<p>In the (<strong>quantity, income</strong>) space, <em>not</em> being everywhere differentiable would mean that we would observe gaps, discontinuous jumps, or kinks, in the corresponding graph of the Marshallian individual demand function.</p>

<p><strong>GAPS:</strong> if there was a gap, it would mean that for a certain range of income, the demand function does not exist. In other words, it is not defined for a certain range of income (but it still is defined with respect to prices). This could be interpreted that for this range, income does not affect quantity demanded. So "differentiable everywhere" excludes such a phenomenon.</p>

<p><strong>DISCONTINUOUS JUMPS:</strong> If there was a discontinuous jump, it would mean that a specific level of income induces a "threshold effect" on quantity demanded. So "differentiable everywhere" necessitates that no such effects exist on quantity demanded.</p>

<p><strong>KINKS:</strong> If there was a kink, we would have a "second-order" threshold effect, where quantity demanded would change at a discontinuous rate with respect to income. So "differentiable everywhere" excludes such phenomena.</p>

<p>These should now be "mathematicized" and projected back to the properties of the utility function (<em>if</em>, that is, they are relevant to it).</p>

<p><strong>Note:</strong> In many utility maximization problems, income does not enter the utility function, but it is part of the problem's constraints. In others, we postulate a utility function as a bivariate function of one good and income.</p>

CC BY-SA 3.0
2017-10-08T14:26:17.320Z
2017-10-08T14:26:17.320Z
Alecos Papadopoulos
<p>Suppose you know that the (Marshallian) demand function $x(p,m)$ that satisfies the consumer's problem of utility maximization is such that $\frac{\partial x_i(p,m)}{\partial m}$ is always well-defined for each good $i\in\{1,\dots,L\}$. In other words, the demand function is differentiable with respect to income.</p>

<p>What does this <strong>necessarily implies</strong> about the properties of the utility function $U(x_1,\dots,X_L)$?</p>

CC BY-SA 3.0
2017-10-03T09:17:29.820Z
For simplicity, we can restrict the analysis to the case of non-negative goods, that is $x_i(p,m)\in R^+, \forall p_i\geq 0, m\geq 0, \forall i\in\{1,\dots,L\}$
GabMac
<microeconomics><utility>
Implications of differentiable demand function on the utility function properties
GabMac
# Implications of differentiable demand function on the utility function properties

(Asked by: username_0 on 2017-10-03)

Suppose you know that the (Marshallian) demand function $x(p,m)$ that satisfies the consumer's problem of utility maximization is such that $\frac{\partial x\_i(p,m)}{\partial m}$ is always well-defined for each good $i\in\{1,\dots,L\}$. In other words, the demand function is differentiable with respect to income.

What does this **necessarily implies** about the properties of the utility function $U(x\_1,\dots,X\_L)$?

**Tags:** microeconomics, utility
> **Comments:**
> **username_0**: For simplicity, we can restrict the analysis to the case of non-negative goods, that is $x_i(p,m)\in R^+, \forall p_i\geq 0, m\geq 0, \forall i\in\{1,\dots,L\}$  

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## 1 answers

### Answer by username_1 on 2017-10-08. Score: 1 | Accepted answer

In the (**quantity, income**) space, *not* being everywhere differentiable would mean that we would observe gaps, discontinuous jumps, or kinks, in the corresponding graph of the Marshallian individual demand function.

**GAPS:** if there was a gap, it would mean that for a certain range of income, the demand function does not exist. In other words, it is not defined for a certain range of income (but it still is defined with respect to prices). This could be interpreted that for this range, income does not affect quantity demanded. So "differentiable everywhere" excludes such a phenomenon.

**DISCONTINUOUS JUMPS:** If there was a discontinuous jump, it would mean that a specific level of income induces a "threshold effect" on quantity demanded. So "differentiable everywhere" necessitates that no such effects exist on quantity demanded.

**KINKS:** If there was a kink, we would have a "second-order" threshold effect, where quantity demanded would change at a discontinuous rate with respect to income. So "differentiable everywhere" excludes such phenomena.

These should now be "mathematicized" and projected back to the properties of the utility function (*if*, that is, they are relevant to it).

**Note:** In many utility maximization problems, income does not enter the utility function, but it is part of the problem's constraints. In others, we postulate a utility function as a bivariate function of one good and income.


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