Cheat Sheet: This is what you need to know!!
Cost minimization:
max − r k − w L(k, q) (L1)
k
How does k change with r? Own-price effects negative!
Verbal logic: when the rental price of capital rises, each unit of capital is more expensive,
so the choice of capital must fall.
Formal logic: The objective in (L1) satisfies increasing differences in (k,−r). So capital
is nonincreasing in its own price.
Notice the verbal logic applies for anything which makes increasing capital by a unit
more expensive: likewise, so does the formal logic.
How does k change with q? When is capital a normal input?
Verbal logic: Consider the rate of technical substitution, which is the rate at which labor
can be decreased in response to a unit increase in capital, while still producing q. If
higher quantities lead to a situation where a unit increase in capital allows a savings of
more labor than before, then the firm wants to substitute capital for labor when q
increases. Then, capital increases with q.
Formal logic: The objective function in (L1) satisfies increasing differences in (k,q) if
−L(k,q) satisfies increasing differences in (k,q). We can interpret this directly as a rate of
technical substitution (try it!). If L is differentiable, we know that
−Lk(k,q)=Fk(k,l)/Fl(k,l) l = L(k,q) , which is nondecreasing in q if the rate of technical
substitution increases as we move to higher isoquants (higher q).
Two step problem: minimize costs and then maximize profits.
l ** (x, w) = l (∃k) (k,l) ∈ arg min rk + wl .
k, l s.t . F(k,l )= x
The second step is:
max px − C(x,w)
x
When does the choice of quantity (x) decrease with w? If labor is a normal input!
Verbal logic: Higher wages lead to higher marginal costs of output (and thus lower
choices of output) exactly when labor is a normal input. That is, if producing more
output leads you to use more labor, then higher wages make it more expensive to produce
output.
Formal logic: px − C(x,w) has decreasing differences in (x,w) when C(x,w) has
increasing differences. Since the marginal effect of w on C(x,w) is l**(x,w) by the
envelope theorem, w increases the marginal cost of output when l**(x,w) is increasing in
x.
Profit Maximization (all in 1 step):
max π (k,l;r, w) ≡ pF(k,l) − rk − wl (L3)
k, l
We break the firm’s maximization problem into two stages, as follows:
l
{
lˆ(k;r, w) = sup arg maxπ (k,l;r,w)
l
} (L4)
k
{ k
}
kˆ( r, w) = sup arg max π ( k, lˆ (k ; r, w); r, w) (L5)
Fact: l(k ; r, w ) l(k ; w ) , since capital is fixed when l is chosen.
Own price effects:
Verbal logic: An increase in r makes each unit of k more expensive, so the firm chooses
less k.
Formal logic: Choice of capital solves max pF(k, lˆ(k;w)) − rk − wlˆ(k;w) . This satisfies
k
increasing differences in (k,−r). Thus k is nonincreasing in r.
Cross price effects:
Definition: k and l are complements if F(k,l) satisfies increasing differences in (k,l).
Substitutes if F(k,l) satisfies decreasing differences in (k,l).
Verbal logic: An increase in r always leads to a (weak) decrease in k. But if capital and
labor are complements, a decrease in k always leads to a decrease in l, since capital
increases the incremental returns to labor. Thus, when capital and labor are
complements, an increase in r will lead to a (weak) decrease in both k and l.
In contrast, when capital and labor are substitutes, a decrease in k leads to a weak
increase in l, and so capital and labor will move in opposite directions. Thus, an increase
in r will lead to a (weak) decrease in k and a (weak) increase in l.
Formal logic: Applying the comparative statics theorems, if F(k,l) satisfies increasing
differences, then lˆ(k; w) is nondecreasing in k; if F(k,l) satisfies decreasing differences,
(;, ) lkw
then lkrw (; ) is nonincreasing in k.
Changes in input prices on maximum profits:
Verbal logic: When the price of capital goes up by a small amount, a maximizing firm’s
profits change according to the amount of capital which was currently in use, since given
a smooth profit function, the adjustments to capital and labor in response to a small price
change are second order.
Formal logic: The objective is differentiable in r. If in addition, the production function
is continuous in k, the envelope theorem tells us that ∂∂r π *(r,w) exists almost everywhere,
and further that ∂∂r π * (r,w) = − kˆ (r,w) where it exists.
LeChatlier
Consider the profit maximization problem above.
Verbal logic (case of complements): In the short run, an increase in the wage leads to a
decrease in the choice of labor. However, labor is chosen at the old (long-run) choice of
capital. In the long run, capital can adjust, and since capital and labor are complements,
the higher wage will lead to lower levels of both capital and labor. When comparing the
short and long run choices of labor, note that the short run decision was made at a higher
level of capital: since they are complements, the optimal choice of labor was higher in the
short run. Thus, labor adjusts (goes down) by more in the long run than in the short run.
Formal logic: By the comparative statics theorems, if F has increasing differences, our
above results tell us that: l(k ; w ) is nondecreasing in k and nonincreasing in w and, k(w )
is nonincreasing. Thus, for all w>w0, k(w0 ) ! k( w ) , and l(k(w0 ); w ) l(k(w ); w ) = lLR(w).
Verbal logic (case of substitutes): In the short run, an increase in the wage leads to a
decrease in the choice of labor. However, labor is chosen at the old (long-run) choice of
capital. In the long run, capital can adjust, and since capital and labor are substitutes, the
higher wage will lead to lower levels of labor and higher levels of capital. When
comparing the short and long run choices of labor, note that the short run decision was
made at a lower level of capital: since they are substitutes, the optimal choice of labor
was higher in the short run. Thus, labor adjusts (goes down) by more in the long run
than in the short run.
Formal logic: By the comparative statics theorems, if F has increasing differences, our
above results tell us that: l(k ; w ) is nonincreasing in both arguments and, k(w ) is
nondecreasing. Thus, for all w>w0, kˆ(w0 ) < kˆ(w) , and l(k(w0 ); w ) l(k(w ); w ) = lLR(w).
Finally: note that without the complements or substitutes assumption, a discrete change
in wneed not lead to a larger long run than short run adjustment. In our counterexample,
an increase in the price of oil caused the firm to shut down in the short run, but in the
long run new fuel-efficient capital allowed the firm to operate using less oil.
Good luck -- don’t stress!!
