Zoe Goggles is a deep sea underwater photographer. Her camera and lenses are valued at $4,000. There is a chance of 1/20 that she will lose her equipment on a dive over the course of the year. Her wealth, including the value of her equipment, is $30,000. Zoe’s utility function is U(W)= ln W. Zoe wishes to purchase insurance against the risk of losing her equipment on a dive. The price per dollar of coverage is γ.
a)Write an equation to represent Zoe’s net wealth in the state of the world where she does not lose her equipment and she has purchased K dollars of coverage. Call this w1.
b)Write an equation to represent Zoe’s net wealth in the state of the world where she does lose her equipment and she has purchased K dollars of coverage. Call this w2.
c)Combine your equations from parts (a) and (b) to write her budget constraint. What is the price of a claim on one dollar of wealth in state of the world 1? What is the price of a claim on one dollar of wealth in state of the world 2? What is the slope of her budget constraint, if w2 is on the vertical axis and w1 on the horizontal axis?
d)Provide an expression for Zoe’s marginal rate of substitution, MRS = - dw1/ dw2.
If γ = 0.10, how much insurance coverage K will Zoe buy? What is her total premium γK in that case?
If, instead, the price of insurance is actuarially fair, show that Zoe will purchase full insurance: K = $4,000.
Draw a diagram depicting Zoe’s budget constraint and her optimal choice of w1 and w2.
To answer these questions, let's break them down one by one:
a) Zoe's net wealth in the state of the world where she does not lose her equipment and she has purchased K dollars of coverage, denoted as w1, can be calculated as follows:
w1 = (1 - 1/20) * (W - K) + K
Since there is a 1/20 chance of losing her equipment, the remaining wealth after deducting the coverage amount from her total wealth (W) is multiplied by (1 - 1/20). Then, the coverage amount (K) is added to it.
b) Zoe's net wealth in the state of the world where she does lose her equipment and she has purchased K dollars of coverage, denoted as w2, can be calculated as follows:
w2 = (1/20) * (W - K) + K
Since there is a 1/20 chance of losing her equipment, the remaining wealth after deducting the coverage amount from her total wealth (W) is multiplied by 1/20. Then, the coverage amount (K) is added to it.
c) Zoe's budget constraint can be represented by combining the equations from parts (a) and (b):
w1 = (1 - 1/20) * (W - K) + K
w2 = (1/20) * (W - K) + K
The price of a claim on one dollar of wealth in state of the world 1 is γ / (1 - 1/20).
The price of a claim on one dollar of wealth in state of the world 2 is γ / (1/20).
The slope of her budget constraint is -1 / (γ / (1 - 1/20)), which simplifies to -(1 - 1/20) / γ.
d) The marginal rate of substitution (MRS) measures the trade-off between state 1 and state 2. It can be calculated as the negative derivative of w1 with respect to w2:
MRS = -dw1/dw2 = -[(1 - 1/20) / γ] * [∂(W - K) / ∂(W - K)] = -(1 - 1/20) / γ
If γ = 0.10, we can substitute this value into the equations to find K:
w1 = (1 - 1/20) * (W - K) + K
w2 = (1/20) * (W - K) + K
By solving these two equations simultaneously, we can find the optimal value of K and subsequently calculate the total premium γK.
If the price of insurance is actuarially fair, it means that the premium equals the expected value of the claim, i.e., γK = (1/20) * (W - K). In this scenario, Zoe will purchase full insurance, where the coverage amount (K) is equal to the value of her camera and lenses ($4,000).
To draw a diagram depicting Zoe's budget constraint and her optimal choice of w1 and w2, you can plot w1 on the horizontal axis and w2 on the vertical axis. The slope of the budget constraint is determined by the equation from part (c), and the point of tangency between the budget constraint and the utility curve (U=W) represents Zoe's optimal choice of w1 and w2.