1[2 {"id":"C1","text":"Closed-form Marshallian demands are x1*=alpha*m/p1=16.8 and x2*=(1-alpha)*m/p2=19.5 at the given parameters.","results_keys":["x1_demand","x2_demand"],"status":"supported","caveat":"Requires 0<alpha<1 and p1,p2,m>0 (interior, unique optimum)."},3 {"id":"C2","text":"Indirect utility at the optimum is 2.9182520128635305, and an independent numerical maximisation on the budget line reproduces it.","results_keys":["indirect_utility","indirect_utility_numerical"],"status":"supported","caveat":"The numerical validation covers only the single parameter point given."},4 {"id":"C3","text":"The budget constraint binds at the optimum (residual 0).","results_keys":["budget_identity_residual"],"status":"supported","caveat":"Also confirmed by SLSQP with budget as an inequality (slack ~3e-14)."},5 {"id":"C4","text":"At the optimum MRS equals the price ratio (0.625) and the expenditure share on good 1 equals alpha (0.35).","results_keys":["mrs_at_optimum","price_ratio","expenditure_share_good1"],"status":"supported","caveat":"Tangency condition holds for interior solutions only."}6]7