The learning curve depicted in Figure 42 can be represented

The learning curve depicted in Figure 4.2 can be represented algebraically in the following equation: Average time to produce x units = ax-B where x is the total number of units produced by a firm in its history, a is the amount of time it took a firm to produce its first unit, and B is a coefficient that describes the rate of learning in a firm. Suppose it takes a team of workers 45 hours to assemble their first product (a=45) and 40.5 hours to assemble their second product. When a firm doubles its production (in this case, from one to two units) and cuts its production time (in this case from 45 hours to 40.5 hours), learning is said to have occurred (in this case 40.5/45 or 90% learning curve). The B for a 90% learning curve is .3219. Thus, this firm’s learning curve is: Average time to produce x units = 45x-.3219. What is the average amount of time it will take this firm to produce 6 products? (Hint: Simply plug “6” in for “X” in the equation and solve). What is the total time it took this firm to produce these 6 products? (Hint: Simply multiply the number of units produced “6” by the average time it will take to produce these 6 products). What is the average time it will take this firm to produce 5 products? What is the total time it will take this firm to produce 5 products? So, what is the total time it will take this firm to produce its sixth product? (Hint: Subtract the total time needed to produce five produces form the total time needed to produce six products). Suppose a new firm is going to start producing these same products. Assuming this new firm doesn’t learn anything from established firms, what will the cost disadvantage for this new firm be when it assembles its first product? (Hint: Compare the costs of the experienced firm’s sixth product with the cost of the new firm’s first product).

Solution

What is the average amount of time it will take this firm to produce 6 products?

Average time to produce x units = 45x^-0.3219

With x = 6, Average time to produce 6 units =  45 x 6^-0.3219 = 25.277 hours.

What is the total time it took this firm to produce these 6 products?

Total time = 25.277 x 6 = 151.662 hours

What is the average amount of time it will take this firm to produce 5 products?

Average time to produce x units = 45x^-0.3219

With x = 5, Average time to produce 5 units =  45 x 5^-0.3219 = 26.805 hours.

What is the total time it took this firm to produce these 5 products?

Total time = 26.805 x 5 = 134.024 hours

What is the total time it will take this firm to produce its sixth product?

151.662 hours - 134.024 hours = 17.638 hours

What will the cost disadvantage for this new firm be when it assembles its first product?

The new firm will start with the first product made in 45 hours. Whereas the same product can be made by the old firm in 17.638 hours.

So, the time savings for the first part will be 45 - 17.638 = 27.362 hours. Multiplied to this, the cost per hour (which is not given in the question) will give the cost savings figure.