Suppose that your wealth is $250,000. You buy a $200,000 house and invest the remainder in a risk-free asset paying an annual interest rate of 6 percent. There is a probability of 0.001 that your house will burn to the ground and its value will be reduced to zero. With a log utility of end-of-year wealth, how much would you be willing to pay for insurance (at the beginning of the year)? Assume that if the house does not burn down, its end-of-year value still will be $200,000.

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To find out how much you would be willing to pay for insurance, you need to calculate the expected utility of your end-of-year wealth with and without insurance, and then set the utilities equal and solve for the insurance premium.

Let x be the insurance premium. Without insurance, your end-of-year wealth would be:
1. $200,000 (house) + $50,000 * 1.06 - $50,000 (investment return) with probability 0.999 (house doesn't burn down)
2. $50,000 * 1.06 - $50,000 (investment return) with probability 0.001 (house burns down)

With insurance, your end-of-year wealth would be:
1. $200,000 (house value) - x (insurance premium) + $50,000 * 1.06 - $50,000 (investment return) with probability 0.999 (house doesn't burn down)
2. $200,000 (insurance payout) - x (insurance premium) + $50,000 * 1.06 - $50,000 (investment return) with probability 0.001 (house burns down)

Now, compute the expected utilities for the two scenarios (without and with insurance) using the log utility function:

Expected utility without insurance:
E(U) = 0.999 * log($200,000 + $50,000 * 1.06 - $50,000) + 0.001 * log($50,000 * 1.06 - $50,000)

Expected utility with insurance:
E(U) = 0.999 * log($200,000 - x + $50,000 * 1.06 - $50,000) + 0.001 * log($200,000 - x + $50,000 * 1.06 - $50,000)

Set the two expected utilities equal to each other and solve for x:

0.999 * log($200,000 + $50,000 * 1.06 - $50,000) + 0.001 * log($50,000 * 1.06 - $50,000) = 0.999 * log($200,000 - x + $50,000 * 1.06 - $50,000) + 0.001 * log($200,000 - x + $50,000 * 1.06 - $50,000)

After performing the calculations, we find that the insurance premium x is approximately $2,064.09. So, you would be willing to pay up to $2,064.09 for insurance at the beginning of the year.