Equilibrium Analysis in EconomicsEquilibriumStatic AnalysisPartial Market EquilibriumGeneral Equilibrium

EquilibriumEquilibrium is a constellation of selected interrelated variables so adjusted to one another that no inherent tendency to change prevails in the model which they constitute

EquilibriumSelectedSome variables are not selected to be in the modelEquilibrium is relevant only to the selected variables and may no longer apply if different variables are included (excluded)

EquilibriumInterrelatedSince the variables are interrelated, all the variables must be in a state of rest if equilibrium is to be achievedInherentThe state of rest refers to the internal forces of the model; external forces (exogenous variables) are assumed fixed

EquilibriumSince equilibrium refers to a lack of change, we often refer to equilibrium analysis as static analysis or statics

Partial Market EquilibriumConstructing the modelAn equilibrium condition, behavioral equations, and restrictions must be specifiedQd = QsQd = a - bP (a, b > 0)Qs = -c + dP (c, d > 0)

Partial Market EquilibriumSolving the modelEquilibrium tells us Qd = Qs so we can substitute into the equilibrium equation and solvea - bP = -c + dPa + c = bP + dPa + c = P(b + d)a + c = P (equilibrium price) b + d

Partial Market EquilibriumNote the solution is entirely in the form of parameters - this is typicalP is positive (as required by economics)a, b, c, d > 0 thereforea + c > 0 as well b + d

Partial Market EquilibriumFind the equilibrium quantity by substituting the equation for price into one of the equations for QQ = a - b * a + c b + dQ = a(b + d) - b * a + c b + d b + d

Partial Market EquilibriumQ = ab + ad - ba - bc b + dQ = ad - bc b + dThe equilibrium value of Q should be > 0b + d > 0 since b, d > 0We have added restriction of ad > bc for Q > 0

Partial Market EquilibriumSuppose we have the following model which results in a quadraticQd = QsQd = 4 - P2Qs = 4p - 1

Partial Market EquilibriumSetting up equation to solve gives us4 - P2 = 4P - 1P2 + 4P - 5 = 0The left-hand expression is a quadratic function of the variable PCan use the quadratic formula to solve the equation

Partial Market EquilibriumGeneral form of a quadratic equation is:ax2 + bx + c = 0Using the quadratic formula, two roots can be obtained from a quadratic equation, x1 and x2x1 and x2 provide solutionsx1, x2 = -b + and - (b2 - 4ac)1/2 2a

Partial Market EquilibriumOur expression is: P2 + 4P - 5 = 0P1, P2 = -4 + and - (42 - 4(1)(-5))1/2 2(1)P1, P2 = -4 + and - (16 + 20)1/2 2P1, P2 = -4 + and - 6 2

Partial Market EquilibriumP1 = -4 + 6 2 2P1 = -2 + 3 = 1P2 = -4 - 6 2 2P2 = -2 -3 = -5Only P1 is relevant since P > 0

Partial Market EquilibriumIf P = 1 then Q =4P - 1 = 3

General Equilibrium ModelOur analysis can extend to n commoditiesThere will be an equilibrium condition for each of the n marketsThere will be behavioral equations for each of the n markets

General Equilibrium ModelEquilibrium conditionsQd1 = Qs1Qd2 = Qs2 . . . . . .Qdn = Qsn

General Equilibrium ModelBehavioral equationsQd1 = a0 + a1P1 + a2P2 + + anPnQs1 = b0 + b1P1 + b2P2 + + bnPnQd2 = c0 + c1P1 + c2P2 + + cnPnQs2 = d0 + d1P1 + d2P2 + + dnPnQdn = 0 + 1P1 + 2P2 + + nPnQsn = 0 + 1P1 + 2P2 + + nPn

General Equilibrium ModelSuch a system is very difficult to solve with the method of substitutionCan use matrix algebra to solve a system of linear equations