Calculating the impact of a per-unit Subsidy given to Consumers? I am somewhat confused about the concept of calculating the prices payed by consumers and producers after a subsidy has been applied. 

For example, say a good is being traded at 100 dollar at the market equilibrium. If the government were to give a 50 dollar subsidy to the consumer, would the price the consumer actually pays now (post-subsidy) be greater than 50$ due to differing elasticities? Would some of that subsidy be translated over to the price the producer receives? Or, would the price the consumer pays always be the (Equilibrium Price - Subsidy)? 

Is the price reduction for the consumer (as a result of the subsidy) a function of the elasticity of demand or would the subsidy always fully fall on the consumer?
This is a great question which economists have a pretty good answer to. Let's first suppose the market is in an equilibrium with no intervention at price $P^*$. 

If there is a tax of value $t$ we have a new equilibrium with a consumer price $P^*_c$ and producer price $P^*_s$. These prices surely do not equal each other any more, and the difference of the two is not necessarily equal to the value of the tax. 

You can think of a subsidy as a negative tax. 

The change in consumer price due to a marginal tax or subsidy is called the pass through rate, $\rho$. For a sufficiently small tax, the relationship is $P^*_c=P^*+\rho\cdot t$. 

Clearly, if the pass-through rate is one the consumer completely pays for the tax. If it is zero, the consumer does not at all pay for the tax. It is not impossible for $\rho$ to be negative or greater than one. 

In perfectly competitive markets the pass through rate depends on the elasticity of demand, $\varepsilon_D$, and the elasticity of supply, $\varepsilon_S$. $$\rho=\frac{1}{1+\frac{\varepsilon_D}{\varepsilon_S}}$$

From this we can see that the pass through rate is larger when demand is more inelastic, and smaller when supply is more inelastic. This is where we get the adage "the more inelastic side of the market bares the burden of a tax."

For pass through to be equal to one, we either need (a) perfectly inelastic demand or (b) perfectly elastic supply. A lot of the time (b) can be satisfied by assuming constant marginal cost of production. 

For imperfectly competitive situations, the pass through rate depends on firm conduct as well as the curvature of demand. I suggest Weyl and Fabinger "Pass through as an economic tool" 2013, Journal of Political Economy, publication for the details.