<p>This is an example of quasilinear utility. In general, a quasilinear utility function takes the form 
<span class="math-container">$$u(x,y)=f(x)+y.$$</span>
The perfect substitutes utility function, for example, is a special case of the quasilinear utility function. One feature of quasilinear utility is that the MRS is independent of at least one of the two goods:
<span class="math-container">$$\frac{\frac{\partial u(x,y)}{\partial x}}{\frac{\partial u(x,y)}{\partial y}}=f'(x),$$</span>
meaning the slope of indifference curves is independent of the quantity of good <span class="math-container">$y$</span>. </p>

<p>The Marshallian demand functions satisfy the equations:
<span class="math-container">$$ f'(x)=\frac{P_x}{P_y}$$</span>
<span class="math-container">$$I=P_xx+P_yy,$$</span></p>

<p>which come from the first-order conditions of the constrained maximization problem. We can solve for the Marshallian demand function for <span class="math-container">$x$</span> directly from the first equation:
<span class="math-container">$$x^*=f'^{-1}(\frac{P_x}{P_y}).$$</span>
Substituting this into your second equation gives 
<span class="math-container">$$I=P_xf'^{-1}(\frac{P_x}{P_y})+P_yy$$</span>
<span class="math-container">$$y^*=\frac{I-P_xf'^{-1}(\frac{P_x}{P_y})}{P_y}.$$</span></p>

<p>In your example, we get
<span class="math-container">$$x^*=(\frac{P_y}{P_x})^2.$$</span>
<span class="math-container">$$y^*=\frac{I-P_x(\frac{P_y}{P_x})^2}{P_y}.$$</span></p>

<p>There is a fundamental problem with your solution. You write a demand function for <span class="math-container">$x$</span> that depends on the quantity of <span class="math-container">$y$</span>. Demand functions depend either on <span class="math-container">$(P_x,P_y,I)$</span>, if Marshallian, or <span class="math-container">$(P_x,P_y,U)$</span>, if Hicksian.</p>

CC BY-SA 4.0
2024-03-09T17:31:55.263Z
What about the corner solutions?
deps_stats
CC BY-SA 4.0
2019-03-24T03:19:52.153Z
2019-03-24T04:17:02.107Z
2019-03-24T04:17:02.107Z
dlnB
<p>How do I derive the demand function from</p>

<p><span class="math-container">$$U=y + 2\sqrt{x}$$</span></p>

<p>Currently I have <span class="math-container">$MRS = \frac{1}{\sqrt{x}} = \frac{P_x}{P_y}$</span>, so, <span class="math-container">$P_y = P_x\sqrt{x}$</span></p>

<p>Using the budget line: <span class="math-container">$$I = xP_x + yP_y \implies I = xP_x +y(P_x\sqrt{x})$$</span></p>

<p>Which simplifies to <span class="math-container">$$x = \frac{2I + y^2 P_x \pm y\sqrt{Px(4I+y^2P_x)}}{2P_x}$$</span></p>

<p>I'm not sure if this is correct, should I have did <span class="math-container">$MRS = \frac{1}{\sqrt
{x}} = \frac{P_x}{P_y}$</span>, so, <span class="math-container">$P_x = \frac{P_y}{\sqrt{x}}$</span> instead and substitute <span class="math-container">$\frac{P_y}{\sqrt{x}}$</span> into budget line? How do I determine what to substitute into the budget line?</p>

CC BY-SA 4.0
2019-03-25T17:16:34.047Z
Questions about my solution? Please accept if not.
dlnB
CC BY-SA 4.0
2022-10-02T10:12:26.837Z
See:  https://economics.stackexchange.com/a/16475/11824
Amit
<microeconomics>
Deriving demand function from utility function
Gai Weo
# Deriving demand function from utility function

(Asked by: username_0 on 2019-03-24)

How do I derive the demand function from

$$U=y + 2\sqrt{x}$$

Currently I have $MRS = \frac{1}{\sqrt{x}} = \frac{P\_x}{P\_y}$, so, $P\_y = P\_x\sqrt{x}$

Using the budget line: $$I = xP\_x + yP\_y \implies I = xP\_x +y(P\_x\sqrt{x})$$

Which simplifies to $$x = \frac{2I + y^2 P\_x \pm y\sqrt{Px(4I+y^2P\_x)}}{2P\_x}$$

I'm not sure if this is correct, should I have did $MRS = \frac{1}{\sqrt
{x}} = \frac{P\_x}{P\_y}$, so, $P\_x = \frac{P\_y}{\sqrt{x}}$ instead and substitute $\frac{P\_y}{\sqrt{x}}$ into budget line? How do I determine what to substitute into the budget line?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Questions about my solution? Please accept if not.  
> **username_2**: See:  https://economics.stackexchange.com/a/16475/11824  

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## 1 answers

### Answer by username_1 on 2019-03-24. Score: 2

This is an example of quasilinear utility. In general, a quasilinear utility function takes the form
$$u(x,y)=f(x)+y.$$
The perfect substitutes utility function, for example, is a special case of the quasilinear utility function. One feature of quasilinear utility is that the MRS is independent of at least one of the two goods:
$$\frac{\frac{\partial u(x,y)}{\partial x}}{\frac{\partial u(x,y)}{\partial y}}=f'(x),$$
meaning the slope of indifference curves is independent of the quantity of good $y$.

The Marshallian demand functions satisfy the equations:
$$ f'(x)=\frac{P\_x}{P\_y}$$
$$I=P\_xx+P\_yy,$$

which come from the first-order conditions of the constrained maximization problem. We can solve for the Marshallian demand function for $x$ directly from the first equation:
$$x^\*=f'^{-1}(\frac{P\_x}{P\_y}).$$
Substituting this into your second equation gives
$$I=P\_xf'^{-1}(\frac{P\_x}{P\_y})+P\_yy$$
$$y^\*=\frac{I-P\_xf'^{-1}(\frac{P\_x}{P\_y})}{P\_y}.$$

In your example, we get
$$x^\*=(\frac{P\_y}{P\_x})^2.$$
$$y^\*=\frac{I-P\_x(\frac{P\_y}{P\_x})^2}{P\_y}.$$

There is a fundamental problem with your solution. You write a demand function for $x$ that depends on the quantity of $y$. Demand functions depend either on $(P\_x,P\_y,I)$, if Marshallian, or $(P\_x,P\_y,U)$, if Hicksian.

> **Comments:**
> **username_3**: What about the corner solutions?  

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