# W3 1-d

Given utility function,

$$
u(q_1, q_2)=q_1 +\sqrt{q_2},
$$

price $(p_1,p_2)=(6,2)$ and income $y=120$, how to find the optimal bundle?

First, try to find the interior solution. We can obtain

$$
MRS = - \frac{u_1}{u_2} = -\frac{1}{\frac{1}{2}q_2^{-\frac{1}{2}}} = - 2 \sqrt{q_2}.
$$

and

$$
MRT = - \frac{p_1}{p_2} = -3.
$$

The interior solution must satisfies
$$
MRS=MRT \implies q_2 = \frac{9}{4} = 2.25,
$$

substitute it into budget the constraint,

$$
q_1 = \frac{1}{p_1} (120 - \frac{9}{4} \cdot p_2 )= 19.25
$$

In this case, the interior solution is available, so it is the optimal bundle.

However, if income is less than $4.5$, the interior solution will locate in infeasible region, i.e., negative amount of good $1$. In that case, the agent will sepnd all money on good $2$.

The below blue line is the budget constraint, and the red curve is the indifference curve at the optimla bundle. The green curve is another indifference curve with higher utility level.


The indifference curve actually intersects with two axes, but one intercept is too large that we cannot show in the above plot.