Question

3. Marvin is one of the few remaining snowboard producers in Canada. He produces his snowboards in a small factory in Guelph, Ontario. However, price competition in the market for snowboards has much intensified over the last 10 years, and Marvin knows that he needs to considerably cut his own production costs to stay in business Marvin is considering procuring the raw boards for his snowboards from an external supplier, instead of producing them in his own factory. He knows that he first needs an accurate estimate of how much it actually costs him to produce the raw boards himself. He then needs to find a suitable supplier that could sell him the raw boards at a lower price Marvin has a detailed record of his total cost of producing the raw boards in each of the past 24 months (available on the course website; note that q denotes the number of raw boards produced in each month) a) Using the cost data, estimate the monthly total cost function for the production of raw boards in Marvin's factory. Show your regression results and carefully explain how you identified the total cost function. (Hint: For all estimated coefficients carry 2 non-zero digits to the right of the decimal.) (7 points) b) Use your estimated cost function from part (a) to derive expressions for Marvin's (i) average total cost (ATC), and (ii) his marginal cost (MC)·(2 points) c) Download the Excel spreadsheet 'Marvin's Snowboard Factory - Cost Simulation' from the course website. The spreadsheet contains one column with all possible output levels for Marvin's factory (ranging from 1 to 90 raw boards), and three more columns labeled TC (total cost), ATC (average total cost), and MC (marginal cost). Use your expressions for Marvin's total cost (TC), average total cost (ATC), and marginal cost (MC) from parts (a) and (b), to calculate TC, ATC, and MC, for all output levels in the spreadsheet (using your formulas in the corresponding columns in the spreadsheet). Then print the spreadsheet and include it in your assignment. (3 points) d) According to your cost simulation from part (c), what is the average total cost (ATC) and marginal cost (MC) when Marvin produces 20 raw boards? Briefly explain the economic meaning of these two numbers. (4 points) e) According to your cost simulation from part (c), for which output range does Marvin's production of raw boards exhibit economies of scale? Briefly explain. (2 points) f) Marvin eventually found Snow-Tech, a small manufacturer of raw boards located in Peterborough. Snow-Tech offered Marvin to sell him raw boards at a price of $120 per unit. According to your cost simulation from part (c), for which range of q is it optimal for Marvin to buy the raw boards from Snow-Tech (instead of producing the raw boards himself)? Briefly explain. (3 points)

Answer 1

(a). Given in the data, we have values of production for the past 24 months: the total cost and quantity. If we plot the data on a graph, we get a scatter-plot. Using regression, our aim is to find a straight line that satisfies most of the points on the graph.

To find the best fit line, we use the linear regression method. First, we find the sum of the x-values.
x_sum = 35+89+56+53+33+38+52+78+54+60+32+18+17+28+56+70+73+44+54+56+79+14+62+39 = 1190
Similarly, y_sum = 1,51,137.4

Next, we find the xy_sum (product of x and y summed for each month)
xy_sum = (35*3455.5) + (89*17040.82).... = 91,10,108.4

Next, we find the sum of squares of all elements of x and y.
x_square = 68504
y_square = 975578117.4

N = 24
If equation of line is y = mx + c
then, the value of m is
m = (N * xy_sum) - (x_sum * y_sum) / (N * x_square) - (x_sum * x_sum)
m = (24 * 9110108.4) - (1190 * 151137.4) / (24 * 68504) - (1190 * 1190)
m = 170.13

The intercept for the best-fit line can be obtained from :

c = (x_square * y_sum ) - (x_sum * xy_sum) / (N * x_square) - (x_sum * x_sum)
c = (68504 * 151137.4) - (1190 * 9110108.4) / (24 * 68504) - (1190 * 1190)
c = -2138.25

Therefore, the equation of the line is
TC = 170.13 q - 2138.25

(b) Average Total Cost (ATC) = TC/q
ATC = 170.13 - (2138.25/q)
Marginal Cost (MC) = dTC/dq
MC = 170.13

(c). Using the formulae calculated above, we can fill the given table for all levels of production,

For q = 15,
TC = (170.13*15) - 2138.25 = 413.7
MC = 170.13
ATC = 413.7/15 = 27.58
And so on, for all other values of q.

(d). For q = 20,
ATC = 63.2175
MC = 170.13
The average total cost implies that when 20 boards are being produced, the cost of production of one board is 63.22. When 15 boards were produced, the cost of production of one board was 27.58.
A marginal cost of 170.13 implies that when the company was producing 19 snowboards, they were incurring some cost. To increase the level of production by 1 more, to reach 20, they had to take on an additional cost of 170.13.

Before q = 12. TC values end up negative, this implies that it is not up to scale, manufacturing at that level of production is not sustainable.

Hope this helped!