Derivation of a Demand Equation

A consumer's utility function is $$U(x,y)=\sqrt x + y$$. Assuming we have an interior solution, I need to show that the demand for $$x$$ does not depend on income.

I know that the consumer consumes where the marginal utilities per dollar of $$x$$ and $$y$$ are equal, so I have found the following two equations:

1. $$\frac 1 {2P_x\sqrt x}$$ = $$\frac 1 {P_y}$$
2. $${P_x}x + {P_y}y = I$$

Then, I am stuck. I do not see how I can get rid of both $$y$$ and $$I$$ in this case? And how does knowing that we have an interior solution help?

• You have 2 equations and 2 unknowns, right? What is keeping you from solving for $x$ and $y$? Seems like you can get $x$ right away from the first equation. Failing to do that, try to express $x$ and $y$ as functions of $p_x,p_y$ and $I$, but not each other. Feb 9 '20 at 15:44
• Okay. I took a break and tried again. I got the demand function of $x$ to be $x = \frac {(P_y)^2} {4(P_x)^2}$. Is this correct? I literally just saw that I could obtain $x$ directly from the first equation... It must have been a long day for me. Feb 9 '20 at 15:52

Hint:

Rearrange your equation 1. to express optimal consumption $$x$$ as a function of $$P_x$$ and $$P_y$$!

Does the rhs now contain $$I$$?

• Not sure why I did not see this when I posted the question... perhaps it was a long day! But thank you :) Feb 11 '20 at 14:39

hint:

Solve for optimal $$x^*,y^*$$ then substitute the solutions into budget constraint and then solve the budget constraint for $$x^*,y^*$$ respectively and you will get the demands as a function of the parameters and of course conditional on the optimal consumption of the other variable.

Knowing that we have an interior solution helps because that means we know demand(s) is(are) not zero.

• But how do I solve for the optimal basket if I do not know the consumer's budget constraint? Do I just use the general equation for a budget line? Feb 9 '20 at 15:25
• @EthanMark yes that’s the budget constraint - you can call it budget line if it’s linear like in your case but normally I would call it budget/budget constraint (it’s the constraint that you use in your constrained optimization (Lagrangian) problem
– 1muflon1
Feb 9 '20 at 15:27
• I see. Thank you! I found the demand of $x$ to be $x = \frac {(P_y)^2} {4(P_x)^2}$. Is this correct? Feb 9 '20 at 16:12
• @EthanMark no the correct answer will be function of $I$ common you are almost there. If you want demand for x just play optimum y into budget and solve for x, this is as direct hint as it goes only if thing left is actually solving it
– 1muflon1
Feb 9 '20 at 17:22
• But the question is asking me to show that the demand of $x$ does not depend on income. If my final answer for the demand of $x$ is a function of $I$, aren’t I contradicting the question? Feb 9 '20 at 17:31