Under the US tax system, capital that is employed in the corporate sector is taxed at a higher rate than capital in the noncorporate sector. This problem will analyze the excess burden of the differential taxation of capital.

Assume that there are two sectors, corporate and noncorporate. The value of marginal product of capital in the corporate sector, \(V M P_{c}\) is given by \(V M P_{c}=100-K_{c}\) where \(K_{c}\) is the amount of capital in the corporate sector, and the value of the marginal product of capital in the noncorporate sector, \(K_{n}\), is given by \(V M P_{n}=80-2 K_{n}\), where \(K_{n}\) is the amount of capital in the noncorporate sector. Altogether there are 50 units of capital in society.

a. In the absence of any taxes, how much capital is in the corporate sector and how much in the noncorporate sector? (Hint: Draw a sketch along the lines of Figure 15.9 to organize your thoughts.)

b. Suppose that a unit tax of 6 is levied on capital employed in the corporate sector. After the tax, how much capital is employed in each sector? What is the excess burden of the tax?