5  ECONOMETRICS CHAPTER Yi = B1 + B2 ln(Xi2) + ui Functional Forms and Regression Models The X and Y variables do not have to be linear. (The model must be linear in parameters.)  Yi = B1 + B2 ln(Xi2) + ui This is okay, even though it is not linear in the explanatory variable.

Y = bX  10,000 = 104 logbY = X  log1010,000 = 4 100 = e4.6051 Review of Logarithms Y = bX  10,000 = 104 logbY = X  log1010,000 = 4 100 = e4.6051 loge100 = ln(100) = 4.6051 ln(AB) = ln(A) + ln(B) ln(Ak) = k ln(A) ln(1) = 0

ln(Yi) = ln(A) + B2ln(Xi) The Log-Linear Model Yi = AXiB2 We ignore the error term, ui, for the time being. We can take the natural log of both sides: ln(Yi) = ln(AXiB2) ln(Yi) = ln(A) + B2ln(Xi) ln(Yi) = B1 + B2ln(Xi)

Demand curve w/ constant slope Price and Quantity 20 40 60 80 100 120 25 50 75 125 150 Price ($) Quantity (100s) Demand curve w/ constant slope Q = 150 – 1.25 P slope = -1.25 % ΔQD % ΔP Price elasticity of demand = = slope x P/Q Elasticity = -1.25 x 40/100 = -0.5 Elasticity = -1.25 x 100/25 = -5.0

Constant price elasticity of demand 20 40 60 80 100 120 Price ($) Quantity (100s) Constant price elasticity of demand Q = 64,000 P-2 slope = -128,000 P-3 Price elasticity of demand = slope x P/Q Elasticity = -0.25 x 80/10 = -2 Note: ln(Q) = ln(64,000) – 2 ln(P) ln(Q) = 11.07 - 2 ln(P) Elasticity = -2 x 40/40 = -2

Qi = B1 + B2 Pi elasticity = B2 x Pi / Qi ln(Qi) = B1 + B2ln(Pi) Price and Quantity Linear (constant slope) model: If B2 = 2,150… $1 increase in price 2,150 more units. Qi = B1 + B2 Pi elasticity = B2 x Pi / Qi Log-linear (constant elasticity) model: If B2 = 0.45… 10% increase in price 4.5% increase in units ln(Qi) = B1 + B2ln(Pi) elasticity = B2

ln(Yi) = B1 + B2ln(X2i) + B3ln(X3i) + ui Multiple Log-Linear Models ln(Yi) = B1 + B2ln(X2i) + B3ln(X3i) + ui Cobb-Douglas Production Function Output = A (LaborB1)(CapitalB2) ln(Yi) = B1 + B2ln(Labori) + B3ln(Capitali) + ui

Double labor or capital, output increases but does not double. Constant returns to scale Double labor or capital, output increases but does not double. Double both labor and capital, output doubles.

ln(Yi) = B1 + B2ln(Labori) + B3ln(Capitali) + ui Cobb-Douglas Production Function ln(Yi) = B1 + B2ln(Labori) + B3ln(Capitali) + ui Constant returns to scale: B2 + B3 = 1 Decreasing returns to scale: B2 + B3 < 1

ln(Yi) = B1 + B2X2i + B3X3i + ui The Semilog Model Log variable(s) only on one side of the equation. Most useful for variables that grow exponentially. Pop1 = 100 Pop2 = 100 x 1.04 = 104 Pop3 = 104 x 1.04 = 108.16 Pop4 = 1.0816 x 1.04 = 112.49

ln(Popi) = 4.63 + .0385 ti + ui A one year increase in time results in a 3.85% change in population