<p>Probably just an issue with the phrasing of the problem you're tackling. I'd assume they just want you be using the Slutsky equation for own-price changes, so you should be good just differentiating what you derived with respect to px and then carrying on with the income effect.</p>

CC BY-SA 4.0
2020-01-15T19:45:11.583Z
2020-01-15T19:45:11.583Z
Emil
<p>Just take the total derivative:</p>

<p><span class="math-container">$Dh_x = \frac{dh_x}{dp_x}dp_x + \frac{dh_x}{dp_y}dp_y$</span></p>

<p>Notice that this "contains" both partial derivatives, which, by definition, are each defined only when the other variable is held constant (e.g., <span class="math-container">$p_y = 0$</span>).</p>

CC BY-SA 4.0
2020-02-14T20:52:42.587Z
2020-02-14T22:16:28.763Z
2020-02-14T22:16:28.763Z
heh
<p>We have the utility function <span class="math-container">$$U(x,y)=x + y$$</span> and we have to derive the substitution and income effects using Slutsky equation. But after I derive the Hicksian demand functions for e.g. x:</p>

<p><span class="math-container">$$h_x= \frac{I}{p_x+\frac{p_y^3}{p_x^2}}$$</span></p>

<p>do we derive this only with respect to x in order to account for impacts of changes in <span class="math-container">$p_x$</span> or do I do the same derivation with respect to y and sum both up?</p>

<p>The confusing part is that I don't know what prices change, so it would seem most generally to assume that both change. </p>

<microeconomics><slutsky-equation><nonexpert>
How to derive substitution and income effect using Slutsky equation if we don't know which of the prices change?
Svit
# How to derive substitution and income effect using Slutsky equation if we don't know which of the prices change?

(Asked by: username_0 on 2020-01-01)

We have the utility function $$U(x,y)=x + y$$ and we have to derive the substitution and income effects using Slutsky equation. But after I derive the Hicksian demand functions for e.g. x:

$$h\_x= \frac{I}{p\_x+\frac{p\_y^3}{p\_x^2}}$$

do we derive this only with respect to x in order to account for impacts of changes in $p\_x$ or do I do the same derivation with respect to y and sum both up?

The confusing part is that I don't know what prices change, so it would seem most generally to assume that both change.

**Tags:** microeconomics, slutsky-equation, nonexpert


---

## 2 answers

### Answer by username_1 on 2020-01-15. Score: 0

Probably just an issue with the phrasing of the problem you're tackling. I'd assume they just want you be using the Slutsky equation for own-price changes, so you should be good just differentiating what you derived with respect to px and then carrying on with the income effect.


---

### Answer by username_2 on 2020-02-14. Score: 0

Just take the total derivative:

$Dh\_x = \frac{dh\_x}{dp\_x}dp\_x + \frac{dh\_x}{dp\_y}dp\_y$

Notice that this "contains" both partial derivatives, which, by definition, are each defined only when the other variable is held constant (e.g., $p\_y = 0$).


---

