Finding the % of the total consummation by category using elasticity and MPC Given a total consummation of 1000 and a total revenue of 2000 with 

Cat 1 => elasticity of 0.5; MPC of 0.1

Cat 2 => elasticity of 1, MPC of 0.3

How would I go about to calculating the % of the total revenue for each category?



I tried using both these formulas





and tried to figure out the first category by assuming the variation of revenue to be 2000 (2000-0) and the initial consummation to be 0.  



which gives me a value of 500 for c2. But then, using the MPC formula, it does not seem to make any sense
Notice that the elasticity can be written as 

$$\frac{\Delta C}{\Delta Y}\cdot\frac{Y}{C}$$

So for category $1$ you have $$\frac{\Delta C_1}{\Delta Y}\cdot\frac{Y}{C_1}=0.5$$ and $$MPC_1=\frac{\Delta C_1}{\Delta Y}=0.1$$

This gives you 
$$0.1\cdot\frac{Y}{C_1}=0.5$$
which by reordering yields to
$$\frac{C_1}{Y}=\frac{1}5$$

Similarly $$\frac{C_2}{Y}=\frac{3}{10}$$

Finally notice that $C=C_1+C_2=1000$ and $Y=2000$, hence we must have that $\frac{C}{Y}=\frac{C_1+C_2}{Y}=0.5$ which is satisfied since
$$\frac{1}5+\frac{3}{10}=0.5$$

(Since we have that $2C=Y$ and $100\%$ of revenue comes from $C$ is straightforward to see that the $\%$ of revenue associated with each category are twice this numbers)