<p>Consider a quasilinear utility function:  <span class="math-container">$u(x, y) = x+4 y^{.5} \quad \text{s.t.} \, I=P_{x}x+P_{y}y$</span>. I know how to calculate the demand for good <span class="math-container">$x$</span>, beginning with
<span class="math-container">\begin{align*}
&amp;\cfrac{\partial}{\partial x}\left(x + 4 y^{.5}\right)=1 &amp;  \cfrac{\partial}{\partial y}\left(x + 4 y^{.5}\right)=\cfrac{2}{y^{.5}}
\end{align*}</span>
and substituting the solution for <span class="math-container">$y$</span> into our budget constraint
<span class="math-container">\begin{align*}
\cfrac{\cfrac{2}{y^{.5}}}{1}=\cfrac{P_y}{P_x} \Rightarrow y=\cfrac{4P_{x}^2}{P_{y}^2} \Rightarrow I=P_{x}x+P_{y}\cfrac{4P_{x}^2}{P_{y}^2}.
\end{align*}</span>
Marshallian demand for <span class="math-container">$x$</span> works out to
<span class="math-container">\begin{align*}
&amp;x_{M}=\frac{IP_{y}-4P_{x}^2}{P_{x}P_{y}}.
\end{align*}</span>
But, what about the demand function for good <span class="math-container">$y$</span>? No econ book that I have (Varian's Microecon, and others) really gives a straightforward answer (that I've yet found) for the full demand system. What would the first steps be for deriving demand for good <span class="math-container">$y$</span>?</p>

CC BY-SA 4.0
2023-01-28T04:30:26.017Z
This might help: https://economics.stackexchange.com/a/51756/11824
Amit
CC BY-SA 4.0
2023-01-28T06:57:21.840Z
You can use the budget constraint and the demand for $X$ ($x^d$)to determine the demand for $Y$ in this case. $y^d= \dfrac{I-p_Xx^d}{p_Y}$
Amit
CC BY-SA 4.0
2023-01-28T14:30:35.370Z
@ Jared Greathouse. Be careful when you take a derivative, the derivative with respect to $y$ is not correct, it is $20y^4$.
BakerStreet
CC BY-SA 4.0
2023-01-28T16:07:58.007Z
@BakerStreet Incorrect. $y^.5 = y^{1/2}$. By the power rule, this is $0.5y^{-.5}$, and when we rearrange the negative exponent, we get what I got. It would be $20y^4$ if the exponent was 5, though.
Jared Greathouse
CC BY-SA 4.0
2023-01-28T16:33:21.587Z
Why $y^5= y^{1/2}$?? In your text, in the utility function, there is $y^5$, the exponent is $5$.
BakerStreet
CC BY-SA 4.0
2023-01-28T16:37:47.900Z
Maybe there is a typo, in the utility function the exponent should be $0.5$, not $5$
BakerStreet
CC BY-SA 4.0
2023-01-28T16:49:44.283Z
No, in the original utility function, it's $.5$. Unless someone edited it after I wrote it. But, you're right, I should've been clearer and put $0.5$.
Jared Greathouse
CC BY-SA 4.0
2023-01-28T17:08:51.873Z
Ah, there is point before $5$, this is the reason.  But honestly it is impossible to see it without a microscope :-). It is not your fault, it is the lateX here that makes the point very small.
BakerStreet
<microeconomics><mathematical-economics><utility>
Deriving Demand for Exponent Term for Quasilinear Utilities
Jared Greathouse
# Deriving Demand for Exponent Term for Quasilinear Utilities

(Asked by: username_0 on 2023-01-27)

Consider a quasilinear utility function: $u(x, y) = x+4 y^{.5} \quad \text{s.t.} \, I=P\_{x}x+P\_{y}y$. I know how to calculate the demand for good $x$, beginning with
\begin{align\*}
&\cfrac{\partial}{\partial x}\left(x + 4 y^{.5}\right)=1 & \cfrac{\partial}{\partial y}\left(x + 4 y^{.5}\right)=\cfrac{2}{y^{.5}}
\end{align\*}
and substituting the solution for $y$ into our budget constraint
\begin{align\*}
\cfrac{\cfrac{2}{y^{.5}}}{1}=\cfrac{P\_y}{P\_x} \Rightarrow y=\cfrac{4P\_{x}^2}{P\_{y}^2} \Rightarrow I=P\_{x}x+P\_{y}\cfrac{4P\_{x}^2}{P\_{y}^2}.
\end{align\*}
Marshallian demand for $x$ works out to
\begin{align\*}
&x\_{M}=\frac{IP\_{y}-4P\_{x}^2}{P\_{x}P\_{y}}.
\end{align\*}
But, what about the demand function for good $y$? No econ book that I have (Varian's Microecon, and others) really gives a straightforward answer (that I've yet found) for the full demand system. What would the first steps be for deriving demand for good $y$?

**Tags:** microeconomics, mathematical-economics, utility

> **Comments:**
> **username_1**: This might help: https://economics.stackexchange.com/a/51756/11824  
> **username_1**: You can use the budget constraint and the demand for $X$ ($x^d$)to determine the demand for $Y$ in this case. $y^d= \dfrac{I-p_Xx^d}{p_Y}$  
> **username_2**: @ username_0. Be careful when you take a derivative, the derivative with respect to $y$ is not correct, it is $20y^4$.  
> **username_0**: @username_2 Incorrect. $y^.5 = y^{1/2}$. By the power rule, this is $0.5y^{-.5}$, and when we rearrange the negative exponent, we get what I got. It would be $20y^4$ if the exponent was 5, though.  
> **username_2**: Why $y^5= y^{1/2}$?? In your text, in the utility function, there is $y^5$, the exponent is $5$.  
> **username_2**: Maybe there is a typo, in the utility function the exponent should be $0.5$, not $5$  
> **username_0**: No, in the original utility function, it's $.5$. Unless someone edited it after I wrote it. But, you're right, I should've been clearer and put $0.5$.  
> **username_2**: Ah, there is point before $5$, this is the reason.  But honestly it is impossible to see it without a microscope :-). It is not your fault, it is the lateX here that makes the point very small.  

---

## 0 answers

