Consider a family farm you inherit with the following production function:
$q = f(k,l) = k^{\frac{1}{2}}l^{\frac{1}{2}}$,
where $k$ and $l$ represent tractors/machines and labor/workers, respectively, and
$q$ is the quantity of corn produced. Your objective at the farm is maximize
profits:
$\pi = pq - [wl + vk]$,
where $w$ and $v$ are the rental prices of labor and machines, respectively, and
$p$ is the price of corn. Hence, you need to decide on the amount of corn to
produce and the amount and optimal mix of labor and machines to use. To
tackle this problem you first decide break it into manageable pieces. You fix
the target output to what was produced last year, say $q_0$ [Note that $q_0$ is
simply a number], and then ask: what are the mix of workers and machines
that would allow me to produce this at the very lowest cost? This problem
translates to:
subject to
$\min_{l,k} wl + vk$
$k^{\frac{1}{2}}l^{\frac{1}{2}} = q_0$
Show all workings!
1. Assume that $q_0 = 9$ (keep this assumption for all parts). Suppose further
that $w = v = 1$. Set up the Lagrangian and solve for the optimal amounts
of workers and machines that would give the lowest cost of production: $l^*$
and $k^*$. Use these amounts to find the actual (lowest) cost of producing
$q_0 = 9$.
2. Suppose further that wages increase so that $w = 9$ and the price of
machines stay the same $v = 1$. Set up the Lagrangian and solve for
the optimal amounts of workers and machines that would give the lowest
cost of production: $l^*$ and $k^*$. Use these new amounts to find the actual
(lowest) cost of producing $q_0 = 9$.
3. Compare the optimal amounts of $l^*$ and $k^*$ for the two cases in parts 1
and 2. How do things change? What's the intuition for this?
