$\backslash [$

Y$\_\{$i$\}=$A$\_\{$i$\}$ L$\_\{$i$\}^\{\backslash$gamma$\}$

$\backslash ]$

Suppose we have the production function above. Firm $1$ has $\backslash ($ A$=1\backslash ),$ while firm $2$ has $\backslash ($ A$=2\backslash ).$ Both firms have $\backslash (\backslash$mathrm$\{$p$\}=1\backslash ).$ Suppose the parameter gamma is $0\mathrm{.}5$ and $\backslash ($ w$=1\backslash ).$

$7.$ Solve for the labour demanded by each firm. $[1$pt$]$

$8.$ If there are $10$ total units of labour in the economy, solve for the labour allocation across firms that an output maximizing social planner would choose, the resulting level of output, and the total factor productivity for the economy $($measured in output per worker$).[1$pt$]$

$9.$ Repeat the exercise above only this time assume that the social planner allocates an equal amount of labour to each firm. $[1$pt$]$

$10.$ What wage subsidy would you have to give to firm $1$ to achieve the outcome in part $9$ in an environment with profit$-$maximizing firms? $[1$pt$]$

$11.$ Suppose we did not know the value of the parameter gamma and wanted to estimate it$.$ We have data on the output of each firm and the labour input of each firm. Provide a linear regression equation you could use to estimate this parameter $($hint: take logs of the labour demand equation in $7)$ and discuss the direction of bias if you estimated this by OLS for a cross$-$section of firms with different productivity parameters. $[1$pt$]$

$12.$ Bonus: Using a method we discussed in the class $($other than an RCT$)$ how could you get an unbiased estimate of this parameter?