Ari runs a party venue where he employs DJs (L, expressed in DJ-hours) and karaoke machines (K, expressed
in machine-hours). The amount of partying he can host can be expressed as
q =K^1/3L^1/3
He currently owns 8 karaoke machines and currently employs 4 DJs. Each machine cost B when it was
initially purchased, but Ari does not need to pay any more to use it at a party. There is also a rental market
for karaoke machines, which can be rented on the market for r per day. However, the equipment is dicult
to transport, and it takes a month to move between venues. DJs can also be hired for a daily wage of w,
and can be hired by the day (or portion of the day).
(For this problem, do not worry about limiting your answers to whole numbers.)

(a) Describe Ari's accounting costs, opportunity costs. Next, describe Ari's variable costs and sunk costs.
Which are relevant for his production decision? Explain why the cost of karaoke machines B doesn't
factor into prot maximization.
(b) What time frame describes the long run? Write Ari's short-run production function.
(c) In the short-run, what is the xed cost of production? (Hint: What is K in the short run?)
(d) In the short run, what is the variable cost of production? (Hint: How much L is needed to produce a
given amount of q? At what price is L purchased?)
(e) Compute the average total cost and average variable cost curves when L is the only variable factor.
(f) Show that short-run average variable cost is equal to the inverse of the average product of labor times
the wage rate. (Hint: First, compute the average product of DJ-hours. Then, replace q with a function
of L in the short-run average variable cost expression and rearrange terms.)
(g) Compute the marginal cost of production when L is the only variable factor.

(h) How does the short-run marginal cost relate to the short-run marginal product of labor? (You do not
need to show it mathematically, but it is useful practice for the exam.)
(i) On a single graph, plot the short-run marginal cost, average total cost, and average variable cost when
when L is the only variable factor and when w = 3 and r = 5. Draw the curves to scale; you may need
to use a calculator or website to do so.
(j) Write Ari's long-run production function.
(k) Write an expression for a long-run isoquant curve generally in terms of w, r, and some output level q.
(l) Set up Ari's cost minimization problem as a Lagrangian. That is, write the Lagrangian that corresponds
to Ari minimizing the cost of production subject to producing a xed amount q.
(m) Solve the cost-minimization problem to compute K and L as a function of w, r, and q.
(n) Multiply the optimal levels of K and L by their respective prices to compute the long-run cost curve.
(o) On a new graph, plot the long-run average cost when both factors are variable and when w = 3 and
r = 5. On the same graph, plot the short-run average total cost curve, when only L is variable, at the
same prices when K
is 5, 8, and 11. Draw the curves to scale; you may need to use a calculator or
website to do so.
(p) Explain the relationship between the short-run and long-run average cost functions you plotted above.