The sum of the incomes of A and B is more than that of C and D taken together. The sum of the incomes of A and C is the same as that of B and D taken together. Moreover, A earns half as much as the sum of the incomes of B and D. Whose income is the highest?

Let the incomes of A, B, C, and D be represented by $A,B,C,$ and $D$ respectively.
From the problem statements, we form the following relationships:

1. $$A+B>C+D\text{— (Equation 1)}$$

2. $$A+C=B+D\text{— (Equation 2)}$$

3. $$A=\frac{1}{2}(B+D)⟹2A=B+D\text{— (Equation 3)}$$

Substitute Equation (3) into Equation (2):

$$A+C=2A⟹C=A$$

Since $C=A$, let's rewrite Equation (3) as:

$$B+D=2A⟹B+D=A+C$$

(which matches Equation 2).

Now substitute $C=A$ into Equation (1):

$$A+B>A+D⟹B>D$$

Since $B+D=2A$ and $B>D$, it must be true that:

$$B>A>D$$

Since $C=A$, we can combine everything:

$$B>A=C>D$$

Thus, B has the highest income.