Wellbeing and Macroeconomics: A SAGE Approach

Overview

Standard macroeconomic models treat welfare as preference satisfaction: an agent is as well off as the utility of the bundle they choose. This lecture takes a different premise, namely that personal welfare depends on more than consumption, and that some of its other determinants are shaped by economic choices in ways a standard model does not record.

To make that idea operational we use the SAGE framework of Lima de Miranda and Snower [2020], in which wellbeing has separate dimensions:

	dimension	how it enters the model
S	Social cohesion	a public good Q that agents contribute time to
A	Agency	how much of a person’s labour effort becomes income
G	material Gain	the standard consumption-leisure motive
E	Environment	not modelled here; it belongs to the research versions

We build toward this in stages: start from models you already know, then add one idea at a time.

1. The neoclassical growth model, the deterministic core of optimal saving.

2. The real business cycle model, which adds a productivity shock and a labour choice.

3. A SAGE-RBC, which gives the representative agent agency and social cohesion.

4. The SAGE-Bewley model, which replaces the representative agent with a distribution of households facing uninsurable risk, and asks what an activation policy does to the separate dimensions of their wellbeing, and to whom.

The first two rungs are deliberately brief, because the QuantEcon lectures cover them in depth (optimal growth, finite Markov chains); the destination is the fourth. The background assumed is dynamic programming at the level of value function iteration, and enough Julia to read a loop. Every figure below is produced by the code shown. We begin by loading the packages and the model engine, fetching the engine from the repository if the file is not next to the notebook, so the notebook runs anywhere.

using QuantEcon, Plots, Statistics, Printf, Downloads
isfile("sage_engine.jl") ||
    Downloads.download("https://raw.githubusercontent.com/conway1521/sage-bewley/main/sage_engine.jl",
                       "sage_engine.jl")
include("sage_engine.jl")
using .SAGEBewley
gr(fmt = :png, size = (820, 300))
C1, C2, C3, C4 = "#1f77b4", "#ff7f0e", "#2ca02c", "#9467bd"
BLUE, ORANGE = C1, C2;          # low income, high income (thesis colours)

Two of the engine’s diagnostics are worth distinguishing before we use them. frac_constrained reports the mass sitting on the lowest asset node, which is useful for checking that the borrowing constraint binds but is a grid cell rather than an economic set, so it shrinks as the asset grid is refined. hand_to_mouth reports the share holding less than four weeks of mean labour income in wealth, the threshold Kaplan et al. [2014] use, which does not move with the grid. It is the second one we report.

The Neoclassical Growth Model

A representative household maximises \(\sum_{t} \beta^t \log c_t\) subject to the resource constraint

\[ c_t + k_{t+1} = k_t^{\alpha} + (1-\delta) k_t . \]

We solve it by value function iteration on a grid for capital, and keep the section to a minimum: QuantEcon’s optimal growth lecture develops the model and the method properly, and its only job here is to set the pattern the next two models reuse. The Bewley model at the end switches to policy iteration.

function solve_ncgm(; α=0.33, β=0.96, δ=0.10, nk=400, tol=1e-8, maxit=2000)
    kstar = (α / (1/β - 1 + δ))^(1/(1-α))
    kgrid = range(0.1*kstar, 1.6*kstar, length=nk)
    V = zeros(nk); Vnew = similar(V); g = zeros(Int, nk)
    for _ in 1:maxit
        for i in 1:nk
            resources = kgrid[i]^α + (1-δ)*kgrid[i]
            best = -Inf; bestj = 1
            for j in 1:nk
                c = resources - kgrid[j]
                c <= 0 && break
                val = log(c) + β*V[j]
                if val > best; best = val; bestj = j; end
            end
            Vnew[i] = best; g[i] = bestj
        end
        maximum(abs.(Vnew .- V)) < tol && (V .= Vnew; break)
        V .= Vnew
    end
    kgrid, kgrid[g], kstar, α, δ
end

kgrid, kpol, kstar, αn, δn = solve_ncgm();

Starting from a capital stock below the steady state, the economy saves and converges. Capital and consumption both rise monotonically along the transition.

T = 60
kp = fill(0.4*kstar, T); cp = zeros(T)
for t in 1:T-1
    i = argmin(abs.(kgrid .- kp[t]))
    kp[t+1] = kpol[i]
    cp[t] = kp[t]^αn + (1-δn)*kp[t] - kp[t+1]
end
cp[T] = kp[T]^αn - δn*kp[T]
p1 = plot(1:T, kp, c=C1, lw=2, legend=false, title="Capital path", xlabel="t")
hline!(p1, [kstar], c=:gray, ls=:dot)
p2 = plot(1:T, cp, c=C1, lw=2, legend=false, title="Consumption path", xlabel="t")
plot(p1, p2, layout=(1,2))

The Real Business Cycle Model

Now total factor productivity z follows a persistent AR(1) process, discretised into a Markov chain with the Rouwenhorst method [Rouwenhorst, 1995] (QuantEcon’s finite Markov chains lecture covers the toolkit), and the household chooses labour n. The state is (k, z) and the calibration is the standard quarterly one. We fold the intratemporal labour choice into the per-period reward, so value iteration stays a simple maximisation over next-period capital, and after convergence we recover continuous policies between grid points. That last step matters for what follows: the labour response to a shock is a fraction of a percent, smaller than the steps a discrete policy takes on any affordable grid, so without the recovery the labour figure would be showing the grid, not the model. The same routine will serve the SAGE-RBC once we switch on agency αa (the share of labour income retained) and a social-cohesion pull Λ B away from work.

function solve_rbc(; α=1/3, β=0.984, δ=0.025, ψ=0.25, ρ=0.979, σ=0.0072,
                     αa=1.0, Λ=0.0, B=1.0, n_target=0.2, nk=180, tol=1e-6, maxit=1500)
    mc = rouwenhorst(7, ρ, σ); zv = exp.(mc.state_values); Π = mc.p
    rk_ss = 1/β + δ - 1
    k_ss  = (α/rk_ss)^(1/(1-α)) * n_target
    y_ss  = k_ss^α * n_target^(1-α)
    φ     = (1-α)*y_ss/n_target * (1/(y_ss - δ*k_ss)) * n_target^(-ψ)
    function best_n(k, z, kp)               # labour from the static first-order condition
        f(n) = begin
            y = z*k^α*n^(1-α)
            c = (αa*(1-α)+α)*y + (1-δ)*k - kp
            c <= 0 && return (-Inf, c)
            (c^(-1)*(αa*(1-α)+α)*(1-α)*y/n - φ*n^ψ - Λ*B, c)
        end
        lo, hi = 1e-4, 0.999
        for _ in 1:60
            mid = (lo+hi)/2
            f(mid)[1] > 0 ? (lo = mid) : (hi = mid)
        end
        n = (lo+hi)/2; n, f(n)[2]
    end
    kgrid = range(0.5*k_ss, 1.6*k_ss, length=nk)
    R = fill(-Inf, nk, 7, nk); N = zeros(nk, 7, nk)
    for iz in 1:7, ik in 1:nk, jk in 1:nk
        n, c = best_n(kgrid[ik], zv[iz], kgrid[jk])
        c <= 0 && continue
        R[ik,iz,jk] = log(c) - φ*n^(1+ψ)/(1+ψ) + Λ*B*(1-n); N[ik,iz,jk] = n
    end
    V = zeros(nk,7); Vnew = similar(V); gk = zeros(Int,nk,7)
    for _ in 1:maxit
        EV = V * Π'
        for iz in 1:7, ik in 1:nk
            best = -Inf; bj = 1
            for j in 1:nk
                val = R[ik,iz,j] + β*EV[j,iz]
                if val > best; best = val; bj = j; end
            end
            Vnew[ik,iz] = best; gk[ik,iz] = bj
        end
        maximum(abs.(Vnew .- V)) < tol && (V .= Vnew; break)
        V .= Vnew
    end
    # recover continuous policies: golden-section on k' between the discrete
    # optimum's neighbours, continuation value interpolated linearly. On the
    # grid alone the labour policy is a step function and the labour response
    # inherits the steps; this is the same recovery the Bewley engine uses.
    EV = V * Π'
    EVi(kp, iz) = begin
        j = clamp(searchsortedlast(kgrid, kp), 1, nk-1)
        t = (kp - kgrid[j]) / (kgrid[j+1] - kgrid[j])
        (1-t)*EV[j,iz] + t*EV[j+1,iz]
    end
    reward(k, z, kp) = begin
        n, c = best_n(k, z, kp)
        c <= 0 ? -Inf : log(c) - φ*n^(1+ψ)/(1+ψ) + Λ*B*(1-n)
    end
    kpol = zeros(nk,7); gn = zeros(nk,7); gold = (sqrt(5)-1)/2
    for iz in 1:7, ik in 1:nk
        j = gk[ik,iz]
        lo = kgrid[max(j-1,1)]; hi = kgrid[min(j+1,nk)]
        obj(kp) = reward(kgrid[ik], zv[iz], kp) + β*EVi(kp, iz)
        c = hi - gold*(hi-lo); d = lo + gold*(hi-lo); fc = obj(c); fd = obj(d)
        for _ in 1:40
            if fc > fd
                hi = d; d = c; fd = fc; c = hi - gold*(hi-lo); fc = obj(c)
            else
                lo = c; c = d; fc = fd; d = lo + gold*(hi-lo); fd = obj(d)
            end
        end
        kp = (lo+hi)/2
        kpol[ik,iz] = kp; gn[ik,iz] = best_n(kgrid[ik], zv[iz], kp)[1]
    end
    (; kgrid, zv, kpol, gn, α, δ, αa, ρ, k_ss)
end

function irf(m; shock=0.01, T=60)
    (; kgrid, zv, kpol, gn, α, δ, αa, ρ) = m
    nz = length(zv); nk = length(kgrid)
    function itp(P, k, z)
        ik = clamp(searchsortedlast(kgrid, k), 1, nk-1)
        tk = clamp((k-kgrid[ik])/(kgrid[ik+1]-kgrid[ik]), 0.0, 1.0)
        iz = clamp(searchsortedlast(zv, z), 1, nz-1)
        tz = clamp((z-zv[iz])/(zv[iz+1]-zv[iz]), 0.0, 1.0)
        (1-tk)*(1-tz)*P[ik,iz]+tk*(1-tz)*P[ik+1,iz]+(1-tk)*tz*P[ik,iz+1]+tk*tz*P[ik+1,iz+1]
    end
    kss = m.k_ss; for _ in 1:3000; kss = itp(kpol, kss, 1.0); end
    n_ss = itp(gn, kss, 1.0); y_ss = kss^α*n_ss^(1-α); i_ss = δ*kss
    c_ss = (αa*(1-α)+α)*y_ss - δ*kss
    zp = [exp(log(1+shock)*ρ^(t-1)) for t in 1:T]
    k=fill(kss,T+1); y=zeros(T); c=zeros(T); n=zeros(T); iv=zeros(T)
    for t in 1:T
        n[t] = itp(gn,k[t],zp[t]); k[t+1] = itp(kpol,k[t],zp[t])
        y[t]=zp[t]*k[t]^α*n[t]^(1-α); iv[t]=k[t+1]-(1-δ)*k[t]
        c[t]=(αa*(1-α)+α)*y[t]+(1-δ)*k[t]-k[t+1]
    end
    d(x,b)=100 .*(x ./ b .- 1)
    (; y=d(y,y_ss), c=d(c,c_ss), n=d(n,n_ss), iv=d(iv,i_ss), n_ss, c_ss, y_ss)
end

m_rbc = solve_rbc(); r_rbc = irf(m_rbc)
plot(title="RBC: impulse responses to a 1% productivity shock", xlabel="t", ylabel="percent")
plot!(1:length(r_rbc.y), r_rbc.y, lw=2, label="output", c=C1)
plot!(1:length(r_rbc.c), r_rbc.c, lw=2, label="consumption", c=C2)
plot!(1:length(r_rbc.iv), r_rbc.iv, lw=2, label="investment", c=C3)
plot!(1:length(r_rbc.n), r_rbc.n, lw=2, label="labour", c=C4)
hline!([0], c=:black, lw=1, label=false)

Output rises, consumption is smooth, labour rises, and investment is the most volatile component. This is the textbook business cycle, the benchmark for what follows.

A SAGE-RBC

Before the full heterogeneous-agent model, it helps to see what the SAGE ingredients do even with a single representative agent. We give that agent agency αa < 1, so only part of labour income is retained, and a social-cohesion reward Λ B (1 - n) for time spent out of work. These are the same two ideas that drive the household problem later. We solve the same model with the switches on and compare the responses.

m_sage = solve_rbc(αa=0.85, Λ=0.30, B=0.85)
r_sage = irf(m_sage)
@printf("steady-state hours: RBC %.4f, SAGE-RBC %.4f (%+.1f%%)\n",
        r_rbc.n_ss, r_sage.n_ss, 100*(r_sage.n_ss/r_rbc.n_ss - 1))
function compare(field, title)
    a = getfield(r_rbc, field); b = getfield(r_sage, field)
    pl = plot(1:length(a), a, lw=2, label="RBC", c=C1, title=title, xlabel="t")
    plot!(pl, 1:length(b), b, lw=2, label="SAGE-RBC", c=C2)
    hline!(pl, [0], c=:black, lw=1, label=false); pl
end
plot(compare(:y,"Output"), compare(:c,"Consumption"),
     compare(:iv,"Investment"), compare(:n,"Labour"),
     layout=(2,2), size=(820,520))

steady-state hours: RBC 0.2000, SAGE-RBC 0.1906 (-4.7%)

The result is a near miss. The SAGE agent works about five percent fewer hours in steady state, because the social pull makes time out of work valuable, and earns less from each hour, because agency is below one. But the shape of its response to the shock is almost the shape of the RBC’s response. With a single representative agent, agency and social cohesion move levels and leave the cycle nearly alone. Whatever these forces do to an economy, they do it through who is affected and how differently, and a representative agent has no one to be affected differently. So we give the model a distribution of households.

The SAGE Framework

In the SAGE framework an agent pursues several decision objectives at once, one for each dimension of wellbeing, rather than maximising a single index. We keep two of them. Material gain is the familiar consumption-leisure trade-off. Social cohesion is the benefit from a public good that everyone contributes time to: an agent who works effort e contributes q = 1 - e to society, and the public good is the sum of those contributions.

Agency, denoted α, is the degree to which a person can turn effort into income. It stands in for labour-market security, health, and skills, and it will differ across households. Following the thesis behind this lecture [Conway, 2020], agents maximise utility as usual and the wellbeing dimensions are read off separately at the optimum. The utility an agent maximises and the wellbeing we report are deliberately different objects, and keeping them apart is what lets the model show material gain and social cohesion moving in opposite directions.

The SAGE-Bewley Model

We now place these ideas in a Bewley-Aiyagari economy [Aiyagari, 1994, Bewley, 1986]. Readers who know QuantEcon’s Aiyagari lecture will recognise the bones of what follows; the differences are the labour-effort margin, agency, and the public good. A unit mass of households faces uninsurable idiosyncratic income risk, indexed by a productivity state z that is low or high and follows a two-state Markov chain. Each period a household chooses consumption c, next-period assets \(a' \ge 0\), and labour effort \(e \in [0,1]\), with \(q = 1 - e\) the social contribution.

Utility is additively separable between material gain and the social term. The separable form matters: it keeps the wealth effect on labour supply, and that wealth effect is the model’s engine, because households with more assets have a lower marginal utility of consumption, work less, and contribute more time:

\[ u(c, e; z) \;=\; \underbrace{\Gamma\left(\frac{c^{1-\gamma}}{1-\gamma} - \phi\,\frac{e^{1+\psi}}{1+\psi}\right)}_{\text{material gain } U^c} \;+\; \underbrace{\Lambda\, B_z\, Q}_{\text{social cohesion } U^s}, \]

and the flow budget applies agency to labour income,

\[ c + a' \;=\; \alpha_z\, e\, z + R\, a, \qquad a' \ge 0 . \]

The public good is the average contribution across the stationary distribution, \(Q = \mathbb{E}[\,1 - e\,]\).

The interest rate \(R\) is taken as given, calibrated to a long-run real rate of about two percent, rather than cleared by an asset market. This is an Aiyagari household block in partial equilibrium: the equilibrium object we solve for is the social good \(Q\), through its fixed point, not the interest rate.

The Income Process

Productivity \(z\) follows a two-state Markov chain, the discretised primitive of the idiosyncratic risk that everything downstream inherits. We show it first, because the shape of the wealth distribution is a consequence of it.

p = country_params("FR")   # the calibrated French row, as in the papers
zv, Π = income_process(p)
πstat = vec((Π^1000)[1, :])
@printf("productivity states  z = [%.3f, %.3f]\n", zv[1], zv[2])
@printf("transition matrix    Π = [%.2f  %.2f ;  %.2f  %.2f]\n", Π[1,1], Π[1,2], Π[2,1], Π[2,2])
@printf("stationary fractions   = [%.2f, %.2f]\n", πstat[1], πstat[2])

productivity states  z = [0.775, 1.225]
transition matrix    Π = [0.95  0.05 ;  0.05  0.95]
stationary fractions   = [0.50, 0.50]

Two states is a modelling choice, not a numerical economy. The states stand for the lower- and higher-education groups of the underlying thesis, which is why agency \(\alpha_z\) and the belonging taste \(B_z\) will differ across them, and why we speak of groups rather than shocks when reading results. The engine discretises with the same Rouwenhorst tool used for the RBC and solves finer chains unchanged; the education pair is the configuration every result below uses.

Every default in SAGEParams is sourced. Risk aversion is 2, an elasticity of intertemporal substitution of one half, the meta-analytic consensus of Havranek [2015]. The inverse Frisch is 2, a labour-supply elasticity of one half, from the quasi-experimental consensus of Chetty et al. [2011]. The discount factor and interest rate are standard annual values. And the effort disutility is set so the work share of committed time matches the French time-use figure of 0.53, where paid work averages 3h24 a day against 3h01 of unpaid domestic and associative work [INSEE, 2010]. Two facts the model was never aimed at then come out close to data: about a third of households live hand to mouth [Kaplan et al., 2014], and the wealth Gini is 0.55.

Solving the Model

The solver lives in sage_engine.jl, and its structure is worth seeing, because it is the Aiyagari algorithm with one twist at each end. Four steps.

Step 1: fold effort into the reward. For each state \((a, z)\) and each candidate next-asset level \(a'\), choose effort on a grid to maximise within-period utility. After this inner choice the household problem has a single action, next assets, exactly the shape of QuantEcon’s Aiyagari lecture.

Step 2: solve the discrete problem. The result is a sparse state-action-pair DiscreteDP (the QuantEcon.jl idiom), solved by policy iteration.

Step 3: recover continuous policies. Effort comes from the discrete joint optimum. Next assets are then re-optimised continuously against the linearly interpolated continuation value, golden-section between the discrete optimum’s neighbours. This is the same recovery step the RBC section used, for the same reason: without it the policies inherit the steps of the grid and so does the wealth distribution.

Step 4: aggregate, and close the loop. The stationary distribution uses the lottery of Young [2010]: each household’s continuous savings choice is split across the two bracketing grid nodes, the implied sparse transition matrix is iterated to its fixed point, and the public good is read off as \(Q = E[1-e]\). When the social term is behavioural, so that \(Q\) feeds back into the reward, an outer damped fixed point on \(Q\) wraps the whole solve.

Abridged from sage_engine.jl, the skeleton is:

# 1. effort folded into the reward: one action (next assets) remains
for i_z in 1:nz, i_a in 1:na, k in 1:na            # state (a, z), action a'
    best = -Inf
    for e in e_grid                                 # inner static effort choice
        c = R*a[i_a] + α[i_z]*e*z[i_z] - a[k]
        c <= 0 && continue
        best = max(best, u_c(c, e) + social(e, B[i_z], Q))
    end
    record_state_action_pair!(best)                 # sparse triplets
end

# 2. policy iteration on the sparse DiscreteDP (QuantEcon.jl)
ddp = DiscreteDP(Rvec, Qsparse, β, s_ind, a_ind)
σ   = solve(ddp, PFI).sigma

# 3. continuous savings against the interpolated continuation value
a′ = golden_max(ap -> u_c(resources - ap, e⋆) + β * EV_interp(ap, z), lo, hi)

# 4. Young (2010) lottery -> stationary λ -> Q = E[1-e];
#    outer damped loop on Q when the social term is behavioural

With the solver in hand, we solve the default calibration and read off the decision rules.

sol = solve_model(p)
a = sol.a_grid
ix = findfirst(>(15.0), a); ix = ix === nothing ? length(a) : ix
p1 = plot(a, sol.c[:,1], c=BLUE, lw=2, label="low", title="Consumption c(a,z)", xlabel="assets a")
plot!(p1, a, sol.c[:,2], c=ORANGE, lw=2, label="high")
p2 = plot(a, sol.e[:,1], c=BLUE, lw=2, label="low", title="Labour effort e(a,z)",
          xlabel="assets a", ylims=(0,1.02))
plot!(p2, a, sol.e[:,2], c=ORANGE, lw=2, label="high")
p3 = plot(a[1:ix], sol.a_next[1:ix,1], c=BLUE, lw=2, label="low",
          title="Savings rule a'(a,z)", xlabel="assets a")
plot!(p3, a[1:ix], sol.a_next[1:ix,2], c=ORANGE, lw=2, label="high")
plot!(p3, a[1:ix], a[1:ix], c=:gray, ls=:dash, label="45 degrees")
plot(p1, p2, p3, layout=(1,3), size=(980,300))

All three rules behave the way a Bewley model should. Consumption rises with assets. The savings rule \(a'(a,z)\) starts at the borrowing constraint and rises, hugging the 45-degree line from just below: the near-unit \(\beta R\) gives a high target, so households accumulate slowly and the gap from 45 degrees is what produces the long right tail of the distribution. At the low end the low-income state sits against the constraint \(a' = 0\). Those households are hand to mouth, though the corner is not itself the measure of it: the share reported above counts households holding less than four weeks of income in wealth, which is a set that does not move when the grid is refined. That the stationary distribution puts zero weight on the top grid point, reported below, confirms the grid is wide enough that this tail is resolved rather than truncated. The effort policy carries the model’s central intuition: effort falls with wealth, because wealthier households have a lower marginal utility of consumption and turn their time toward social contribution. Under the evidence-based curvature this wealth effect is strong enough that the higher-agency group, being richer on average, works somewhat less and contributes somewhat more time overall, the pattern participation data actually show.

The wealth distribution itself has the usual Bewley shape, a large mass of low-income households at the borrowing constraint trailing into a thin right tail. The Lorenz curve summarises the inequality it implies, and the bar shows which income group ends up supplying the social good.

function lorenz(vals, w)
    o = sortperm(vals); cw = cumsum(w[o]); cv = cumsum(vals[o] .* w[o])
    vcat(0.0, cw ./ cw[end]), vcat(0.0, cv ./ cv[end])
end
imax = something(findfirst(>(20.0), a), length(a))
pa = plot(a[1:imax], sol.λ[1:imax,1], c=BLUE, lw=2, label="low",
          title="Stationary wealth distribution", xlabel="assets a", ylabel="mass")
plot!(pa, a[1:imax], sol.λ[1:imax,2], c=ORANGE, lw=2, label="high")
lx, ly = lorenz(vec(repeat(a, 1, p.nz)), vec(sol.λ))
pl = plot(lx, ly, c=C3, lw=2, legend=false, title="Lorenz curve (wealth)",
          xlabel="population share", ylabel="wealth share")
plot!(pl, [0,1], [0,1], c=:gray, ls=:dash)
sh = public_good_shares(sol)
pb = bar(["low","high"], sh, c=[BLUE,ORANGE], legend=false,
         title="Who supplies Q", ylims=(0,1))
plot(pa, pl, pb, layout=(1,3), size=(980,300))

@printf("Gini (wealth)          = %.3f\n", wealth_gini(sol))
@printf("Hand to mouth          = %.1f%%  (wealth below four weeks of income)\n", 100*hand_to_mouth(sol))
@printf("Public good size Q     = %.3f, supplied %.0f%% by low income\n",
        sol.Q, 100*public_good_shares(sol)[1])
@printf("Mass at top grid point = %.1e  (a numerical check: ~0 means the asset\n",
        sum(sol.λ[end, :]))
@printf("                          grid is wide enough not to truncate the tail)\n")

Gini (wealth)          = 0.552
Hand to mouth          = 31.8%  (wealth below four weeks of income)
Public good size Q     = 0.470, supplied 48% by low income
Mass at top grid point = 1.9e-165  (a numerical check: ~0 means the asset
                          grid is wide enough not to truncate the tail)

Under the sourced calibration the higher-income group supplies slightly more of the public good, about 53 percent, because the wealth effect dominates: better-rewarded households are richer, and richer households can afford to give time. This matches the well-documented fact that volunteering and associative participation rise with income and education. The model’s distributional tension is then a double disadvantage: the lower-income group both supplies less of the social fabric and, with \(B_{low} < B_{high}\), enjoys what exists less per unit.

The Wellbeing Dashboard

We now read off the two decision objectives at the optimum and display them as a balanced dashboard, by wealth quartile within each income group. Material gain \(U^c\) rises with wealth. Social cohesion \(U^s = \Lambda B Q\) is flat within a group and higher for high-income households, who enjoy the public good more even though they supply less of it. One detail to expect in the figure: the bottom two quartiles of the lower group print the same bar, because the poorest half of that group sits at the borrowing constraint, so its quartiles coincide.

function dashboard(sol)
    # exact quartile means: each grid point's mass occupies an interval of the
    # within-group CDF, and a quartile receives the part that overlaps it, so
    # the atom at the borrowing constraint is split across quartiles correctly
    Uc = zeros(4,2); Us = zeros(4,2)
    for z in 1:2
        w = sol.λ[:,z] ./ sum(sol.λ[:,z]); cw = cumsum(w)
        lo = vcat(0.0, cw[1:end-1])
        for qi in 1:4
            qa, qb = (qi-1)/4, qi/4
            ov = max.(min.(cw, qb) .- max.(lo, qa), 0.0)
            ww = ov ./ sum(ov)
            Uc[qi,z] = sum(ww .* sol.Uc[:,z]); Us[qi,z] = sum(ww .* sol.Us[:,z])
        end
    end
    Uc, Us
end
function gbar(cols; title)
    m = 4; w = 0.4; pl = plot()
    bar!(pl, (1:m).-w/2, cols[:,1], bar_width=w, label="low",  color=BLUE)
    bar!(pl, (1:m).+w/2, cols[:,2], bar_width=w, label="high", color=ORANGE)
    plot!(pl, xticks=(1:m, ["Q1","Q2","Q3","Q4"]), title=title); pl
end
Uc, Us = dashboard(sol)
plot(gbar(Uc; title="Material gain by quartile"),
     gbar(Us; title="Social cohesion by quartile"), layout=(1,2))

A Policy Experiment and the Decoupling

A government wants people in work, so it subsidises work: a make-work-pay subsidy of 20 percent on labour income, financed by a lump-sum tax so that the budget balances. We run it under behavioural social cohesion (the warm-glow form, in which the act of contributing enters utility, so effort responds). Real governments run exactly this policy. In the model it produces a decoupling: the subsidy raises the return to work, households work more and contribute less, so consumption rises while the public good falls.

function solve_financed(p; subsidy = 0.0, tol = 1e-5, maxit = 40)
    z_vals, _ = SAGEBewley.income_process(p)
    T = 0.0
    sol = solve_model(update(p; subsidy = subsidy, lumptax = T))
    for _ in 1:maxit
        paid = sum(sol.λ[ia, iz] * subsidy * p.α[iz] * sol.e[ia, iz] * z_vals[iz]
                   for iz in 1:p.nz, ia in 1:p.na)
        abs(paid - T) < tol && break
        T = 0.5 * T + 0.5 * paid
        sol = solve_model(update(p; subsidy = subsidy, lumptax = T))
    end
    sol
end
pw   = update(p; ne = 320, social_mode = :warmglow, social_strength = 1.0)
s0   = solve_financed(pw; subsidy = 0.0)
s1   = solve_financed(pw; subsidy = 0.20)
agg(s, M) = sum(s.λ .* M)
@printf("consumption: %.3f to %.3f (%+.1f%%)\n",
        agg(s0, s0.c), agg(s1, s1.c), 100*(agg(s1,s1.c)/agg(s0,s0.c)-1))
@printf("public good: %.3f to %.3f (%+.1f%%)\n", s0.Q, s1.Q, 100*(s1.Q/s0.Q-1))

consumption: 0.436 to 0.461 (+5.6%)
public good: 0.495 to 0.470 (-5.1%)

avg(s, M, z) = sum((s.λ[:,z]./sum(s.λ[:,z])) .* M[:,z])
dUc = [100*(avg(s1,s1.Uc,z) - avg(s0,s0.Uc,z))/abs(avg(s0,s0.Uc,z)) for z in 1:2]
dUs = [100*(pw.Λ*pw.B[z]*s1.Q - pw.Λ*pw.B[z]*s0.Q)/abs(pw.Λ*pw.B[z]*s0.Q) for z in 1:2]
pl = plot()
bar!(pl, [1,2].-0.18, dUc, bar_width=0.36, label="Material gain", color=C3)
bar!(pl, [1,2].+0.18, dUs, bar_width=0.36, label="Social cohesion", color=C4)
plot!(pl, xticks=([1,2],["low income","high income"]), ylabel="percent change", size=(560,320),
      title="Who gains and who loses from the subsidy")
hline!(pl, [0], c=:black, lw=1, label=false); pl

Aggregate consumption rises. But the lump-sum financing is the same for everyone while the subsidy is proportional to labour income, so the lower-income group’s material gain falls outright while the higher-income group’s rises, and both groups lose part of the social dimension as the public good shrinks. The lower-income group loses on every dimension at once. A planner reading a single consumption index sees the aggregate gain and adopts the policy. A planner reading the dashboard sees the social cost and who bears it. The model makes no claim that the policy is wrong; the claim is about measurement, because a single index cannot see what the policy trades away or from whom.

Note

A word on numerical accuracy. Effort is chosen from a discrete grid, so the public good \(Q = E[1-e]\) inherits that grid’s resolution and any small change in it is read off steps rather than a smooth function. The policy cell above therefore raises ne to 320 from the default 40, at which point the magnitudes have settled; at the default the direction is right but the size of the fall in \(Q\) is understated. The default is left in place elsewhere, where the quantities of interest are levels rather than small differences in effort. A smaller sensitivity to the alignment of the asset grid survives even then, worth about a quarter of a percentage point on \(Q\), so read the fall in the public good as five percent rather than as its third digit. The same alignment moves the hand-to-mouth share reported above from 0.33 at this asset grid to about 0.31 on much finer ones. Two other checks are worth stating: the Euler-equation errors on the interior of the state space are below \(10^{-3}\) in consumption units, and the stationary distribution reproduces itself under the transition to machine precision. A companion note in the model repository records these checks in full.

Country Calibrations

The defaults above are the French calibration. The engine carries seven country rows calibrated to one standardised pipeline: OECD How’s Life for the agency and belonging gradients, the World Inequality Database for the wealth Gini, Kaplan et al. [2014] for the hand-to-mouth share, the Multinational Time Use Study for the work share, and the World Values Survey for the participation gradient, with the full sourcing documented in the research repository.

println("countries available: ", join(sort(collect(keys(COUNTRIES))), ", "))

countries available: CN, CO, DE, FR, IT, US, ZA

To switch the model to a country, call country_params(code). Below, we solve every country and report the calibrated and untargeted moments side by side.

function row(code)
    p = country_params(code)
    s = solve_model(p)
    tg = country_targets(code)
    (code = code, Q = s.Q, wg = wealth_gini(s), htm = hand_to_mouth(s), tg = tg)
end
results = [row(c) for c in ("FR","DE","IT","US","CO","ZA","CN")]
@printf("%-3s | %-12s | %-12s | %-12s | %-12s\n",
        "cty","work share","wealth Gini","HtM share","Q (fabric)")
for rw in results
    @printf("%-3s |  %.2f / %.2f |  %.2f / %.2f |  %.2f / %.2f |  %.3f\n",
            rw.code, 1 - rw.Q, rw.tg.work_share,
            rw.wg, rw.tg.wealth_gini, rw.htm, rw.tg.htm, rw.Q)
end

cty | work share   | wealth Gini  | HtM share    | Q (fabric)  
FR  |  0.53 / 0.53 |  0.55 / 0.68 |  0.32 / 0.30 |  0.470
DE  |  0.55 / 0.55 |  0.54 / 0.78 |  0.32 / 0.32 |  0.449
IT  |  0.55 / 0.55 |  0.52 / 0.61 |  0.28 / 0.41 |  0.448
US  |  0.60 / 0.60 |  0.52 / 0.85 |  0.27 / 0.31 |  0.398
CO  |  0.60 / 0.60 |  0.40 / 0.81 |  0.10 / 0.55 |  0.395
ZA  |  0.56 / 0.55 |  0.36 / 0.95 |  0.07 / 0.60 |  0.444
CN  |  0.64 / 0.65 |  0.41 / 0.74 |  0.11 / 0.50 |  0.355

The work share hits its target country by country, by construction. The hand-to-mouth share is in the right neighbourhood for the OECD members and undershoots for the developing countries, where the one-asset Bewley structure cannot simultaneously deliver a very high hand-to-mouth share and a very high wealth Gini. The wealth Gini itself is a known limitation of one-asset models: standard calibrations of this class reach 0.38 or so; this one reaches 0.55 for the European calibrations, which is closer to data without yet matching the very high Ginis of the United States or South Africa. Closing that gap is a two-asset extension along Kaplan et al. [2014].

# a country-switch demo: re-run the policy experiment with the German calibration
de   = update(country_params("DE"); social_mode = :warmglow, social_strength = 1.0)
s0de = solve_financed(de; subsidy = 0.0)
s1de = solve_financed(de; subsidy = 0.20)
@printf("Germany under the financed 20%% subsidy:\n")
@printf("  consumption %+.1f%%   public good %+.1f%%\n",
        100*(agg(s1de, s1de.c)/agg(s0de, s0de.c) - 1),
        100*(s1de.Q/s0de.Q - 1))

Germany under the financed 20% subsidy:
  consumption +6.1%   public good -5.5%

Switch the country by changing "DE". Every country row is one call to country_params(code).

Exercises

Everything needed for these is already loaded. Try each one before opening the solution.

Exercise 1

The SAGE-RBC switched on two things at once, agency and the social pull. Separate them: solve the model with agency alone (αa = 0.85) and with the social pull alone (Λ = 0.30, B = 0.85). Which switch moves the level of steady-state hours, and which moves the response of hours to the shock?

Solution to Exercise 1

r_agency = irf(solve_rbc(αa = 0.85))
r_social = irf(solve_rbc(Λ = 0.30, B = 0.85))
@printf("steady-state hours:  RBC %.4f | agency only %.4f | social only %.4f | both %.4f\n",
        r_rbc.n_ss, r_agency.n_ss, r_social.n_ss, r_sage.n_ss)
pl = plot(title = "Labour response by channel", xlabel = "t", ylabel = "percent")
plot!(pl, r_rbc.n, lw = 2, label = "RBC", c = C1)
plot!(pl, r_agency.n, lw = 2, label = "agency only", c = C3)
plot!(pl, r_social.n, lw = 2, label = "social pull only", c = C4)
plot!(pl, r_sage.n, lw = 2, label = "both", c = C2)
hline!(pl, [0], c = :black, lw = 1, label = false); pl

steady-state hours:  RBC 0.2000 | agency only 0.2001 | social only 0.1907 | both 0.1906

The level and the cycle separate cleanly. The social pull cuts steady-state hours by almost five percent, because it prices time out of work. Agency alone barely moves them: under log utility the income and substitution effects of a lower return nearly cancel, a classic result. And all four responses to the shock are similar, which repeats the section’s message from a different angle: with a representative agent these forces live in the levels, and it takes heterogeneity to make them bite.

Exercise 2

The Bewley calibration sets the discount factor and the interest rate jointly, with \(\beta R\) well inside one. Re-solve the baseline with R = 1.03 and compare the share of households at the constraint and the wealth Gini. What should happen as \(\beta R\) approaches one?

Solution to Exercise 2

solR = solve_model(update(p; R = 1.03))
@printf("R = 1.02: Gini %.3f, hand to mouth %.1f%%\n",
        wealth_gini(sol), 100*hand_to_mouth(sol))
@printf("R = 1.03: Gini %.3f, hand to mouth %.1f%%\n",
        wealth_gini(solR), 100*hand_to_mouth(solR))
@printf("mass at top grid point: %.1e\n", sum(solR.λ[end, :]))

R = 1.02: Gini 0.552, hand to mouth 31.8%
R = 1.03: Gini 0.510, hand to mouth 22.0%
mass at top grid point: 1.2e-125

A higher return strengthens the saving motive, so fewer households sit at the constraint and more wealth accumulates. In the limit \(\beta R \to 1\) the target asset level diverges and the stationary distribution starts piling mass on whatever ceiling the grid imposes, which is why the top-grid check exists and why the calibration keeps the product safely inside one.

Exercise 3

Run the financed 20 percent subsidy for the United States row and compare the decoupling with France. The rows differ in the agency gap, the belonging tastes, and the cohesion weight. Which effects change size?

Solution to Exercise 3

for code in ("FR", "US")
    pc = update(country_params(code); social_mode = :warmglow, social_strength = 1.0)
    b0 = solve_financed(pc; subsidy = 0.0)
    b1 = solve_financed(pc; subsidy = 0.20)
    @printf("%s: consumption %+.1f%%, public good %+.1f%%\n", code,
            100*(agg(b1, b1.c)/agg(b0, b0.c) - 1), 100*(b1.Q/b0.Q - 1))
end

FR: consumption +6.0%, public good -5.5%
US: consumption +6.6%, public good -6.8%

The signs are the portable result: every calibration decouples the same way. The sizes move with the row, and tracing a size difference back to a parameter difference is a useful way to learn what each dial does.

Final Remarks

This lecture compares stationary economies rather than full transitions, uses the simplest behavioural form of the social term, and holds the interest rate fixed. Natural extensions run in three directions: a discrete participation margin in the tradition of discrete choice with social interactions, where activation policy is amplified by the social feedback; a homophily channel in which belonging is drawn from one’s own group, which segregates cohesion across education groups; and the general-equilibrium close in which \(R\) clears the capital market.

On provenance: the model is from my master’s thesis [Conway, 2020], which operationalised the SAGE framework of Lima de Miranda and Snower [2020] in a heterogeneous-agent economy. This page and its vendored solver live at github.com/conway1521/sage-bewley; the maintained engine, the calibration data and its sourcing, and an interactive Pluto notebook live in the research repository, github.com/conway1521/sage_macro.

References

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Truman Bewley. Stationary monetary equilibrium with a continuum of independently fluctuating consumers. In Contributions to Mathematical Economics in Honor of Gérard Debreu. North-Holland, 1986.

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Raj Chetty, Adam Guren, Day Manoli, and Andrea Weber. Are micro and macro labor supply elasticities consistent? a review of evidence on the intensive and extensive margins. American Economic Review, 101(3):471–475, 2011.

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Alessandro Conway. Wellbeing and macroeconomics: a SAGE approach. Master's thesis, Sciences Po Paris, 2020.

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Tomas Havranek. Measuring intertemporal substitution: the importance of method choices and selective reporting. Journal of the European Economic Association, 13(6):1180–1204, 2015.

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[KVW14] (1,2,3,4)

Greg Kaplan, Giovanni L. Violante, and Justin Weidner. The wealthy hand-to-mouth. Brookings Papers on Economic Activity, Spring:77–138, 2014.

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Katharina Lima de Miranda and Dennis J. Snower. Recoupling economic and social prosperity. Global Perspectives, 2020.

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K. Geert Rouwenhorst. Asset pricing implications of equilibrium business cycle models. In Frontiers of Business Cycle Research. Princeton University Press, 1995.

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Eric R. Young. Solving the incomplete markets model with aggregate uncertainty using the krusell-smith algorithm and non-stochastic simulations. Journal of Economic Dynamics and Control, 34(1):36–41, 2010.