ECON2101 - Tutorial 7 Week 8 - Production and
Costs
For this week's tutorial, let the production function for a firm be $F(K,L) = K^{1/3}L^{2/3}$.
The tutorial, broadly, is to work through the week's material for this particular production
function.
1. Does this firm exhibit increasing, diminishing, or constant marginal product in each
of its inputs?
2. Does this firm exhibit increasing, decreasing, or constant returns to scale?
3. Draw the isoquants for this firm.
4. What is the rate of technical substitution at $K = 6, L = 3$?
5. What is the rate of technical substitution at generic levels of capital and labour?
6. For $r = 1, w = 4$, and target quantity $Q = 12$, what is the cost-minimising level of
capital and labour (in the long-run). How much does this cost?
7. For generic $r, w$, and $Q$, what is this cost-minimising level of capital and labour
(in the long-run). What is the Long-Run Total Cost function? Graph this in Q-$
space.
a) Check that this function makes sense by thinking about increases in $r$ and $w$.
8. What are the (Long-run) Average Cost and Marginal Cost functions? Graph these.
9. Fix the input prices at $r = 1, w = 4$ and level of capital at $K_0 = 24/\sqrt{2}$. What is
the short-run optimal labour input for a given $Q$. What are the short-run total
cost, average cost, and marginal cost functions (for generic $Q$). Add the short-run
average cost and marginal cost with $K = 24/\sqrt{2}$ to your graph from part 8.
10. If demand for this firm's product falls, and they can now sell only 10 units of the
good (instead of 12), what would you advise this firm to do?
Homework: work through a similar exercise for $F(K, L) = K^{1/2}L^{1/2}$. Your results should
be a little neater, but the process follows the same pattern.
