Why is the conditional mean of the reduced form error zero? For example, we have a simultaneous equation model of supply and demand:
Supply: $$s(p)=\alpha_{s}+\beta_{s}p+\epsilon_{s}$$
Demand: $$d(p)=\alpha_{d}-\beta_{d}p+\epsilon_{d}$$
Market clearing condition: $$q = d(p) = s(p)$$
It is obvious that $E(\epsilon_{s}|p) \neq 0$ and $E(\epsilon_{s}|p) \neq 0$, as price $p$ is endogenous.
The reduced form of this model is:
$$p = \frac{\alpha_{d}-\alpha_{s}}{\beta_{d}+\beta_{s}}+\frac{\epsilon_{d}-\epsilon_{s}}{\beta_{d}+\beta_{s}}$$
$$q = \frac{\alpha_{d}\beta_{s}+\alpha_{s}\beta_{d}}{\beta_{d}+\beta_{s}}+\frac{\beta_{d}\epsilon_{s}+\beta_{s}\epsilon_{d}}{\beta_{d}+\beta_{s}}$$
It follows that the reduced form errors are linear combinations of the structural errors. My question is, how can one derive that the conditional means of the reduced form errors, given the constants, are zero?
 A: Of course they are not. You write
$$p = c + h(\epsilon_{d},\epsilon_{s})$$
where $h(\epsilon_{d},\epsilon_{s})$ is (supposed) to be the "reduced form error".
How could it be that $E[ h(\epsilon_{d},\epsilon_{s})\mid p] = E[ h(\epsilon_{d},\epsilon_{s})\mid c + h(\epsilon_{d},\epsilon_{s})] =0?$
Moreover, what you name "reduced form of the model" is the theoretical solution of a reduced form after incorporating the market clearing condition (this reduced form being your first two equations where we replace $s(p)$ and $d(p)$ by $q$). So what you call "reduced form errors", really are not. Note that this solution is not estimable, since the RHS contains only unobservables. Assume now that you specify a regression
$$q=\gamma_0+\gamma_1p+u$$
which is an attempt to formulate a reduced-form model. But if you insert the solution for $p$ into the solution for $q$, you obtain
$$q = \frac{\alpha_{d}\beta_{s}+\alpha_{s}\beta_{d}}{\beta_{d}+\beta_{s}}+\frac{\beta_{d}\epsilon_{s}+\beta_{s}\epsilon_{d}}{\alpha_{d}-\alpha_{s}+\epsilon_{d}-\epsilon_{s}}p$$
$$\Rightarrow q = \gamma_0 + \gamma_1(\epsilon_{d},\epsilon_{s})\cdot p$$ 
This last equation, which is the "final" reduced form of the model, has no additive "error term" and this makes the inclusion of the reduced-form error $u$ shaky on theoretical grounds-what kind of error/disturbance does it represent, given the theoretical model? All such distrubances are captured by the two structural errors. So postulating the existence of some reduced form error $u$ is a misspecification.
The last equation tells us that the coefficient of price that reflects its effect on quantity is a non-linear function of the structural disturbances (without even containing a constant term around which it fluctuates due to the structural errors) : the reduced form model is one of stochastic and time-varying coefficients, and not of the variety for which some estimation approaches have been developed (that usually assume that the coefficient is the sum of a constant term and a time-varying stochastic term).
And this is why we approach the problem through the Instrumental Variables (IV) method (of which this example is the standard econometrics textbook one to introduce IV estimation), in which essentially we will eventually run a regression without using $p$ as a regressor.
