# How to show consumer subsidy (rebate) on supply and demand diagram?

I am currently working on an assignment and needed some help.

My assignment requires me to find an article relating to a form of government intervention such as taxes or subsidies and analyse the article using a supply and demand diagram.

For my essay, I have chosen an article that essentially talks about a rebate that should be removed. So far I have written that the consumer subsidy causes negative externalities and the removal of these rebates would help reallocate resources for incentivizing other methods. I know that rebates are essentially subsidies given to consumers to increase consumption, so they would affect the demand curve. However, I am stuck on the diagram as I am confused on which way the demand curve shifts and why. Would you please help me?

Let $$x = D(p)$$ be the demand for a good if the price is equals to $$p$$. The inverse demand curve (as you would draw it) is then given by $$p = D^{-1}(x)$$. It gives the price as a function of the quantity.
If there is a rebate of $$r$$ and if $$p$$ is the price, then the consumer only pays $$p^\ast = p - r$$. Then the demand is given by: $$x = D(p^\ast) = D(p - r).$$ Inverting this gives: $$p - r = D^{-1}(x),\\ \to p = D^{-1}(x) + r$$ So the inverse demand curve (as you would draw it) shifts up by an amount $$r$$.