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Unformatted text preview: Principles of Economics I
Economics 101 Class 11 Announcements Readings for today Chapter 6 Notes available on CTools Discussion Sections this week Assignment is available on CTools Estimating the Demand Curve
P P1 1 P2 P3 P4 2 3 4 P5 5 Q1 Q2 Q3 Q4 Q5 Q Ordinary least squares estimation Suppose we knew (or believed) the demand function to have the form Qd = a + b P + a + b P : deterministic (predictable) variation in Qd : random (unpredictable) variation in Qd This term explains why the data don't appear to fall on a linear demand curve P P1 1 Qd = a + b P P2 P3 P4 P5 5 a Q 4 2 Estimating the Demand Curve Make use of available information i.e. observed pricequantity pairs Attempt to estimate the deterministic portion of demand i.e. which deterministic, linear demand curve is most likely to have generated the observed data? Estimated function: ^ ^ ^ Q( P) = a + b P ^ ^ a >0 b< Where ( and ) 0 Estimated Functions and Residuals Create an estimated demand curve by choosing any values . . ^ a>0 ^ b<0 So that ^ ^ ^ Q( P) = a + b P
^ ei = Qi  Q( Pi ) Residuals: A residual is measured for each observation, (Qi, Pi) Measures how inaccurate the estimate is A good estimate is associated with small residuals P e1 Residuals
e3 P1 P2 P3 P4 P5 e4 1 slope = ^ b
Q ^ a> 0 ^ ^ ^ Q( P) = a + b P Estimated Functions and Residuals An estimated function generates a residual for every observation Measure of how inaccurate the estimate is Smaller residuals are better But how do we aggregate the residuals? P e1
P1 Residuals
e2 e3 P2 P3 P4 P5 e4 e5
Q Sum of Residuals
P e3 = 50 30
3 20 2 e1 =  50
1 10 50 100 150 200 300 Q Alternative Aggregations of Residuals Sum of Absolute Residuals Sum of Squared Residuals Preferred method Generates Ordinary Least Squares (OLS) estimators of the intercept and slope parameters Implies aversion to realizing large residuals OLS an example Real (but unknown) demand fn: Qdi = a + b Pi + i Three observations:
1) 2) 3) (Qd1, P1) = (1, 6) (Qd2, P2) = (8, 4) (Qd3, P3) = (3, 6) P OLS an example e = 1
3 6 1 3 Slope = 1/3 2
4 e = 2 e1 = 1
2 Qd = 20 3 P 6 1 2 3 8 20 Q OLS with many explanatory variables If the real (but unknown) demand function has the form:
Qd = a + b1 P + b2 x2 + ...+ bn xn + Estimate demand using:
^ ^ ^ ^ ^ Q d = a + b1 P + b2 x 2 + + bn x n where parameters are chosen to minimize the sum of all squared residuals. Ordinary Least Squares To calculate residuals: Each observation tells us Qd i.e. the value of the dependent variable; and P, x2..., xn i.e. values of the independent variables Evaluate the estimated function using the observed values of the independent variables: ^ ^ ^ ^ ^ Q d = a + b1 P + b2 x 2 + + bn x n Find the difference between the observed and estimated quantities demanded, using the same independent variable values: ^ ^ ^ ^ e = Q d  [a + b1 P + b2 x 2 + + bn x n ] Why Linear Estimators? Demand functions are unlikely to be linear But linear estimators... are simple are often not bad approximations for reality can sometimes be nonlinear estimation in disguise And nonlinear estimators... are complicated often require some knowledge of the precise nature of nonlinearity Linear estimators are certainly a good place to start analysis Linear estimator as approximation to nonlinear function
P Q Linear estimators might be nonlinear estimators in disguise Suppose we knew the demand function had the form log Qd = a + b log P + Could estimate the linear function: where p = log P and qd = log Qd ^ ^ ^ qd = a + b p i.e. simply take logarithms of all variables and perform OLS on the transformed data Supply Functions Also work primarily with linear supply functions in this course Simple case: price is the only explanatory variable QS = a + b P b : inverse of the supply curve slope a : intercept on the horizontal axis Always positive May be positive or negative Q = 10 + 5 P
s
P Slope = 1/5 6 10 40 Q Q = 10 + 5 P
s
P Slope = 1/5 6 10 20 Q Infinitely elastic supply
P Supply function is undefined Slope = 0 P0 Q1 Q2 Q Relatively elastic supply
P Qs = a + belastic P
0 < belastic Slope = P/Q = 1/belastic > 0 P2 P1 P Q Q1 Q2 Q Relatively inelastic supply
P P2 Qs = a + binelastic P
0 < binelastic < belastic P Slope = P/Q = 1/binelastic > 0 P1 Q1 Q Q2 Q Perfectly inelastic supply
P P2 Qs = a b = 0 Slope is not defined P1 a Q2 Q Supply as a function of many variables Example: Qs = 100 + 2 P + 10 x 12 y + 2P: quantity supplied increases as P increases i.e. supply curve is upward sloping Every $1 increase in price > 2 unit increase in quantity supplied Coefficient is a measure of sensitivity of quantity supplied to changes in P Supply as a function of many variables Example: Qs = 100 + 2 P + 10 x 12 y + 10 x : quantity supplied increases as x increases What might x be? Increasing x will shift the supply curve to the right Coefficient is a measure of sensitivity: a large coefficient > large shift in the supply curve Productivity index Measure of technology Index of some favorable environmental factor Level of training for workforce etc. Supply as a function of many variables Example: Qs = 100 + 2 P + 10 x 12 y 12 y: quantity supplied decreases y increases What might y be? i.e. supply curve shifts leftward as y increases Coefficient is a measure of sensitivity: a large coefficient (in magnitude) > large shift in the supply curve Cost parameter Price of an alternative product Unfavorable environmental factor Supply as a function of many variables Example: Qs = 100 + 2 P + 10 x 12 y 100 + 10 x 12 y: supply curve intercept on the horizontal axis Increasing x will increase this intercept (i.e. rightward shift in supply curve) Increasing y will decrease this intercept (i.e. leftward shift in the supply curve) What are supply and demand functions good for? Provide quantitative predictions about market outcomes Examples: Predict prices/sales volumes Predict sizes of changes in price/sales volume due to market shocks Example Demand: Supply: Qd = 100 4 P Qs = 20 + 2 P Equilibrium: Qd = Qs 100 4 P = 20 + 2P 6 P = 120 P = 20 Q = 20 Example
P Supply: Q = 20 + 2P Slope = 1/2 25 20 Demand: Q = 100 4P Slope = 1/4 10 20 20 100 Q Comparative Statics
P Supply: Qs = a + bP Q/P = b P1 P P0 Q0 Q Q1 Q ...
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This note was uploaded on 05/16/2010 for the course ECON Section 40 taught by Professor Hogan during the Winter '09 term at University of Michigan.
 Winter '09
 Hogan
 Economics, Microeconomics

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