer could sub away from expensive goods and maintain v0 for a lower cost than p1q0
L(q0,p1,p0) = p1q0/p0q0 >/ (c(v0,p1)/c(v0,p0)) = T(v0,p1,p0), this holds as consumer is always able to rearrange their spending to minimise costs, meaning c(v0,p1)< p1q0
What is substitution bias with L
Occurs because the L index assumes that consumers continue to buy the same fixed bundle of goods, even if some goods become more expensive, whereas in reality, consumers would adjust.
- means the actual increase in the cost of living is lower than what the L index suggests
Paasche index
- measures the cost of buying the new bundle at old prices
P(q1,p1,p0) = p1q1/p0q1 = 1/SUM(1i(pi0/pi1)) - if P>1, means that the cost of purchasing the current bundle at the new prices is greater than the cost of purchasing it at the old prices, if the consumers total expenditure remains the same, then they must be worse off, as their money doesn’t go as far as before
Substitution bias in the Paasche index
A LB on the true cost of living index
- underestimates true cost of living, as it assumes consumers have already adjusted their purchases optimally, but may not have in realty, so true cost of living may be even higher.
Compare P,T,L
- what if preferences are Homothetic?
P < T < L
- if homothetic, budget shares remain constant across different income levels, so P = T = L
How is the new compensated demand function constructed given the price change?
So where the new price meets the uncompensated demand function, the compensated demand function must also pass through that point.
- to get EV act as though the price change occurred merely on the new compensated demand function, and that loss in utility is EV
