$TITLE M6-3: Economy with two households and a public good $ONTEXT How do we model a public good that is non-excludable and non-rivaled? 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 PARAMETER TAX Value-added tax rate; NONNEGATIVE 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 of (cost of producing) the public good GOVT Budget restriction for government CONS1 Income definition for CONS1 CONS2 Income definition for CONS2 LGP Endowment of public good received by each consumer; 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; * Zero profit conditions: PRF_X.. 80*PL * (1+TAX) =G= 100*PX; PRF_Y.. 80*PL * (1+TAX) =G= 100*PY; PRF_G.. 40*PL * (1+TAX) =G= 50*PG; PRF_W1.. 125*PX**(50/125) * PY**(50/125) * (PG1/0.5)**(25/125) =G= 125*PW1; PRF_W2.. 125*PX**(50/125) * PY**(50/125) * (PG2/0.5)**(25/125) =G= 125*PW2; * Market clearing conditions: MKT_X.. 100*X =G= 50*W1*PW1/PX + 50*W2*PW2/PX ; MKT_Y.. 100*Y =G= 50*W1*PW1/PY + 50*W2*PW2/PY; MKT_L.. 200 =G= (80*X + 80*Y + 40*G); MKT_W1.. 125*W1 =G= CONS1 / PW1; MKT_W2.. 125*W2 =G= CONS2 / PW2; MKT_G.. 50*G =G= GOVT/ PG; MKT_G1.. 50*LGP =G= 25 * W1 * PW1/PG1; MKT_G2.. 50*LGP =G= 25 * W2 * PW2/PG2; * 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; MODEL PUBGOOD /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 /; 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 =0.25; PUBGOOD.ITERLIM = 0; SOLVE PUBGOOD USING MCP; PUBGOOD.ITERLIM = 2000; SOLVE PUBGOOD USING MCP; * The following counterfactuals check that the original * benchmark is indeed an optimum by * raising/lowering the tax TAX = 0.10; SOLVE PUBGOOD USING MCP; TAX = 0.40; SOLVE PUBGOOD USING MCP;