OLS bias in demand estimation: the bias always underestimate the demand's elasticity? Some papers argue that OLS can produce less bias than IV estimation depending on the quality of your instruments. Suppose we consider a demand estimation equation. 

Suppose the demand elasticity is negative in OLS. By my intuition weak instruments should produce biased estimates towards OLS, but no less negative. Can you guys produce an example? I cannot really grasp how it could lead to a more biased estimation with IV estimation.
Usually, $\hat{\beta_1^{IV}} = \beta_1 + \frac{cov(z,u)}{cov(z,x)}$.  The denominator will go to zero. 

That is true unless there is some correlation between the instrument and the error term, and the nominator is the strength of the relationship between the instrument and the endogenous variable. The smaller the denominator gets, the greater the bias $\left[\frac{cov(z,u)}{cov(z,x)}\right]$.

In addition,  weak instrument will have no precision, so that the variance will have a big upward bias. 
\begin{eqnarray} var(\hat{\beta_1}) &amp;\to_p&amp; \frac{\sigma^2}{n \sigma^2_x} \nonumber \\
\hat{\beta_1^{IV}} &amp;=&amp; \frac{\sum (z_i - \bar{z})y_i}{\sum(z_i - \bar{z})x_i} = \beta_1 + \frac{\sum (z_i - \bar{z})u_i}{\sum(z_i - \bar{z})x_i} \nonumber \\
var(\hat{\beta_1^{IV}} &amp;=&amp; var \left( \frac{\sum (z_i - \bar{z})u_i}{\sum(z_i - \bar{z})x_i} \right) \nonumber\\
var(u | z) &amp;=&amp; \sigma^2 \nonumber\\
var(\hat{\beta_1^{IV}}) &amp;=&amp; \frac{\sigma^2 \frac{1}{n} \sum (z_i - \bar{z})}{n[ \frac{1}{n} \sum (z_i - \bar{z})(x_i - \bar{x})]^2} \nonumber
\end{eqnarray}

As $n \to \inf$
\begin{eqnarray} var(\hat{\beta_1^{IV}}) &amp;\to_p&amp; \frac{\sigma^2 \sigma_z^2 }{\sigma^2_{zx}} \nonumber\\
var(\hat{\beta_1^{IV}}) &amp;\to_p&amp;  \sigma^2 \frac{1}{n \sigma^2_x} \frac{1}{\rho^2_{xz}}\nonumber \\
\rho^2_{xz} &amp;=&amp; \frac{[\sigma^2 _{xz}]^2}{\sigma_x^2 \sigma_z^2} \textit{for} \rho \in [0,1]\nonumber
\end{eqnarray}

That is why if your instrument is weak, then you  may be better off running an OLS regression.