Multiple Equilibria – Theory “Big Push” model

Advanced – Recitation, Week #5

Endogenous Growth Theory [2/2]

Claire Palandri [email protected]

February 17, 2021

1 / 12 Multiple Equilibria – Theory “Big Push” model Outline

Multiple Equilibria

1 Theory

2 “Big Push” model

2 / 12 Multiple Equilibria – Theory “Big Push” model Multiple Equilibria

Main idea of endogenous growth literature: having increasing returns to a given factor of production means that the is no longer strictly concave. ⇒ the sf (k) curve and the (δ + n)k curve will intersect in more than one point. ⇒ There may be multiple steady-states (= equilibria).

1 Causes of multiple equilibria: They may arise through network complementarities: situations where: The more people are in the network, the higher the payoff from joining the network. (“Network” = undertaking a specific action. E.g., acquiring a QWERTY keyboard.)

2 Consequences of multiple equilibria: History matters: where you start out affects where you end up (not the case in Solow model). Poor countries may get stuck in a poverty trap (a bad equilibrium). Potential rationale for government intervention: a one-time intervention can give the economy a jump from a low-level steady state into the basin of attraction of a higher-level equilibrium. “Big Push”

3 / 12 Multiple Equilibria – Theory “Big Push” model Multiple Equilibria

Analysis: find all the steady-states (= equilibria) identify which are stable vs. unstable, and their basin of attraction

4 / 12 Multiple Equilibria – Theory “Big Push” model Outline

Multiple Equilibria

1 Theory

2 “Big Push” model

5 / 12 Multiple Equilibria – Theory “Big Push” model Big Push model

How do firms decide when to upgrade their production technology?

Set-up: J symmetric firms. Each firm j:

employs Lj labor produces Dj , and this supply meets demand can decide between two technologies: * traditional technology that produces 1 unit of with 1 unit of labor; no fixed cost * modern technology that produces 1 unit of output with µ units of labor (0 < µ < 1); initial fixed cost of F m ≡ endogenous number of firms that choose the modern technology ( ⇐⇒ J − m keep the traditional technology)

total labor force: L = L1 + L2 + L3 + ... + LJ all goods have price p = 1, and workers earn wage w = 1.

6 / 12 pDj − wLj ratio of output to labor is 1 to 1 → need Dj workers to produce Dj units of output → Lj = Dj T =⇒ π = 1 × Dj − 1 × Dj = 0 M Modern technology: πj = pDj − wLj − F ratio of output to labor is 1 to µ, → need µ × Dj workers to produce Dj units of output → Lj = µDj ? M =⇒ π = Dj − µDj − F = (1 − µ)Dj − F S 0

2. Aggregate economy Assume that all income (wages & profits) + fixed in the modern technology is spent in equal proportion on the J goods: J X Dj = (aggregate income + investments) j=1

 T M  J × Dj = profits + profits + wages + investments

1  T M  Dj = J (J − m)π + mπ + L + mF 1 M = J (mπ + L + mF )

Multiple Equilibria – Theory “Big Push” model Big Push model

1. Behavior of each firm j The firm chooses the technology that maximizes its profit = Tot Revenue - Tot Cost. T Traditional technology: πj =

7 / 12 pDj − wLj − F ratio of output to labor is 1 to µ, → need µ × Dj workers to produce Dj units of output → Lj = µDj ? M =⇒ π = Dj − µDj − F = (1 − µ)Dj − F S 0

2. Aggregate economy Assume that all income (wages & profits) + fixed investments in the modern technology is spent in equal proportion on the J goods: J X Dj = (aggregate income + investments) j=1

 T M  J × Dj = profits + profits + wages + investments

1  T M  Dj = J (J − m)π + mπ + L + mF 1 M = J (mπ + L + mF )

Multiple Equilibria – Theory “Big Push” model Big Push model

1. Behavior of each firm j The firm chooses the technology that maximizes its profit = Tot Revenue - Tot Cost. T Traditional technology: πj = pDj − wLj ratio of output to labor is 1 to 1 → need Dj workers to produce Dj units of output → Lj = Dj T =⇒ π = 1 × Dj − 1 × Dj = 0 M Modern technology: πj =

7 / 12   profitsT + profitsM + wages + investments

1  T M  Dj = J (J − m)π + mπ + L + mF 1 M = J (mπ + L + mF )

Multiple Equilibria – Theory “Big Push” model Big Push model

1. Behavior of each firm j The firm chooses the technology that maximizes its profit = Tot Revenue - Tot Cost. T Traditional technology: πj = pDj − wLj ratio of output to labor is 1 to 1 → need Dj workers to produce Dj units of output → Lj = Dj T =⇒ π = 1 × Dj − 1 × Dj = 0 M Modern technology: πj = pDj − wLj − F ratio of output to labor is 1 to µ, → need µ × Dj workers to produce Dj units of output → Lj = µDj ? M =⇒ π = Dj − µDj − F = (1 − µ)Dj − F S 0

2. Aggregate economy Assume that all income (wages & profits) + fixed investments in the modern technology is spent in equal proportion on the J goods: J X Dj = (aggregate income + investments) j=1

J × Dj =

7 / 12 Multiple Equilibria – Theory “Big Push” model Big Push model

1. Behavior of each firm j The firm chooses the technology that maximizes its profit = Tot Revenue - Tot Cost. T Traditional technology: πj = pDj − wLj ratio of output to labor is 1 to 1 → need Dj workers to produce Dj units of output → Lj = Dj T =⇒ π = 1 × Dj − 1 × Dj = 0 M Modern technology: πj = pDj − wLj − F ratio of output to labor is 1 to µ, → need µ × Dj workers to produce Dj units of output → Lj = µDj ? M =⇒ π = Dj − µDj − F = (1 − µ)Dj − F S 0

2. Aggregate economy Assume that all income (wages & profits) + fixed investments in the modern technology is spent in equal proportion on the J goods: J X Dj = (aggregate income + investments) j=1

 T M  J × Dj = profits + profits + wages + investments

1  T M  Dj = J (J − m)π + mπ + L + mF 1 M = J (mπ + L + mF )

7 / 12 M 1 M ⇐⇒ π = (1 − µ) J (mπ + L + mF ) − F M 1 M 1 ⇐⇒ π = (1 − µ) J mπ + (1 − µ) J (L + mF ) − F M  1  1 ⇐⇒ π 1 − (1 − µ) J m = (1 − µ) J (L + mF ) − F ⇐⇒ πM [J − (1 − µ)m] = (1 − µ)(L + mF ) − JF (1 − µ)(L + mF ) − JF ⇐⇒ πM = J − (1 − µ)m

Network complementarities: the profit a firm will get if it adopts the modern technology is a function of m: it depends on how many firms have already adopted it.

∂πM (m) 1,2 Is it increasing/decreasing/saturating... in m? → check the sign of ∂m .

Multiple Equilibria – Theory “Big Push” model Big Push model

Solve for πM : ( M π = (1 − µ)Dj − F 1 M Dj = J (mπ + L + mF )

1 ∂πM M ∂πM M ∂m < 0 would mean π is ↓ in m, ∂m > 0 would mean π is ↑ in m. 2 numerator Or notice that numerator ↑ in m, denominator ↓ in m, =⇒ denominator ↑ in m. 8 / 12 ∂πM (m) 1,2 Is it increasing/decreasing/saturating... in m? → check the sign of ∂m .

Multiple Equilibria – Theory “Big Push” model Big Push model

Solve for πM : ( M π = (1 − µ)Dj − F 1 M Dj = J (mπ + L + mF )

M 1 M ⇐⇒ π = (1 − µ) J (mπ + L + mF ) − F M 1 M 1 ⇐⇒ π = (1 − µ) J mπ + (1 − µ) J (L + mF ) − F M  1  1 ⇐⇒ π 1 − (1 − µ) J m = (1 − µ) J (L + mF ) − F ⇐⇒ πM [J − (1 − µ)m] = (1 − µ)(L + mF ) − JF (1 − µ)(L + mF ) − JF ⇐⇒ πM = J − (1 − µ)m

Network complementarities: the profit a firm will get if it adopts the modern technology is a function of m: it depends on how many firms have already adopted it.

1 ∂πM M ∂πM M ∂m < 0 would mean π is ↓ in m, ∂m > 0 would mean π is ↑ in m. 2 numerator Or notice that numerator ↑ in m, denominator ↓ in m, =⇒ denominator ↑ in m. 8 / 12 Multiple Equilibria – Theory “Big Push” model Big Push model

Solve for πM : ( M π = (1 − µ)Dj − F 1 M Dj = J (mπ + L + mF )

M 1 M ⇐⇒ π = (1 − µ) J (mπ + L + mF ) − F M 1 M 1 ⇐⇒ π = (1 − µ) J mπ + (1 − µ) J (L + mF ) − F M  1  1 ⇐⇒ π 1 − (1 − µ) J m = (1 − µ) J (L + mF ) − F ⇐⇒ πM [J − (1 − µ)m] = (1 − µ)(L + mF ) − JF (1 − µ)(L + mF ) − JF ⇐⇒ πM = J − (1 − µ)m

Network complementarities: the profit a firm will get if it adopts the modern technology is a function of m: it depends on how many firms have already adopted it.

∂πM (m) 1,2 Is it increasing/decreasing/saturating... in m? → check the sign of ∂m .

1 ∂πM M ∂πM M ∂m < 0 would mean π is ↓ in m, ∂m > 0 would mean π is ↑ in m. 2 numerator Or notice that numerator ↑ in m, denominator ↓ in m, =⇒ denominator ↑ in m. 8 / 12 Multiple Equilibria – Theory “Big Push” model Big Push model

(1 − µ)(L + mF ) − JF u(m) πM (m) = = J − (1 − µ)m v(m)

∂πM (m) u0(m)v(m) − u(m)v 0(m) =⇒ = ∂m v(m)2 (1 − µ)F [J − (1 − µ)m] + [(1 − µ)(L + mF ) − JF ](1 − µ) = [J − (1 − µ)m]2 FJ − (1 − µ)mF + (1 − µ)L + (1 − µ)mF − JF = (1 − µ) [J − (1 − µ)m]2 (1 − µ)L = (1 − µ) [J − (1 − µ)m]2 (1 − µ)2L = > 0 [J − (1 − µ)m]2

πM is increasing in m.

9 / 12 (1 − µ)(L + mF ) − JF = 0

JF ⇐⇒ (L + mF ) = 1−µ JF ⇐⇒ mF = 1−µ − L ∗ J L ⇐⇒ m = 1−µ − F

Analysis: 3 equilibria: {m = 0}: no firms adopts. Stable. Basin of attraction = the interval [0; m∗[ {m = m∗}: unstable. πT (m∗) = πM (m∗) = 0: each firm is indifferent. {m = J}: all firms adopt. Stable. Basin of attraction = the interval ]m∗; J]

Multiple Equilibria – Theory “Big Push” model Big Push model

The profit-maximizing firm will adopt the modern technology if/when πM > πT , i.e. πM > 0.

Solve for the value m∗ at which πM = 0: (1 − µ)(L + mF ) − JF πM = = 0 ⇐⇒ J − (1 − µ)m

10 / 12 Multiple Equilibria – Theory “Big Push” model Big Push model

The profit-maximizing firm will adopt the modern technology if/when πM > πT , i.e. πM > 0.

Solve for the value m∗ at which πM = 0: (1 − µ)(L + mF ) − JF πM = = 0 ⇐⇒ (1 − µ)(L + mF ) − JF = 0 J − (1 − µ)m JF ⇐⇒ (L + mF ) = 1−µ JF ⇐⇒ mF = 1−µ − L ∗ J L ⇐⇒ m = 1−µ − F

Analysis: 3 equilibria: {m = 0}: no firms adopts. Stable. Basin of attraction = the interval [0; m∗[ {m = m∗}: unstable. πT (m∗) = πM (m∗) = 0: each firm is indifferent. {m = J}: all firms adopt. Stable. Basin of attraction = the interval ]m∗; J] 10 / 12 Multiple Equilibria – Theory “Big Push” model Big Push model

Network complementarities: The profit from modernization depends on m, s.t. modernization becomes profitable for a subsequent firm iff at least a fraction m∗ of firms have already modernized.

Intuition: if no firm yet has modernized, total income in the economy is low, the first firm to modernize might not reach the scale where average cost is low enough (remember average cost goes down with output) and its profits will be negative. Whereas if many (m∗) firms have modernized, the income in the economy is high enough to drive average costs below unit price, which benefits any “late-adopter”.

Rationale for government intervention: Problem: this m∗ can’t be reached, since no firm wants to be the first to modernize. There is a coordination problem: all firms would benefit if all adopted, but in the low-level equilibrium, no individual firm has the incentive to adopt. If the economy offers no mechanism for firms to coordinate their efforts, there is a rationale for the state to step in and make (at least m∗) firms modernize at the same time.

11 / 12 Multiple Equilibria – Theory “Big Push” model Big Push model – Main takeaways

Model: The production function contains a sunk cost, making the average cost & as output % (the fixed cost is spread among more units of output), allowing for increasing returns to K (for some portion of the graph). In such a context, coordination problems can generate multiple equilibria. Empirical implications – matching theory to data: Multiple-equilibria models may explain cross-country growth patterns, e.g., why some countries exhibit stagnant growth at low levels of income per capita while others race ahead. E.g., it may be that African and Latin American countries are stuck in low-level equilibria (“poverty traps”), while several East Asian countries have transitioned to a higher-level equilibrium. Policy implications: =⇒ “Big Push”: A poor country can be caught in a poverty trap. Can escape it with government intervention that is sufficiently “big” and pushes the economy across the threshold, into the basin of attraction of the better equilibrium, i.e., a “take-off” into sustained growth. Whereas a ’bit by bit’ programme would not suffice.

12 / 12