2. Thor runs a business called Heavy Hammers, which produces some of the heaviest
hammers you've ever seen! He uses two inputs in the production of hammers: labour,
denoted by L, and capital, denoted by K. Heavy Hammers produces hammers, denoted
by H, according to the following production technology: $H = f(L,K) = L^{\frac{1}{2}}K^{\frac{1}{2}}$. Thor
must pay each unit of labour a wage rate $w$ and is able lease his capital equipment at
$r$ dollars per unit. The market price of hammers is $4.
(a) Write down Thor's profit maximization problem.
(b) Write down and equation that sets the marginal product of labour equal to the
wage rate of labour, $w$. Then write down and equation that sets the marginal
product of capital equal to the lease rate, $r$.
(c) Solve the two equations from part (b) for the two unknown inputs $L$ and $K$. Your
solutions give the profit-maximizing choices of labour and capital as functions of
the wage and lease rates, $w$ and $r$.
(d) If the wage is 2, and the lease rate is 1, how many units of labour and capital will
Heavy Hammers employ? How much output will it produce? How much profit
will it make?
