<p>Yes, under some conditions. This is the classic <strong>integrability problem</strong>: for detailed discussion, see some <a href="http://people.hss.caltech.edu/~kcb/Notes/Demand4-Integrability.pdf">excellent notes by Kim Border</a>.</p>

<p>Several other technical conditions are required, but the most economically substantive condition is that the <a href="http://en.wikipedia.org/wiki/Slutsky_equation">Slutsky matrix</a> must always be symmetric and negative semidefinite. To be concrete, if we define the $ij$th element of the Slutsky matrix at $(p,m)$ to be
$$\sigma_{ij}(p,m)=\frac{\partial D_i(p,m)}{\partial p_j}+D_j(p,m)\frac{\partial D_i(p,m)}{\partial m}$$
then we must have $\sigma_{ij}(p,m)=\sigma_{ji}(p,m)$ for all $(p,m)$, and also for any vector $v$ we must have for all $(p,m)$
$$\sum_i \sum_j \sigma_{ij}(p,m)v_iv_j \leq 0$$
The <em>necessity</em> of these conditions follows immediately from basic consumer theory, which shows that if Marshallian demand is derived from constrained maximization of a utility function, then the Slutsky matrix is symmetric and negative semidefinite. But the <em>sufficiency</em> of these conditions (in conjunction with some other technical assumptions) for us to back out a utility function is a more complicated matter, and to get the details I recommend Border's notes or some other advanced micro source.</p>

<p>If, <em>assuming that the Slutsky conditions hold</em>, you want a rough practical way (ignoring the technical subtleties) to back out indifference curves in the typical two-good case, the simplest way is probably to use your knowledge of demand to determine the compensating change in expenditure that is necessary to adjust for a given change in prices. Specifically, for either $i=1,2$ use the identity
$$\frac{\partial e(p,u)}{\partial p_i} = h_i(p,u) = D_i(p,e(p,u))$$
which, given knowledge of the Marshallian demand function $D$, is a differential equation in the expenditure function $e$. Starting with some initial values $(\bar{p},\bar{m})$ that produce some unknown utility $\bar{u}$, we know that $e(\bar{p},\bar{u})=\bar{m}$. Then, varying $p_1$, we can integrate the above differential equation for $i=1$ to obtain $e(p_1,\bar{p}_2,\bar{u})$ for any $p_1$. And then we can get the Hicksian demand vector
$$h(p_1,\bar{p}_2,\bar{u})=D(p_1,\bar{p}_2,e(p_1,\bar{p}_2,\bar{u}))$$
for any $p_1$. </p>

<p>Since these Hicksian demands all correspond to the same utility $\bar{u}$, they are on the same indifference curve. By varying $p_1$, we will be able to trace out many different points on this indifference curve. In fact, if demand is sufficiently well-behaved, then we can trace out the entire indifference curve by varying $p_1$ enough in either direction. (By the way, "tracing indifference curves" is <em>all</em> we can do in any event: since the cardinality of utility is irrelevant to Marshallian demand, we can only retrieve ordinal properties like indifference curves and their ordering.)</p>

CC BY-SA 3.0
2014-12-18T19:55:28.390Z
2014-12-19T07:17:09.420Z
2014-12-19T07:17:09.420Z
nominally rigid
<p>In a two good world, will a marshallian demand function the likes of <code>D(p,m)</code> where p is the price of one good and m the income yield a utility function or indifference curve function? If so, how does one go about solving this? </p>

<microeconomics><utility><consumer-theory>
Is it possible to derive indifference curves given marshallian demand function?
Howard Black
# Is it possible to derive indifference curves given marshallian demand function?

(Asked by: username_0 on 2014-12-18)

In a two good world, will a marshallian demand function the likes of `D(p,m)` where p is the price of one good and m the income yield a utility function or indifference curve function? If so, how does one go about solving this?

**Tags:** microeconomics, utility, consumer-theory

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## 1 answers

### Answer by username_1 on 2014-12-18. Score: 11

Yes, under some conditions. This is the classic **integrability problem**: for detailed discussion, see some [excellent notes by Kim Border](http://people.hss.caltech.edu/~kcb/Notes/Demand4-Integrability.pdf).

Several other technical conditions are required, but the most economically substantive condition is that the [Slutsky matrix](http://en.wikipedia.org/wiki/Slutsky_equation) must always be symmetric and negative semidefinite. To be concrete, if we define the $ij$th element of the Slutsky matrix at $(p,m)$ to be
$$\sigma\_{ij}(p,m)=\frac{\partial D\_i(p,m)}{\partial p\_j}+D\_j(p,m)\frac{\partial D\_i(p,m)}{\partial m}$$
then we must have $\sigma\_{ij}(p,m)=\sigma\_{ji}(p,m)$ for all $(p,m)$, and also for any vector $v$ we must have for all $(p,m)$
$$\sum\_i \sum\_j \sigma\_{ij}(p,m)v\_iv\_j \leq 0$$
The *necessity* of these conditions follows immediately from basic consumer theory, which shows that if Marshallian demand is derived from constrained maximization of a utility function, then the Slutsky matrix is symmetric and negative semidefinite. But the *sufficiency* of these conditions (in conjunction with some other technical assumptions) for us to back out a utility function is a more complicated matter, and to get the details I recommend Border's notes or some other advanced micro source.

If, *assuming that the Slutsky conditions hold*, you want a rough practical way (ignoring the technical subtleties) to back out indifference curves in the typical two-good case, the simplest way is probably to use your knowledge of demand to determine the compensating change in expenditure that is necessary to adjust for a given change in prices. Specifically, for either $i=1,2$ use the identity
$$\frac{\partial e(p,u)}{\partial p\_i} = h\_i(p,u) = D\_i(p,e(p,u))$$
which, given knowledge of the Marshallian demand function $D$, is a differential equation in the expenditure function $e$. Starting with some initial values $(\bar{p},\bar{m})$ that produce some unknown utility $\bar{u}$, we know that $e(\bar{p},\bar{u})=\bar{m}$. Then, varying $p\_1$, we can integrate the above differential equation for $i=1$ to obtain $e(p\_1,\bar{p}\_2,\bar{u})$ for any $p\_1$. And then we can get the Hicksian demand vector
$$h(p\_1,\bar{p}\_2,\bar{u})=D(p\_1,\bar{p}\_2,e(p\_1,\bar{p}\_2,\bar{u}))$$
for any $p\_1$.

Since these Hicksian demands all correspond to the same utility $\bar{u}$, they are on the same indifference curve. By varying $p\_1$, we will be able to trace out many different points on this indifference curve. In fact, if demand is sufficiently well-behaved, then we can trace out the entire indifference curve by varying $p\_1$ enough in either direction. (By the way, "tracing indifference curves" is *all* we can do in any event: since the cardinality of utility is irrelevant to Marshallian demand, we can only retrieve ordinal properties like indifference curves and their ordering.)


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