$TITLE: M4-3b.GMS: modeling health insurance * with moral hazzard, adverse selection modeled as an MCP * COPYRIGHT JAMES R. MARKUSEN $ONTEXT MODELING DEMAND FOR HEALTH INSURANCE |---Sick |--Do not buy Insurance--choose effort --| | |---Healthy Income---| + Type | |---Sick |-- Buy Insurance --choose effort---| |---Healthy $OFFTEXT PARAMETERS M0 income in the first time period MH income in the second time period when healthy MS income in the second time period when sick (before insurance) ACUF acutuarially fairness 1 = actuarily fair ACUF < 1 unfair BETA needed to make the consumption concave(diminishing returns) TYPE1 low-risk type: probability of good health at effort = 0 TYPE2 high-risk type: probability of good health at effort = 0 RISKAV average riskiness at effort = 0 EFFT1 effort of type 1 EFFT2 effort of type 2 PROFIT1 profit from insuring type 1 PROFIT2 profit from insuring type 2 PROFIT profit for insurance company from insuring person 1 and 2; ACUF=1.0; BETA = 0.5; M0 = 10; MH = 10; MS = 4; TYPE1 = 0.5; TYPE2 = 0.5; RISKAV = ((1-TYPE1)+(1-TYPE2))/2; *PROFIT = 0; POSITIVE VARIABLES U1 expected utility type 1 INS1 insurance purchased PNS1 payoff from insurance if sick ALPHA1 probability of good health EFFORT1 effort spent to insure good health: diet exercise and such LAMINS1 Lagrangean multiplier on constraint equation INSURANCE1 LAMEFF1 Lagrangean multiplier on constraint equation MORALHAZ1 U2 expected utility type 2 INS2 insurance purchased PNS2 compensation from purchasing insurance if sick ALPHA2 probability of good health EFFORT2 effort spent to insure good health: diet exercise and such LAMINS2 Lagrangean multiplier on constraint equation INSURANCE2 LAMEFF2 Lagrangean multiplier on constraint equation MORALHAZ2; EQUATIONS UTILITY1 expected utility type 1 INSURANCE1 the amount of insurance puchased MORALHAZ1 good health depends on effort but effort reduces U FOCINS1 first-order condition for insurance purchased FOCPNS1 first-order condition for payoff when sick FOCEFF1 first-order condition for effort1 FOCALP1 first-order condition for alpha1 UTILITY2 expected utility type 2 INSURANCE2 the amount of insurance puchased MORALHAZ2 good health depends on effort but effort reduces U FOCINS2 first-order condition for insurance purchased FOCPNS2 first-order condition for payoff when sick FOCEFF2 first-order condition for effort2 FOCALP2 first-order condition for alpha2; UTILITY1.. U1 =E= (M0-INS1)**BETA + ALPHA1*MH**BETA + (1-ALPHA1)*(MS+PNS1)**BETA - 0.06*(EFFORT1 + EFFORT1**2); INSURANCE1.. INS1*ACUF =E= PNS1*RISKAV; MORALHAZ1.. TYPE1 + 0.15*EFFORT1 =G= ALPHA1; FOCINS1.. BETA*(M0-INS1)**(BETA-1) =G= LAMINS1*ACUF; FOCPNS1.. LAMINS1*RISKAV =G= (1-ALPHA1)*BETA*(MS+PNS1)**(BETA-1); FOCEFF1.. 0.06 + 0.12*EFFORT1 =G= 0.15*LAMEFF1; FOCALP1.. LAMEFF1 =G= MH**BETA - (MS+PNS1)**BETA; UTILITY2.. U2 =E= (M0-INS2)**BETA + ALPHA2*MH**BETA + (1-ALPHA2)*(MS+PNS2)**BETA - 0.06*(EFFORT2 + EFFORT2**2); INSURANCE2.. INS2*ACUF =E= PNS2*RISKAV; MORALHAZ2.. TYPE2 + 0.15*EFFORT2 =G= ALPHA2; FOCINS2.. BETA*(M0-INS2)**(BETA-1) =G= LAMINS2*ACUF; FOCPNS2.. LAMINS2*RISKAV =G= (1-ALPHA2)*BETA*(MS+PNS2)**(BETA-1); FOCEFF2.. 0.06 + 0.12*EFFORT2 =G= 0.15*LAMEFF2; FOCALP2.. LAMEFF2 =G= MH**BETA - (MS+PNS2)**BETA; MODEL INSURE /UTILITY1.U1, INSURANCE1.LAMINS1, MORALHAZ1.LAMEFF1 FOCINS1.INS1, FOCPNS1.PNS1, FOCEFF1.EFFORT1, FOCALP1.ALPHA1 UTILITY2.U2, INSURANCE2.LAMINS2, MORALHAZ2.LAMEFF2 FOCINS2.INS2, FOCPNS2.PNS2, FOCEFF2.EFFORT2, FOCALP2.ALPHA2/; U1.L = 1; INS1.L =2; PNS1.L = 4; ALPHA1.L = 0.5; EFFORT1.L = 0; U2.L = 1; INS2.L =2; PNS2.L = 4; ALPHA2.L = 0.5; EFFORT2.L = 0; SOLVE INSURE USING MCP; PROFIT1 = INS1.L - (1 - ALPHA1.L)*PNS1.L; PROFIT2 = INS2.L - (1 - ALPHA2.L)*PNS2.L; PROFIT = PROFIT1 + PROFIT2; EFFT1 = EFFORT1.L; EFFT2 = EFFORT2.L; DISPLAY TYPE1, TYPE2, EFFT1, EFFT2, PROFIT1, PROFIT2, PROFIT; * counterfactual: two risk types TYPE1 = 0.55; TYPE2 = 0.45; RISKAV = ((1-TYPE1)+(1-TYPE2))/2; SOLVE INSURE USING MCP; PROFIT1 = INS1.L - (1 - ALPHA1.L)*PNS1.L; PROFIT2 = INS2.L - (1 - ALPHA2.L)*PNS2.L; PROFIT = PROFIT1 + PROFIT2; EFFT1 = EFFORT1.L; EFFT2 = EFFORT2.L; DISPLAY TYPE1, TYPE2, EFFT1, EFFT2, PROFIT1, PROFIT2, PROFIT; * can the insurance companies make money by offering actuarily * unfair insurance? ACUF = 0.9; SOLVE INSURE USING MCP; PROFIT1 = INS1.L - (1 - ALPHA1.L)*PNS1.L; PROFIT2 = INS2.L - (1 - ALPHA2.L)*PNS2.L; PROFIT = PROFIT1 + PROFIT2; EFFT1 = EFFORT1.L; EFFT2 = EFFORT2.L; DISPLAY TYPE1, TYPE2, EFFT1, EFFT2, PROFIT1, PROFIT2, PROFIT;