Abstract: Cobb-Douglas production function has many important usages in the economic research. The traditional one is only concerned with two investment factors, namely labor and capital, and doesn't study the curved features of cost function which corresponds with the C-D production function. This paper mainly studies the interior connections among long-run cost function,short-run cost function and graphs, when the production function is C-D production function with the input elements being k. Based on the above cost function and the known market demand, this paper points out the long-run and short-run equilibrium quantity and the corresponding factor application. Finally, this paper investigates the difference between estimating the parameter in C -D production function and estimating other parameters.