The Sons of Knute had a hunting lodge up on Loon Lake that burned down last winter.They plan to rebuild it this summer and are trying to decide how large the new lodge should be.The organization has 50 members.The marginal rate of substitution of each of them between square feet of hunting lodge and money for other goods is 1.2 - 0.0004y, where y is the size of the hunting lodge in square feet.What is the efficient size for the new hunting lodge?
A)1,000 square feet
B)1,200 square feet
C)2,000 square feet
D)2,400 square feet
E)None of the above.

Question

Quizplus · Accepted Answer

To find the efficient size for the hunting lodge, we need to set the marginal rate of substitution of each member equal to each other. So:

1.2F1S1 - 0.0004y = 1.2F2S2 - 0.0004y
1.2(F1S1 - F2S2) = 0.0004y
F1S1 - F2S2 = (0.0004/1.2)y
Solving for S1, we get:

S1 = (F2S2 - (0.0004/1.2)y) / F1
We know that there are 50 members, so the total size of the lodge is:

50S1 = 50(F2S2 - (0.0004/1.2)y) / F1
50S1 = 50(F2/F1)S2 - (0.02/1.2)y
Solving for S2, we get:

S2 = (F1/50F2)(50S1 + (0.02/1.2)y)

Now, we want to find the size of the lodge that maximizes total welfare, which is:

W = 50(F1S1 + F2S2) - C
where C is the cost of building the lodge. We can assume that C is proportional to the size of the lodge, so:

C = k(y)
where k is a constant. To find the efficient size of the lodge, we want to maximize W with respect to y. So:

dW/dy = 50(0.0004/1.2)(F1/F2)S1 - k'(y)
where k'(y) is the derivative of k with respect to y. Setting this equal to zero and solving for y, we get:

y = (15/2)(F2/F1)(S1/k')
Since we don't know k', we can say that y is proportional to (F2/F1)S1. So the larger F2 is relative to F1, the larger the efficient size of the lodge will be. Since F2 represents the willingness of each member to pay for additional square footage of the lodge, we can assume that F2 > F1 for all members (i.e. everyone would prefer a larger lodge to more money for other goods). Therefore, the efficient size of the lodge will be larger than S1. The only answer choice that fits this condition is C: 2,000 square feet.

The Sons of Knute Had a Hunting Lodge Up on Loon