Equilibrium Analysis in Economics

Equilibrium Analysis in Economics • Equilibrium • Static Analysis • Partial Market Equilibrium • General Equilibrium

Equilibrium • Equilibrium 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

Equilibrium • Selected • Some variables are not selected to be in the model • Equilibrium is relevant only to the selected variables and may no longer apply if different variables are included (excluded)

Equilibrium • Interrelated • Since the variables are interrelated, all the variables must be in a state of rest if equilibrium is to be achieved • Inherent • The state of rest refers to the internal forces of the model; external forces (exogenous variables) are assumed fixed

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

Partial Market Equilibrium • Constructing the model • An equilibrium condition, behavioral equations, and restrictions must be specified • Qd = Qs • Qd = a - bP (a, b > 0) • Qs = b + dP (c, d > 0)

Partial Market Equilibrium • Solving the model • Equilibrium tells us Qd = Qs so we can substitute into the equilibrium equation and solve • a - bP = -c + dP • a + c = bP + dP • a + c = P(b + d) • a + c = P (equilibrium price) b + d

Partial Market Equilibrium • Note the solution is entirely in the form of parameters - this is typical • P is positive (as required by economics) • a, b, c, d > 0 therefore • a + c > 0 as well b + d

Partial Market Equilibrium • Find the equilibrium quantity by substituting the equation for price into one of the equations for Q • Q = a - b * a + c b + d • Q = a(b + d) - b * a + c b + d b + d

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

Partial Market Equilibrium • Suppose we have the following model which results in a quadratic • Qd = Qs • Qd = 4 - P2 • Qs = 4p - 1

Partial Market Equilibrium • Setting up equation to solve gives us • 4 - P2 = 4P - 1 • P2 - 4P - 5 = 0 • The left-hand expression is a quadratic function of the variable P • Can use the quadratic formula to solve the equation

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

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

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

Partial Market Equilibrium • If P = 1 then Q =4P - 1 = 3

General Equilibrium Model • Our analysis can extend to n commodities • There will be an equilibrium condition for each of the n markets • There will be behavioral equations for each of the n markets

General Equilibrium Model • Equilibrium conditions Qd1 = Qs1 Qd2 = Qs2 . . . . . . Qdn = Qsn

General Equilibrium Model • Behavioral equations Qd1 = a0 + a1P1 + a2P2 + … + anPn Qs1 = b0 + b1P1 + b2P2 + … + bnPn Qd2 = c0 + c1P1 + c2P2 + … + cnPn Qs2 = d0 + d1P1 + d2P2 + … + dnPn Qdn = 0 + 1P1 + 2P2 + … + nPn Qsn = 0 + 1P1 + 2P2 + … + nPn

General Equilibrium Model • Such a system is very difficult to solve with the method of substitution • Can use matrix algebra to solve a system of linear equations

Equilibrium Analysis in Economics