Multiple Equilibria { Theory \Big Push" model Advanced Economic Development { 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 production function 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 output 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 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 = 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.