$27.$ The demand for tea is given by the demand function $\backslash ($ Q$\_\{$d$\}=120-$p $\backslash ).$ Tea is supplied by two price$-$taking firms in a perfectly competitive market: an Indian firm and a British firm. Each firm has a cost function given by $\backslash ($ c$($q$)=\backslash$frac$\{1\}\{2\}$ q$^\{2\}\backslash ),$ where $\backslash ($ q $\backslash )$ is the firm's own output level.

$($a$)$ Suppose the government imposes a tax of $\backslash ($ t $\backslash )$ per unit on both the firms. Calculate equilibrium market price and quantity as functions of $\backslash ($ t $\backslash ).$

$($b$)$ What value of $\backslash ($ t $\backslash )$ maximizes national social surplus $($sum of consumers' surplus, profits of the Indian firm only and government tax revenue$)?$

$($c$)$ What value of $\backslash ($ t $\backslash )$ maximizes global social surplus $($sum of consumers' surplus, profits of both firms and government tax revenue$)?$

$($d$)$ Suppose WTO rules prohibit the government from imposing any taxes or subsidies on the foreign $($British$)$ firm. However, it is free to provide a per unit subsidy $\backslash ($ s $\backslash )$ to the domestic $($Indian$)$ firm fully funded by a lump sum tax $\backslash ($ T $\backslash )$ on the same firm $($i$.$e$.,$ the total subsidy payment must be equal to the lump sum tax$).$ Find the optimum $\backslash (($s$,$ T$)\backslash )$ that maximizes national social surplus $($sum of consumers' surplus and the domestic firm's profits$).$