Using Calculus Calculating the Marginal Product of Labor The short-run production function, g = f[L,If],can be written as solely a function of i. because capital is fixed: at = g{L). The calculus definition of the marginal product of labor is the derivative ofthis production function with respect to labor: MP1, = dylfLMdL. In the long run, when both labor and capital are free to vary, the marginal product of labor is the partial derivative of the production function, Equation 5.1 E', q = L, K}, with respect to labor: 3!; 3ftL.K} 3L 3L ' We use the symbol aqutL instead of dqfdl. because we are taking a partial derivative.1 We use partial derivatives when we want to change only one explanatory.' variable in a function that has more than one such variable. Here, a is a function of both labor, L, and capital, K. To obtain a partial derivative with respect to one variable, say L, we differentiate as usual where we treat the other variables {here just K} as constants. QSLA 5.1 For a linear production function g = ffL, K) = 2L + K and a multiplicative production function g = LE, what are the shortrun production functions given that capital is fixed at K = 1m}? "What are the marginal products of labor for these shortrun production functions? Answer 1. Obtain theshorfrun productionfunc'tions by settingf? = 100. The short-run linear production function is g = 2L + 1'00 and the shortrun multiplicative function is q=L x 1|]0=1(iL. 2. Determine the marginal pro ducts ofiairor by di'erentiaiing the shortrun production functions with respect to labor. The marginal product of labor is MPL = d(2L + llde = 2 for the shortrun Iinear production function and MPL = dUULjde = IUD for the shortrun multiplicative production function