$TITLE M6-4.GMS: Economy with two consumers, public good, * optimal provision with an endogenous tax rate, Samuelson rule $ONTEXT Samuelson rule for optimal provision, PG = PG1 + PG2 introduces an auxiliary variable and constraint equation Here is the tax rate is a VARIABLE, set optimally Generalizes M6-3.gms: two consumers with different preferences Production Sectors Consumers Markets| X Y G W1 W2 CONS1 CONS2 GOVT -------------------------------------------------------------- PX | 100 -50 -50 PY | 100 -50 -50 PG | 50 -50 PL | -80 -80 -40 100 100 TAX | -20 -20 -10 50 PW1 | 125 -125 PW2 | 125 -125 PG1 | -25 25 PG2 | -25 25 -------------------------------------------------------------- $OFFTEXT PARAMETERS SHX1, SHY1, SHG1 shares of X Y and G in consumer 1's utility SHX2, SHY2, SHG2 shares of X Y and G in consumer 2's utility; SHG1 = 0.2; SHX1 = 0.5 - SHG1/2; SHY1 = 0.5 - SHG1/2; SHG2 = 0.2; SHX2 = 0.5 - SHG2/2; SHY2 = 0.5 - SHG2/2; POSITIVE VARIABLES X Activity level for sector X, Y Activity level for sector Y, W1 Activity level for sector W1, W2 Activity level for sector W2, G Activity level for government sector, PX Price index for commodity X, PY Price index for commodity Y, PL Price index for primary factor L, PW1 Price index for welfare 1(expenditure function), PW2 Price index for welfare 2(expenditure function), PG1 Private valuation of the public good (consumer 1), PG2 Private valuation of the public good (consumer 2), PG Price (marginal cost) of the public good GOVT Budget restriction for government, CONS1 Income definition for CONS1, CONS2 Income definition for CONS2, LGP Level of government provision TAX Uniform value-added tax rate; EQUATIONS PRF_X Zero profit for sector X PRF_Y Zero profit for sector Y PRF_W1 Zero profit for sector W1 PRF_W2 Zero profit for sector W2 PRF_G Zero profit in government sector MKT_X Supply-demand balance for commodity X MKT_Y Supply-demand balance for commodity Y MKT_L Supply-demand balance for primary factor L MKT_W1 Supply-demand balance for consumer 1 MKT_W2 Supply-demand balance for consumer 2 MKT_G1 Private valuation of the public good (consumer 1) MKT_G2 Private valuation of the public good (consumer 2) MKT_G Supply-demand balance for commodity G I_G Budget restriction for government I_CONS1 Income definition for CONS1 I_CONS2 Income definition for CONS2 A_LGP Auxiliary for government provision A_TAX Auxiliary for government provision; * Zero profit conditions: PRF_X.. 80*PL * (1+TAX) =G= 100*PX; PRF_Y.. 80*PL * (1+TAX) =G= 100*PY; PRF_W1.. 125*PX**(SHX1) * PY**(SHY1) * (PG1/0.5)**(SHG1) =E= 125*PW1; PRF_W2.. 125*PX**(SHX2) * PY**(SHY2) * (PG2/0.5)**(SHG2) =E= 125*PW2; PRF_G.. 40*PL * (1+TAX) =G= 50*PG; * Market clearing conditions: MKT_X.. 100*X =G= 125*SHX1*W1*PW1/PX + 125*SHX2*W2*PW2/PX ; MKT_Y.. 100*Y =G= 125*SHY1*W1*PW1/PY + 125*SHY2*W2*PW2/PY; MKT_W1.. 125*W1 =G= CONS1/PW1; MKT_W2.. 125*W2 =G= CONS2/PW2; MKT_L.. 200 =G= (80*X + 80*Y + 40*G); MKT_G1.. 50 * LGP =G= 125*SHG1 * W1 * PW1/PG1; MKT_G2.. 50 * LGP =G= 125*SHG2 * W2 * PW2/PG2; MKT_G.. 50*G =G= GOVT/ PG; * Income constraints: I_G.. GOVT =G= PL*(80*X + 80*Y + 40*G )*TAX; I_CONS1.. CONS1 =E= 100*PL + 50*LGP*PG1; I_CONS2.. CONS2 =E= 100*PL + 50*LGP*PG2; * Auxiliary constraints: A_LGP.. LGP =E= G; A_TAX.. PG =E= PG1 + PG2; MODEL PUBGOOD2 /PRF_X.X, PRF_Y.Y, PRF_W1.W1, PRF_W2.W2, PRF_G.G, MKT_X.PX, MKT_Y.PY, MKT_L.PL, MKT_W1.PW1, MKT_W2.PW2, MKT_G.PG, MKT_G1.PG1, MKT_G2.PG2, I_G.GOVT, I_CONS1.CONS1, I_CONS2.CONS2, A_LGP.LGP, A_TAX.TAX /; X.L =1; Y.L =1; W1.L =1; W2.L =1; G.L =1; PL.FX =1; PX.L =1; PY.L =1; PG.L =1; PW1.L =1; PW2.L =1; PG1.L =0.5; PG2.L =0.5; CONS1.L =125; CONS2.L =125; GOVT.L =50; LGP.L =1; TAX.L =0.25; PUBGOOD2.ITERLIM = 0; SOLVE PUBGOOD2 USING MCP; PUBGOOD2.ITERLIM = 2000; SOLVE PUBGOOD2 USING MCP; * Change consumer 1's preferences, higher preference for the * public good, which now has a Cobb-Douglas share of 0.3 SHG1 = 0.3; SHX1 = 0.5 - SHG1/2; SHY1 = 0.5 - SHG1/2; *PUBGOOD2.ITERLIM = 0; SOLVE PUBGOOD2 USING MCP; TAX.FX = 0.25; SOLVE PUBGOOD2 USING MCP;