Suppose the production function for flyswatters during a particular period can be represented byTo construct the marginal and average productivity functions of labor (l) for this function, we must assume a particular value for the other input, capital (k). Suppose k = 10. Then the production function is given bywhich diminishes as l increases, eventually becoming negative. This implies that q reaches a maximum value. Setting MPlequal to 0,120,000l – 3,000l2 = 040l = l2l = 40as the point at which q reaches its maximum value. Labor input beyond 40 units per period actually reduces total output. For example, when l = 40, Equation 9.7 shows that q = 32 million flyswatters, whereas when l = 50, production of flyswatters amounts to only 25 million. Average product.To find the average productivity of labor in flyswatter production, we divide q by l, still holding k = 10:which occurs when l = 30. At this value for labor input, Equation 9.12 shows thatAPl= 900,000, and Equation 9.8 shows that MPlis also 900,000. When APlis at amaximum, average and marginal productivities of labor are equal.3Notice the relationship between total output and average productivity that is illustrated by this example. Even though total production of flyswatters is greater with 40 workers (32 million) than with 30 workers (27 million), output per worker is higher in the second case. With 40 workers, each worker produces 800,000 flyswatters per period, whereas with 30 workers each worker produces 900,000. Because capital input (flyswatter presses) is held constant in this definition of productivity, the diminishing marginal productivity of labor eventually results in a declining level of output per worker.
solution: Given production function is q=f(k,l) = 600k^2 l^2- k^3. l^3. In order to study the marginal and the average productivity a particular value for capital is assumed. Suppose k = 10. Then the production function is given by, q= 600*100 .l^2 1000l^3 = 60000l^2 1000l^3. Having fixed the level of capital to 10 units, to calculate the marginal product of labor, we differentiate the output function wrt labor. Thus, MP= 120000l-3000l^2. The function form of MP of labor shows that as the value of l (number of labors) is increasing, the marginal product of labor is falling. This is also consistent because when the level of capital is fixed at a certain level for any firm, initially, the more the number of people, more will be the output but after a certain stage, adding more labor will lead to a fall in the level of output. Thus MP falls as the number of labor increases. It may even fall to be negative. The point where the output is maximized can be found by differentiating the output function wrt labor and setting it equal to zero. This means that the marginal productivity of labor should be set equal to 0. Thus, 120000l-3000l^2=0 ….
Solving this, gives l=40. Thus, max output is produced by setting l=40. This can also be verified from the total product function. When, l=40, output q= 60000l^2 1000l^3 = 32000000. On the other hand if l=50, q=60000l^2 1000l^3 = 25000000. In order to calculate the average product, the total product is divided by the level of quantity. Thus, AP= TP/l = (60000l^2 1000l^3)/l = 60000l-1000l^2. To find the point where the average product of labor reaches a maximum, we differentiate the Avg product wrt l and set it equal to 0. Thus, 60000-2000l=0. On solving this, we get l=30. At a level when the average product is maximum, the marginal product and the average product should be equal. Thus MP at l=30 is 900000. The AP at l=30 is also equal to 900000.