Question

(2) Ralf also enjoys fishing. The total benefits (in dollars) he obtains from time spent fishing are given by TB = 40*ln(h), where ln(.) is the natural log function, and h is hours spent fishing. Because he has to miss work, where he earns a wage of $8 per hour, his marginal cost of time spent fishing is MC = 8.

A.Solve for the optimal quantity of hours that Ralf will spend fishing and show your work. Hint: You may need to Google how to find the derivative of the natural log function. (2 points)

B.Generate a graph in Excel that illustrates the total benefit and total cost functions on one graph as well as the optimum that you found in part a. To generate your graph, I recommend having excel calculate the total benefit and total cost for a range of values between 0 and 10. Once you have done this, you can choose to insert a chart and by choosing the ‘scatter’ option you can generate separate lines for the total benefits and total costs. In the graph, dollars should be on the vertical axis and the quantity between 0 and 10 on the horizontal axis. (3 points)

Answer 1

Solution:

Total benefit function, TB = 40*ln(h), h = hours spent fishing

Marginal cost, MC of fishing = $8 (constant)

Total cost, TC = intergration of MC with respect to h = 8h + c.

For simplicity, taking the constant c = 0, So, TC = 8h

A.) Optimal quantity of fishing hours, h* is where marginal benefit equals the marginal cost of fishing.

Marginal benefit, MB = $\partial TB/\partial h$ = 40/h (since derivative of natural log $\partial ln(x)/\partial x$ = 1/x)

So, under optimality, MB = MC

40/h* = 8

h* = 40/8 = 5

Thus, Ralph will spend 5 hours fishing.

B.) Before the required scatter plot, you'll find a series of images which show how to go with the method.

1. Setting a range for h (column A) from 0 to 10 (I've started from 0.01 as ln(0) goes to (-)infinity, and thus, excel do not generate any value for it), and using the formula, complete the values for total benefit (column B) and total cost (column C). Notice the formula bar for the mentioned selection to see how the formula is written.

2. On the toolbar, under the tab 'Insert', select 'Chart', and select the scatter option as follows:

3. Finally, you click 'OK' to form the scatter. Once, the scatter is obtained, you can change the chart title and label the axis according to your choice, using a '+' (plus) sign on the side of figure. You get a figure such as this:

You can edit the figure,, to mark the optimal quantity of hours spent on fishing (you can edit the snapshot using paint).