Chapter 1: Uncertainty and the Experimental Endowment Effect
II. A simple Model of Aversion to Risk Changes
Now consider another method of evaluating the risk of an action. The risk of the end-states γ (γ’) and δ (δ’) is compared with the original state ρ (ρ’), viewed from ρ (ρ’), so taking this point as a reference point. In the purchase situation (2 A), the resulting com-parisons 2 and 3 yield the same result as the theories cited above, as the starting point is risk-free: The state with the lottery δ contains more risk than the starting point ρ without a lottery. The individual has to be compensated with a risk premium to choose the ac-tion “buy lottery ticket”.
The change this approach brings is visible by applying it to figure B, the sell situa-tion: Comparisons 2’ and 3’ now yield a different result: Comparison 3’ shows that the end-state with lottery ticket, δ’, contains exactly the same risk as the starting point ρ’, where the lottery risk is also included. There is no difference between the two states in any possible state of the world. There is no change in the risk. Comparison 2’ reveals the opposite for state γ’: Seen from the reference point, ρ’, the risk has changed, be-cause the lottery ticket has been sold. The conventional comparison would state that the risk has been reduced. However, with a relative notion of risk, one must state that the risk has nevertheless been changed. The difference γ’ – ρ’ is in itself a lottery: (WTA-H, WTA-L). In comparison with the original state, either the high payoff H is foregone or the low payoff L. By choosing “sell ticket“, the relative result is now “additional risk“.
Now comparing these relative outcomes of δ’ and γ’ shows that the individual has to receive a risk premium to make her sell the lottery, to choose the end-state γ’ without the lottery. This is the opposite result of the one achieved in the buy situation, so WTP and WTA differ.
We need to extend the concept of a reference point as used in Prospect Theory from the domain of wealth to the domain of risk. To model this in a corresponding assump-tion, we will for a moment treat the risk of all current holdings as incorporated and ac-cepted. Formally, the current state is risk-free. By giving up a “positive” risk, this is treated as if acquiring a corresponding “negative” risk.
Assumption: Giving up the lottery (H, L) corresponds to accepting the opposite lot-tery (-H, -L).
This “trick” allows us to proceed with the standard framework and the conventional notion of risk aversion to display “aversion to risk changes”. Later, the conventional ab-solute risk aversion will be reintroduced.
Compared to the other theories that link uncertainty and the endowment effect (Ran-kin, 1990, Sugden, 2003, and Blondel and Lévy-Garboua, 2005), our approach stays closest to Prospect Theory by only making the single assumption stated above.
Formulation of the Model
We model one individual with a given preference set in two different situations: As a buyer and as a seller of a lottery that pays out money. The information the individual has about the good is always the same, so there is no problem of asymmetric informa-tion15. A rational (complete, transitive) preference ordering over lotteries is assumed to exist. “~” displays indifference between two options.
The discrete lottery (H, 0.5; L, 0.5) yields a high payoff H and a low payoff L with even probability of 50 % each. H ≥ L. (Round brackets will be used for lotteries, square brackets for mathematical operations. As the probabilities are always 50 %, they are omitted in all lottery notations: (H, L). To simplify the notation, the current wealth level that has to be added in all states of the world is defined as zero, so it can be left out.
Willingness to pay (WTP) and Willingness to Accept (WTA) are the money values that satisfy the following conditions:
Purchase: (H-WTP, L-WTP) ~ 0; (3.)
Sale: (-H+WTA, -L+WTA) ~ 0 (4.)
The first equation, related to the buy situation, is straightforward: The price one has to pay for the lottery is subtracted from the payoffs in both the high and the low state.
WTP is defined as the price at which the individual is indifferent between buying (left hand side) and not buying (right hand side). The second equation is related to the “aversion to risk changes”. Selling the lottery (H, L) is treated as acquiring the lottery (H, -L). The WTA is the price attached to this lottery that makes the individual indifferent towards the transaction.
It is straightforward to see that in both cases the variance of the lottery is not altered by the price attached to it16. The difference between the two states is H-L ≡ k in the buy as well as in the sell situation. The corresponding lotteries must be identical for two
15 In reality, a seller will most probably have more information than a buyer. In the experiments, this is not the case.
16 The variance is [(H-L)/2]2, the standard deviation is (H-L)/2.
sons: First, they have the same variance (and probabilities) and second, the individual is indifferent between both lotteries and doing nothing (staying at current wealth, which is defined as zero). There cannot exist two different lotteries (a, b) and (a’, b’) such that both fulfill (a, b) ~ 0 ~ (a’, b’) when their payoff difference is the same a-b=k=a’-b’.
They must be the same lotteries, a=a’ and b=b’.
Figure 3: The resulting gap between WTA and WTP.
Consider Figure 3: The lotteries (+H, +L) and (-H, -L) are both shifted towards zero (the reference point of actual wealth) and collapse into the lottery (a, b) that is defined as (a, b) ~ 0.
Being indifferent between these two transformed lotteries and zero means (from 3.
and 4.):
(H-WTP, L-WTP) ~ (-H+WTA, -L+WTA) ~ 0 (5.)
wealth wealth
+L +H
-L
-H -E
+E acquire lottery
give away lottery
+L +H
-L
-H -E
+E
a
b 0 c
+WTA
-WTP x≡0
1. negative and positive lottery
2. assigning prices
Æ lotteries collapse into single lottery
This can only be the case if the high and the low outcomes of the buy- and sell-lottery are identical. The high outcome of the “buy sell-lottery” is H-WTP, while the high outcome of the “sell lottery” is –L+WTA:
H-WTP = -L+WTA and L-WTP = -H+WTA (6.)
yielding in both equations:
WTA+WTP = H+L (7.)
Definition: [H+L]/2 ≡ E
E=E(H, L) is the expected value of the lottery. The last equation can be rewritten as:
[WTA+WTP]/2 = E (8.)
With “aversion to risk changes” only, the midpoint of willingness to accept and will-ingness to pay is the expected payoff of the lottery.
a) Benchmark I: Neutrality towards Risk Changes
As a benchmark, let us briefly consider the case of risk neutrality: The individual is indifferent between the lottery and the payment of its expected payoff: (a, b) ~ E(a, b) = [a+b]/2. If (a, b) ~ 0, this means that a= -b. Applying this to the lotteries in (3. and 4.) we get:
H-WTP =-[L-WTP] and –H+WTA=-[-L+WTA] 17 (9.)
WTP=E=WTA (10.) In case of risk neutrality, WTA and WTP fall together and correspond to the
ex-pected payoff. There is no gap between WTA and WTP.
b) Benchmark II: No Uncertainty
If there is no uncertainty, we have H=L=E. Insert into (5.):
(E-WTP, E-WTP) ~ (-E+WTA, -E+WTA) ~ 0 (11.)
(E-WTP) ~ (-E+WTA) ~ 0 (12.)
This can only be solved for
WTP=E=WTA (13.) When there is no uncertainty, there is no endowment effect.
17 One can also formulate that the expected value has to be zero: 0.5*(L-WTP)+0.5*(H-WTP)=0 and analogously for WTA, yielding the same result.
c) Uncertainty and Aversion to Risk Changes
Now let us consider the case when there is both risk aversion and uncertainty. Risk aversion means that (a, b) ≺ E(a, b)=[a+b]/2 – an individual prefers the payment of the expected payoff to the lottery. This means that for accepting a lottery, there has to be a reward: the expected value of the lottery has to be positive. For a mixed lottery (a, b) with a>0, b<0 such that (a, b) ~ 0, this means that 0.5a+0.5b=c>0. It will depend on the individual risk aversion how the payoffs of the lottery have to be increased to make the lottery just acceptable. For a given payoff variation H-L of a lottery, there is exactly one such value c.
Definition: c is the “normalized” risk premium of the lottery (H, L) such that:
c=E(a, b)=0.5a+0.5b with (a, b) ~ 0, a>0, b<0 and a-b=H-L=k.
We have seen that the payoff variation in both lotteries (purchase and sale) is of the same size H-L=k, so the expected value of both lotteries has to be the same to make them just acceptable.
0.5[H-WTP] + 0.5[L-WTP] = c = 0.5[-H+WTA] + 0.5[-L+WTA] (14.)
WTP = [H+L-2c]/2 and WTA = [H+L+2c]/2 (15.)
WTP = E - c and WTA = E + c (16.)
WTA = WTP + 2c (17.) In case of uncertainty and aversion to risk changes, a gap arises between the
willing-ness to accept and the willingwilling-ness to pay, symmetrically around the expected value of the lottery. WTP is equal to the expected value minus the normalized risk premium of the lottery. WTA is equal to the expected value plus the normalized risk premium of the lottery. Therefore, the size of the WTP/WTA-gap depends positively on the (relative) risk aversion and on the variance of the lottery.
d) Reintroduction of absolute Risk Aversion
Allowing for absolute risk-aversion together with aversion to risk changes simply decreases the value of the lottery in both the buy and the sell situation: The individual has to receive a risk premium r to hold the lottery. Applying this to (16.) yields:
Results:
[WTP + WTA] / 2 = E - r (18.) WTP = E - c - r and WTA = E + c - r (19.)
WTA = WTP + 2c (corresponds to 17.)
Equation (17.) remains unchanged. “Aversion to risk changes” alone cause the en-dowment effect.
The results (14.) to (16.) are applied to the experimental lottery data of van Dijk and van Knippenberg (1996) in Appendix I. The necessary risk aversion to predict the data is the same as empirically found in Tversky and Kahneman (1992, p. 59).
III. Experiment