California is in a drought and the reservoirs are running low. The probability o

Question

California is in a drought and the reservoirs are running low. The probability of rainfall in 1991 is 1/2,
but with probability 1 there will be heavy rainfall in 1992 and any saved water will be useless. The state
uses rationing rather than the price system, and it must decide how much water to consume in 1990 and
how much to save till 1991. Each Californian has a utility function of U = log(w90)+log(w91). Show that
if the discount rate is zero the state should allocate twice as much water to 1990 as to 1991.

Explanation / Answer

Solution

probability of rainfall in 1991 is 0.5

probability heavy rainfall in 1992 and any saved water will be useless = 1

Suppose the total amount of existing water is w units. Expected utility from this water is

U = 0.5log(w90)+0.5[log(w90) + log(w- w90)], since if it rains in 1991 the saved water won't be needed.

Differentiating with respect to w90 and equating to zero gives w90 = ( 2/3 )w.

Thus, two thirds of the water should be consumed the first year.

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