1[2 {"id":"C1","text":"Under 0<alpha<1, p>0, m>0 the unique optimum is x1*=alpha m/p1=16.8 and x2*=(1-alpha)m/p2=19.5.","results_keys":["x1_demand","x2_demand"],"status":"supported","caveat":"Uniqueness is analytical; numerical check at one parameter point."},3 {"id":"C2","text":"Indirect utility at the optimum is 2.9182520128635305 utils.","results_keys":["indirect_utility"],"status":"supported","caveat":""},4 {"id":"C3","text":"The budget binds exactly at the closed-form demands (residual 0).","results_keys":["budget_identity_residual","validation_details"],"status":"supported","caveat":"SLSQP slack also 0."},5 {"id":"C4","text":"At the optimum MRS equals the price ratio (0.625) and the expenditure share of good 1 equals alpha (0.35).","results_keys":["mrs_at_optimum","price_ratio","expenditure_share_good1"],"status":"supported","caveat":""},6 {"id":"C5","text":"An independent numerical maximisation on the budget line reproduces the closed-form utility to machine precision and the maximiser to ~2e-7.","results_keys":["indirect_utility_numerical","validation_details"],"status":"supported","caveat":"Utility agreement is weak evidence due to flatness; covers only the given parameter point."}7]8