{
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    "pe": "https://proofengine.info/proofs/consider-a-sector-with-n-2-symmetric-firms-each-endowed-with-l-task-positions/",
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  "entity": {
    "pe:claim": {
      "prov:type": "pe:Claim",
      "prov:label": "Consider a sector with \\(N \\geq 2\\) symmetric firms, each endowed with L task-positions. Each firm i chooses an automation rate \\(\\alpha_i\\) in [0,1], paying wage w per human-staffed task and cost c per automated task, with integration friction cost \\((k/2) \\cdot L \\cdot \\alpha_i^2\\) where \\(k > 0\\). Aggregate sectoral demand is \\(D = A + \\lambda \\cdot w \\cdot L \\cdot [N - (1-\\eta) \\sum_j \\alpha_j]\\), where \\(A > 0\\) is exogenous demand, \\(\\lambda \\in (0,1]\\) is workers' marginal propensity to consume from wages, and \\(\\eta \\in [0,1)\\) is the fraction of displaced wage income recovered through reemployment. Each firm's revenue is \\(D/N\\) (equal market shares). Define \\(s = w - c > 0\\) and \\(\\ell = \\lambda(1-\\eta)w > 0\\). Each firm i maximizes \\(\\pi_i = D/N - wL(1-\\alpha_i) - cL\\alpha_i - (k/2)L\\alpha_i^2\\). The Nash equilibrium automation rate is \\(\\alpha^{NE} = (s - \\ell/N)/k\\). The cooperative optimum is \\(\\alpha^{CO} = (s - \\ell)/k\\). The difference \\(\\alpha^{NE} - \\alpha^{CO} = \\ell \\cdot (1 - 1/N)/k\\) is strictly positive.",
      "prov:value": "Consider a sector with \\(N \\geq 2\\) symmetric firms, each endowed with L task-positions. Each firm i chooses an automation rate \\(\\alpha_i\\) in [0,1], paying wage w per human-staffed task and cost c per automated task, with integration friction cost \\((k/2) \\cdot L \\cdot \\alpha_i^2\\) where \\(k > 0\\). Aggregate sectoral demand is \\(D = A + \\lambda \\cdot w \\cdot L \\cdot [N - (1-\\eta) \\sum_j \\alpha_j]\\), where \\(A > 0\\) is exogenous demand, \\(\\lambda \\in (0,1]\\) is workers' marginal propensity to consume from wages, and \\(\\eta \\in [0,1)\\) is the fraction of displaced wage income recovered through reemployment. Each firm's revenue is \\(D/N\\) (equal market shares). Define \\(s = w - c > 0\\) and \\(\\ell = \\lambda(1-\\eta)w > 0\\). Each firm i maximizes \\(\\pi_i = D/N - wL(1-\\alpha_i) - cL\\alpha_i - (k/2)L\\alpha_i^2\\). The Nash equilibrium automation rate is \\(\\alpha^{NE} = (s - \\ell/N)/k\\). The cooperative optimum is \\(\\alpha^{CO} = (s - \\ell)/k\\). The difference \\(\\alpha^{NE} - \\alpha^{CO} = \\ell \\cdot (1 - 1/N)/k\\) is strictly positive."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Nash equilibrium rate alpha^NE = (s - ell/N)/k via FOC",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy symbolic differentiation of pi_i w.r.t. alpha_i; solve FOC; verify result equals (w-c-lambda(1-eta)w/N)/k",
      "pe:result": "Confirmed: residual = 0"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Cooperative optimum alpha^CO = (s - ell)/k via joint FOC",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy symbolic differentiation of total profit Pi w.r.t. alpha at symmetric profile; solve FOC; verify result equals (w-c-lambda(1-eta)w)/k",
      "pe:result": "Confirmed: residual = 0"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Gap alpha^NE - alpha^CO = ell*(1 - 1/N)/k > 0",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy symbolic subtraction alpha^NE - alpha^CO; verify equals lambda(1-eta)w(1-1/N)/k; positivity from parameter assumptions",
      "pe:result": "Confirmed: gap formula correct, strictly positive for N >= 2"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: PROVED",
      "prov:value": "PROVED"
    }
  },
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      "prov:type": "pe:VerdictDetermination",
      "prov:label": "Determine verdict from evidence",
      "prov:endTime": "2026-04-16"
    }
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      "prov:label": "Proof Engine",
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    "pe:gen-verdict": {
      "prov:entity": "pe:verdict",
      "prov:activity": "pe:determine-verdict"
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      "prov:generatedEntity": "pe:evidence-A3",
      "prov:usedEntity": "pe:evidence-A1"
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      "prov:usedEntity": "pe:evidence-A2"
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