I am modifying an RTS game (Rise of Nations) and want to make a unit twice as good and expensive as it's previous version so the balance is unaffected.
Without context, this goal is meaningless.
In RTS games, typically you do not have units fight each other one-on-one; instead they fight in groups, and group dynamics tend to dominate.
For example; suppose you have two identical units fighting each other. On average, the outcome is a draw; they either kill each other off simultaneously, or one barely survives because they happened to get the first shot off.
A vs. B -> draw
What if we double one side, so that it's 1 vs. 2? Do we expect that the two sides lose one unit each, leaving the side with 2 units remaining with 1 unit?
A vs. 2B -> B?
No! Because while the single "A" unit is attacking a "B" unit, those two "B" units are simultaneously attacking the "A" unit. The "A" unit loses health at twice the rate, and so we're left with one "B" unit at half health, and the other at full health.
A vs. 2B -> 1.5B
That is, by doubling the number of units, we've made one side 4 times as powerful.
The key here is that all the units are fighting simultaneously, which is common when you have ranged units. For melee units it's different, as you often have large groups of units waiting at the back and not engaged. The way this affects relative strengths is described in Lanchester's Laws. You could, given a ratio of engaged units, estimate how an increase in unit strength affects the group outcome, using these formulae.
But RTS games are usually much more complex than this. A lot of factors affect combat, such as range, attack speed, movement speed, area-of-effect attacks, support powers like healing, and buffs/debuffs. Instead, you should focus on testing and simulations.