$TITLE  Model M34: Closed 2x2 Economy with a public good

SCALAR  TAX     Value-added tax rate;

POSITIVE VARIABLES
        X       ! Activity level for sector X,
        Y       ! Activity level for sector Y,
        W1      ! Activity level for sector W1 (Hicksian welfare index),
        W2      ! Activity level for sector W2 (Hicksian welfare index),
	G	! Activity level for government sector,
	PX      ! Price index for commodity X,
        PY      ! Price index for commodity Y,
        PK      ! Price index for primary factor K,
        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	! Private valuation of the public good
	
	GOVT	! Budget restriction for government,
	CONS1	! Income definition for CONS1,
	CONS2	! Income definition for CONS2,

	LGP	! Level of government provision;

EQUATIONS
        PRF_X   Zero profit for sector X
        PRF_Y   Zero profit for sector Y
        PRF_W1  Zero profit for sector W1 (Hicksian welfare index)
        PRF_W2  Zero profit for sector W2 (Hicksian welfare index)
	PRF_G	Zero profit in government sector
   
        MKT_X   Supply-demand balance for commodity X
        MKT_Y   Supply-demand balance for commodity Y
        MKT_G   Supply-demand balance for commodity G
        MKT_L   Supply-demand balance for primary factor L
        MKT_K   Supply-demand balance for primary factor K
	MKT_W1  Supply-demand balance for aggregate demand consumer 1
	MKT_W2  Supply-demand balance for aggregate demand consumer 2
	MKT_G1  Private valuation of the public good (consumer 1)
	MKT_G2  Private valuation of the public good (consumer 2)

	I_G	Budget restiktion 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..		100 * (PL**(3/8) * PK**(5/8)) * (1+TAX)/(1.25) =G= 100 * PX;

PRF_Y..		100 * (PL**(5/8) * PK**(3/8)) * (1+TAX)/(1.25) =G= 100 * PY;

PRF_G..		50 *  (PL**0.5 * PK**0.5) * (1+TAX)/(1.25) =G= 50 * PG;

PRF_W1..	125 * PX**(30/125) * PY**(70/125) * (PG1/0.5)**(25/125) =E= 125 * PW1;

PRF_W2..	125 * PX**(70/125) * PY**(30/125) * (PG2/0.5)**(25/125) =E= 125 * PW2;

*	Market clearing conditions:

MKT_X..		100 * X =G= 30 * W1 * PW1/PX + 70 * W2 * PW2/PX ;

MKT_Y..		100 * Y =G= 70 * W1 * PW1/PY + 30 * W2 * PW2/PY;
   
MKT_G..		50 * G =G= GOVT/ PG;

MKT_W1..	125 * W1 =G= CONS1 / PW1;

MKT_W2..	125 * W2 =G= CONS2 / PW2;

MKT_L..		50 + 50 =G= (50 * X * PX + 30 * Y * PY + 20 * G * PG)*1.25/(PL*(1+TAX));

MKT_K..		50 + 50 =G= (30 * X * PX + 50 * Y * PY + 20 * G * PG)*1.25/(PK*(1+TAX));

MKT_G1..	50 * LGP =G= 50 * W1 * PW1*0.5/PG1;

MKT_G2..	50 * LGP =G= 50 * W2 * PW2*0.5/PG2;
			
*	Income constraints:

I_G..		GOVT =G= PL * (50 * X + 30 * Y + 20 * G ) * TAX + PK * (30 * X + 50 * Y + 20 * G ) * TAX;

I_CONS1..	CONS1 =E= 50 * PL + 50 * PK + 50 * LGP * PG1;

I_CONS2..	CONS2 =E= 50 * PL + 50 * PK + 50 * LGP * PG2;

*	Auxiliary constraints:

A_LGP..		LGP =E= G;

MODEL ALGEBRAIC /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_K.PK, 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;
PK.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;

ALGEBRAIC.ITERLIM = 0;
SOLVE ALGEBRAIC USING MCP;
ALGEBRAIC.ITERLIM = 2000;

*    The following counterfactuals check that the original
*    benchmark is indeed an optimum by raising/lowering the tax

TAX = 0.20;

SOLVE ALGEBRAIC USING MCP;

TAX = 0.30;

SOLVE ALGEBRAIC USING MCP;