# Budget constraint when one of the goods can be sold

I'm studying for a microeconomics midterm and ran across the following question:

A person consumes two goods, X and Y. The consumer's only source of income is given by the quantity of Y equal to ten, which he can sell on the market at the existing price.

What would the budget constraint look like? What would it look like if the price of Y increases?

I initially drew the curve with a downward slope until it hits Y=10 on the Y-axis then it becomes vertical down to the X-axis. Can it look like this given that the question doesn't specify how much of X is being consumed?

:)

A person is endowed with $(x,y) = (0,10)$ units. Suppose, price of $x$ is $p_x$; price of $y$ is $p_y$ . So, the income of the consumer = $10 p_y$

And the corresponding budget constraint will be: $x p_x + y p_y = 10p_y$

Vertical intercept $(0, y) = (0,10)$ and horizontal intercept $(x, 0) = \left( \displaystyle\frac{10p_y}{p_x}, 0\right)$

Now suppose $p_y$ is increased to $p_y'$.

New budget line will be : $x p_x + y p_y' = 10p_y'$

Vertical intercept $(x = 0, y) = (0,10)$ which is same as the previous case.

And horizontal intercept $(x, 0) = \left( \displaystyle\frac{10p_y'}{p_x}, 0\right) > \left( \displaystyle\frac{10p_y}{p_x}, 0\right)$ because $p_y' > p_y$

Therefore, the new budget line has become flatter with the same vertical intercept.

The graph looks something like this: 