State Prices and Scarcity

Three states, with probabilities and state prices side by side.

A state price is not a raw probability. It is a probability weighted by marginal utility. A dollar in the scarcest state costs the most. The formula below shows exactly that weighting.

For example, a village usually has plenty of 🍞 bread. In a famine year, bread is very scarce. Money to spend in that year is worth far more than money in a normal year.

qs=Ο€s/Esβˆ‘tΟ€t/Etq_s = \frac{\pi_s / E_s}{\sum_t \pi_t / E_t}

Examples & Exercises

With Ο€=(1/3,1/3,1/3)\pi = (1/3, 1/3, 1/3) and E=(6,10,14)E = (6, 10, 14), compute the state prices qsq_s directly.

qs=Ο€s/Esβˆ‘tΟ€t/Etq_s = \frac{\pi_s / E_s}{\sum_t \pi_t / E_t}
E=(6,10,14)E = (6, 10, 14) β€” qβ‰ˆ(0.493,0.296,0.211)q \approx (0.493, 0.296, 0.211).
Show proof

  1. Ο€s/Es=(1/18,Β 1/30,Β 1/42)\pi_s/E_s = (1/18,\ 1/30,\ 1/42). Over the common denominator 630: (35,21,15)/630(35, 21, 15)/630, summing to 71/63071/630.

  2. Normalizing, q=(35/71,Β 21/71,Β 15/71)β‰ˆ(0.493,0.296,0.211)q = (35/71,\ 21/71,\ 15/71) \approx (0.493, 0.296, 0.211).

In the example above, compute qs/Ο€sq_s/\pi_s and explain why the scarcest state's price exceeds its probability.

q1Ο€1β‰ˆ1.479,q2Ο€2β‰ˆ0.887,q3Ο€3β‰ˆ0.634\frac{q_1}{\pi_1} \approx 1.479, \quad \frac{q_2}{\pi_2} \approx 0.887, \quad \frac{q_3}{\pi_3} \approx 0.634
Scarcity premium: q1/Ο€1q_1/\pi_1 is largest.
Show proof

  1. q1/Ο€1=(35/71)/(1/3)=105/71β‰ˆ1.479q_1/\pi_1 = (35/71)/(1/3) = 105/71 \approx 1.479, q2/Ο€2=63/71β‰ˆ0.887q_2/\pi_2 = 63/71 \approx 0.887, q3/Ο€3=45/71β‰ˆ0.634q_3/\pi_3 = 45/71 \approx 0.634.

  2. State 1 is scarcest (E1=6E_1 = 6), hence highest marginal utility β€” so "one dollar in state 1" trades above its probability 1/31/3. State prices are marginal-utility-weighted probabilities, not raw probabilities.

15. General Equilibrium under Uncertainty