Competitive Equilibrium Models in Macroeconomics

Credit Market Equilibrium

Given parameters {β, N, y₁, y₂}, a competitive equilibrium consists of an allocation {c*₁, c*₂, b*₁} and an interest rate R* such that:

• {c*₁, c*₂, b*₁} maximize the household’s utility given R*.
• Markets clear:
  • Bonds: ∑_i=1^N b*_i,1 | = Nb*₁ = 0
  • Consumption goods: Nc*₁ = Ny₁, Nc*₂ = Ny₂

Overlapping Generations (OLG) Model

Given parameters {N, β, δ, α, s₁, {A_t}_t=1^∞}, an equilibrium consists of sequences of prices {w_t, r_t}_t=1^∞, quantities {c_y,t, c_o,t+1, s_t+1}_t=1^∞ for the household, and allocations {K_t, L_t}_t=1^∞ for the firm such that:

• Given {w_t, r_t}_t=1^∞, households make optimal decisions for each generation t ≥ 1, i.e., {c_y,t, c_o,t+1, s_t+1} satisfy equations (10), (11), and (12).
• The initial old generation maximizes its utility given r₁ and chooses c_o,1 = (1 + r₁ − δ)s₁.
• Given {w_t, r_t}_t=1^∞, firms make optimal decisions for all t ≥ 1, i.e., K_t and L_t satisfy equations (6) and (7) for all t ≥ 1.
• All markets clear for all t ≥ 1.

Pension Systems and Fiscal Policy

Given parameters (endowments: k₁; preferences: β, ...; depreciation: δ; production function: ...) and a government policy {G₁, G₂, T, τ_l, τ_k, τ_c}, a competitive equilibrium consists of an allocation {c*_t, l^s*_t, k^s*_t, y*_t, l^d*_t, k^d*_t}_t=1,2 and prices {w*_t, r*_t} such that:

• {c*_t, l^s*_t, k^s*_t}_t=1,2 maximize the household’s utility given prices and taxes.
• {y*_t, l^d*_t, k^d*_t}_t=1,2 maximize the firm’s profits given prices.
• The government policy is budget-feasible.
• Markets clear, i.e., for t = 1, 2:
  • Goods market: y*_t + (1 − δ)k*_t = c*_t + G_t + k*_t+1
  • Labor market: l^s*_t = l^d*_t
  • Capital market: k^s*_t = k^d*_t

Unemployment and Rational Expectations

Given parameters {γ, F, A, M, α, β} and a government policy {b, τ}, a decentralized equilibrium with rational expectations consists of quantities {s*, v*, θ*, u*} and a wage w* such that:

• s* is optimal for the worker given θ* and w*.
• Each firm decides optimally whether to enter or not given θ* and w*.
• Consistency/Rational expectations: θ* = s*/v*.
• w* is a bargaining solution given β, A, b.
• The government’s budget is balanced: τ = u*b, where u* = 1 − m(s*, v*) is the equilibrium unemployment.