The Long Run

Christopher Makler

Stanford University Department of Economics

Econ 50 : Lecture 18

pollev.com/chrismakler

What kind of problem is the condition

\(MRTS = {w \over r}\) used in?

Part I: The Firm in the Long Run

Part II: Long-Run Competitive Equilibrium

  • When do firms choose to leave or enter an industry?
  • What is the effect of entry and exit on the industry?
  • How does the firm cost minimize if it can freely adjust both labor and capital?
  • How are its long-run cost curves different than its short-run cost curves?
  • How is a competitive firm's long-run supply curve different than its short-run supply curve?

Cost Minimization

Cost Minimization Subject to a Utility Constraint

Cost Minimization Subject to an Output Constraint

\min p_1x_1 + p_2x_2 \text{ s.t. } u(x_1,x_2) = U
\min wL + rK \text{ s.t. } f(L,K) = q
\text{solutions}: x_1^c(p_1,p_2,U), x_2^c(p_1,p_2,U)
\text{solutions}: L^c(w,r,q), K^c(w,r,q)

Hicksian Demand

Conditional Demand

Cost Minimization: Lagrange Method

\mathcal{L}(L,K,\lambda)=
wL + rK +
(q - f(L,K))
\lambda
\frac{\partial \mathcal{L}}{\partial L} = w - \lambda MP_L

First Order Conditions

\frac{\partial \mathcal{L}}{\partial K} = r - \lambda MP_K
\frac{\partial \mathcal{L}}{\partial \lambda} = q - f(L,K) = 0 \Rightarrow q = f(L,K)
= 0 \Rightarrow \lambda = w \times {1 \over MP_L}
= 0 \Rightarrow \lambda = r \times \frac{1}{MP_K}

MRTS (slope of isoquant) is equal to the price ratio

\text{Also: }\frac{\partial \mathcal{L}}{\partial q} = \lambda = \text{Marginal cost of producing last unit using either input}
\text{Set two expressions }
\text{for }\lambda \text{ equal:}
\frac{MP_L}{MP_K} = \frac{w}{r}
f(L,K) = L^{1 \over 4}K^{1 \over 4}

Tangency condition: \(MRTS = w/r\)

Constraint: \(q = f(L,K)\)

MRTS = {MP_L \over MP_K} =
= {K \over L}

Conditional demands for labor and capital:

K = {w \over r}L
q = L^{1 \over 4}\left({w \over r}L\right)^{1 \over 4}
\left({r \over w}\right)^{1 \over 4}q = L^{1 \over 2}
L^c(w,r,q) = \sqrt{r \over w} q^2
K^c(w,r,q) = \sqrt{w \over r} q^2
f(L,K) = L^{1 \over 4}K^{1 \over 4}

 Long Run (can vary both labor and capital)

L^c(q) = \sqrt{\frac{r}{w}}q^2
K^c(q) = \sqrt{\frac{w}{r}}q^2
TC^{LR}(q) = wL^c(q) + rK^c(q)
=2\sqrt{wr}q^2

 Short Run with Capital Fixed at \(\overline K \)

L^c(q | \overline K) = {q^2 \over \overline K}
TC(q) = wL^c(q | \overline K) + r \overline K
={wq^4 \over \overline K} + r \overline K

Relationship between
Short-Run and Long-Run Costs

TC(q | \overline{K}) = \text{Cost if capital is fixed at }\overline{K}
TC(q,K^c(q)) = \text{Cost if capital is fixed at optimal }K \text{ for producing }q

What conclusions can we draw from this?

TC(q,\overline{K}) \ge TC^{LR}(q)
= TC^{LR}(q)
TC^{LR}(q) \text{ is the lower envelope of the family of }TC(q)\text{ curves}

Long Run Supply

In the long run, the firm can adjust its inputs more freely.

This additional flexibility makes its supply curve more elastic.

TC^{LR}(w,r,q) = 2\left(\sqrt{wr}\right)q^2

LONG RUN

SHORT RUN

f(L,K) = L^\frac{1}{4} K^\frac{1}{4}
\pi(q) = pq - 2\left(\sqrt{wr}\right)q^2
\pi(q) = pq - \left[w\frac{q^4}{\overline{K}} + r\overline{K}\right]
\pi'(q) = p - 4\left(\sqrt{wr}\right)q = 0
\pi'(q) = p - 4w\frac{q^3}{\overline{K}} = 0
q^*(w,r,p) = \frac{p}{4\sqrt{wr}}
TC(w,r,q) = w\frac{q^4}{\overline{K}} + r\overline{K}
q^*(w,r,p) = \left(\frac{\overline K p}{4w}\right)^\frac{1}{3}
\text{Sanity check: how do these depend on }w, r, p,\overline K?
\text{Step 2. Take the derivative, set it equal to zero, and solve for the optimal choice, }q^*:
\text{Step 1. Write down the expression of profit as a function of the choice variable, }q:

Supply Elasticities

LONG RUN

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
K^c(w,r,q) = w^{1 \over 2}r^{-{1 \over 2}}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

SHORT RUN

L^c(q|\overline K) = {1 \over \overline K}q^4
q^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

LONG RUN

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2

SHORT RUN

L^c(q|\overline K) = {1 \over \overline K}q^4

What is the output elasticity of conditional labor demand in the short run and long run?

Intuitively, why this difference?

\epsilon_{L^c,q}=
\epsilon_{L^c,q}=
2
4

In the long run, the firm uses
both labor and capital to increase output;
in the short run, it only increases labor.

LONG RUN

q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

SHORT RUN

q^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

What is the price elasticity of supply
 in the long run and short run?

Intuitively, why this difference?

\epsilon_{q,p}=
\epsilon_{q_s,p}=
1
{1 \over 3}

In the long run, the firm can adjust its capital to keep its costs down; so its marginal cost rises less steeply, and its supply curve is flatter.

Supply Elasticities

LONG RUN

Work with a partner:
How would the firm respond to a
6% increase in the wage rate in the long run?

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
K^c(w,r,q) = w^{1 \over 2}r^{-{1 \over 2}}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

pollev.com/chrismakler

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
K^c(w,r,q) = w^{1 \over 2}r^{-{1 \over 2}}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

How would the firm respond to a
6% increase in the wage rate in the long run?

A 6% increase in w decreases q by 3%.

and decreases by 6% (due to the -3% change in q),
for a total decrease of 9%.

K increases by 3% (due to the +6% change in w)

\epsilon_{L^c,w}=-{1 \over 2}
\epsilon_{L^c,q}=+2
\epsilon_{q,w}=-{1 \over 2}

L decreases by 3% (due to the +6% change in w)

\epsilon_{K^c,w}=+{1 \over 2}
\epsilon_{K^c,q}=+2

and decreases by 6% (due to the -3% change in q),
for a total decrease of 3%.

L^c(w,r,q) = w^{-{1 \over 2}}r^{1 \over 2}q^2
q^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

How would the firm respond to a
6% increase in the wage rate in the long run?

A 6% increase in w decreases q by 3%.

and decreases by 6% (due to the -3% change in q),
for a total decrease of 9%.

L decreases by 3% (due to the +6% change in w)

L^*(w,r,p) = w^{-{1 \over 2}}r^{1 \over 2}[y^*(w,r,p)]^2
= w^{-{1 \over 2}}r^{1 \over 2}\left[{1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p\right]^2
= {1 \over 16}w^{-{3 \over 2}}r^{-{1 \over 2}}p^2

Note: we can calculate the LR profit-maximizing demand for labor:

\epsilon_{L^c,w}=-{3 \over 2}

What did we just show?

  • If there is a direct causality \(X \rightarrow Y\), elasticity measures how Y responds to X.
  • If there is a chain of causality \(X \rightarrow Y \rightarrow Z\), the elasticity composes just like a function does (like the chain rule for elasticity)

Long-Run Competitive Equilibrium

The "Zero Profit Condition"

\pi_1 = p_1q_1^* - wL_1^*-rK_1^*

Profits in industry 1 when profit maximizing

Profits in industry 2 when profit maximizing

A firm in industry 1 should remain in industry 1 as long as

\pi_1 > \pi_2
p_1q_1^* - wL_1^*-rK_1^* > \pi_2
p_1q_1^* - wL_1^*-rK_1^*- \pi(q_2^*)>0

"Positive economic profit"

SR fixed costs

\pi_2 = p_2q_2^* - wL_2^*-rK_2^*

LR fixed costs

The Effect of Entry and Exit

Industry Short Run:

Number of Firms is Fixed

Industry Long Run:

Firms will enter an industry with positive economic profits; firms will leave an industry with negative economic profits.

In long-run competitive equilibrium, firms in all industries make nonnegative economic profit.

Definition: Minimum Efficient Scale

A firm's minimum efficient scale (MES) is the quantity at which average cost is the lowest.

If MC is increasing, this coincides with the quantity at which MC = AC. 

Increasing, Decreasing, and Constant Cost Industries

Market Supply Curve:

Quantity supplied by firms at every possible price

Industry "Supply Curve":

Locus of (quantity, price) combinations that could arise in long-run competitive equilibrium, given different demand conditions.

Market Supply and Demand

Typical Firm's Cost Curves

MC

y

$ perunit

P

Q

S

1. demand
increases

D'

D

AC

What is the effect of an increase in demand if costs are unaffected by the number of firms?

S'

3. firms

enter

\(S_{LR}\)

Market Supply and Demand

Typical Firm's Cost Curves

MC

y

$ perunit

P

Q

S

demand
increases

D'

D

AC

What is the effect of an increase in demand if costs decrease as firms enter?

S'

firms enter,
costs decrease

\(S_{LR}\)

MC'

AC'

firms enter,
costs decrease

How to Solve for Long-Run Equilibrium

\text{Ind. firm cost }c(q)
\text{Market demand }D(p)
  1. Find marginal cost function, \(MC(q) = c'(q)\).
  2. Set \(p = MC(q)\) and solve for the individual firm supply function, \(q^*(p)\).
  3. Multiply \(q^*(p)\) by the number of firms \(N_F\) to get the market supply curve \(S(p, N_F)\).
  4. Set \(D(p) = S(p,N_F)\) and solve for \(p\) to find the equilibrium market price as a function of \(N_F\)

Part I: Solve for the short-run equilibrium price as a function of the number of firms.

Part II: Find the long-run equilibrium price and solve for the equilibrium number of firms.

\text{Givens: }
  1. Find the average cost function, \(AC(q) = c(q)/q\).
  2. Set \(AC(q) = MC(q)\) and solve for the minimum efficient scale (MES).
  3. The LR price is the MC and AC at the MES; plug this into step 4 above and solve for \(N_F\)!
c(q) = 64 + \tfrac{1}{4}q^2
D(p) = 384 - 24p

What happens to inputs as more firms enter an industry?

Industry too small to affect price of inputs

Inputs get cheaper/faster/better

Inputs are scarce, command higher prices

Story

Industry Type

Industry Supply Curve

"Constant Cost Industry"

Horizontal

"Decreasing Cost Industry"

Downward Sloping

"Increasing Cost Industry"

Upward Sloping

Most important takeaways

Firms optimize by setting MR = MC

Entry and exit forces AR = AC

Constant cost industry = one price in the long run

pollev.com/chrismakler

What does the equals sign in the condition

\(AR = AC\) represent?

What did we learn today?

The long run means more flexibility.

You can shift your production process, or even leave the industry if you want to.

Firms chasing profits will move resources to take advantage of every opportunity. Some of those resources can be moved faster than others.

Macro is going to be all about responses to shocks.