Use the concepts of marginal cost and marginal revenue to derive an optimal capital budget for Company X, which has identified 7 possible investment projects and determined its cost of capital as shown below.

Table A: Alternative Projects, Required Investments, and Expected Rate of Return
Project Investment Required in Millions of Dollars Expected Rate of Return on Investment
A 150 12%
B 300 15%
C 125 10%
D 75 16%
E 50 20%
F 500 14%
G 250 18%

Table B: Cost of Capital by Amount Raised
Block of Funds
(in Millions) Amount of Funds
in Block Cost of Capital for Block
First Block of Funds $500 10%
Second Block of Funds $400 11%
Third Block of Funds $300 12%
Fourth Block of Funds $200 13%
Fifth Block of Funds $100 14%
Sixth Block of Funds $100 15%

To derive an optimal capital budget for Company X using the concepts of marginal cost and marginal revenue, we need to compare the expected marginal cost and marginal revenue for each investment project. Marginal cost refers to the additional cost incurred by investing in an additional project, while marginal revenue represents the additional revenue generated from that project.

To calculate the marginal cost, we need to consider the cost of capital for each block of funds raised, as shown in Table B. The cost of capital represents the minimum rate of return that Company X expects to generate for each amount of funds raised.

To determine the marginal cost for each project, we need to identify the block of funds that includes the required investment for that project. For example, projects A, C, D, and E require investments of $150 million, $125 million, $75 million, and $50 million respectively. These investments fall under the first block of funds, with a cost of capital of 10%. Therefore, the marginal cost for projects A, C, D, and E is 10%.

Similarly, project B requires an investment of $300 million, which falls under the second block of funds with a cost of capital of 11%. Hence, the marginal cost for project B is 11%.

Project F requires an investment of $500 million, which falls under the third block of funds with a cost of capital of 12%. Therefore, the marginal cost for project F is 12%.

Lastly, project G requires an investment of $250 million, which also falls under the third block of funds with a cost of capital of 12%. Therefore, the marginal cost for project G is 12%.

Now, let's calculate the marginal revenue for each project. Marginal revenue is determined by the expected rate of return on investment, which is provided in Table A.

For project A, the expected rate of return is 12%. Therefore, the marginal revenue for project A is 12%.

For project B, the expected rate of return is 15%. So, the marginal revenue for project B is 15%.

Project C has an expected rate of return of 10%. Hence, the marginal revenue for project C is 10%.

Project D has an expected rate of return of 16%, resulting in a marginal revenue of 16%.

Project E has an expected rate of return of 20%. Therefore, the marginal revenue for project E is 20%.

Project F has an expected rate of return of 14%, leading to a marginal revenue of 14%.

Project G has an expected rate of return of 18%, resulting in a marginal revenue of 18%.

By comparing the marginal costs and marginal revenues for each project, we can identify the optimal capital budget. The optimal capital budget would include projects with a higher marginal revenue compared to their marginal costs. In this case, projects E and G have the highest marginal revenues of 20% and 18% respectively, while having a marginal cost of 12%. These projects provide the best return on investment compared to their cost of capital.

Therefore, the optimal capital budget for Company X would include projects E and G with investments of $50 million and $250 million respectively, as they have the highest marginal revenues and a marginal cost of 12%.