Unformatted text preview: Principles of Economics I
Economics 101 Class 10 Announcements Readings for today and Wednesday Chapter 6 Notes available on CTools Discussion Sections this week New assignment is available US Demand for Sugar
$/lb $0.21 $0.08 20.3 US Sugar Demand 24.3 Lbs (billions) The Demand Function Example: Sugar Demand Two points on the demand curve: (P0, Q0) = (0.08, 24.3) (P1, Q1) = (0.21, 20.3) If the demand curve is linear, then the gradient (slope) between any two points is the same Slope = rise (0.21  0.08) 0.13 = = = 0.0325 run (20.3  24.3) 4 US Demand for Sugar
$/lb Slope = 0.13/4 = 0.0325 $0.21 Rise = 0.13 $0.08 Run = 4 20.3 US Sugar Demand 24.3 Lbs (billions) The Demand Function The slope between any two points on the demand curve is the same E.g.. slope between must equal 0.0325
i.e. (P,Qd) (0.08, 24.3) an arbitrary point on the demand curve) rise ( P  0.08) = = 0.0325 run (Qd  24.3) Rearrange this: Qd= 26.76 30.77 P US Demand for Sugar
$/lb P Slope = 0.0325 =  (24.3 Q) P 0.08 $0.21 Rise = P  0.08 Run = Q 24.3 $0.08 Q 20.3 US Sugar Demand 24.3 Lbs (billions) US Demand for Sugar
Qd = 26.76 30.77 P What do 26.76 and 30.77 mean? If P = 0, then Qd = 26.76 i.e 26.76 is the intercept on the horizontal axis If P rises by $1/lb, quantity demanded falls by 30.77 billion pounds i.e. 30.77 is the inverse of the slope of the demand curve US Demand for Sugar
$/lb US Sugar Demand Slope = 1/30.77 = 0.0375 $0.21 $0.08 20.3 24.3 26.76 Lbs (bil) The Linear Demand Function Generically: Qd = a + b P where a = intercept on the Qdaxis b = inverse of the slope (b < 0) Price Coefficient of the Demand Function
Qd = a + bP Negative slope implies b < 0 b reflects the sensitivity of demand to changes in the price If demand does not respond significantly to changes in price then b is very small demand curve is relatively steep If demand responds a lot to changes in price then b is large demand curve is relatively flat Sensitivity of demand to changes in price
P Demand is relatively price elastic, i.e. quantity demanded is sensitive to price changes b is large and negative P0 P1 Q0 Q1 Q Sensitivity of demand to changes in price
P Demand is infinitely or perfectly price elastic, i.e. quantity demanded responds overwhelmingly to changes in price b is not defined P0 Q Elastic Demand Relatively sensitive to price changes Price cofficient in the demand function, b, is a large negative number Typically the result of the availability of close substitutes for the good Demand for domestically produced goods was very elastic when identical exports were available Demand for one brand of rice is likely to be very elastic: If the price of that brand rises, most buyers will substitute to another brand Sensitivity of demand to changes in price
P Demand is relatively price inelastic, i.e. quantity demanded is not sensitive to price changes P0 b is small and negative P1 Q0 Q1 Q Sensitivity of demand to changes in price
P Demand is perfectly price inelastic, i.e. quantity demanded is totally insensitive to price changes b=0 P0 P1 Q0 Q Inelastic Demand Relatively insensitive to price changes Price coefficient in the demand function, b, is a small negative number Typically the result of the scarcity of close substitutes for the good Demand for specific drugs tend to be inelastic Firms often advertise to convince people that their products do not have close substitutes E.g. Insulin Less elastic demand curves will allow high prices without as great a loss in sales The Qaxis intercept Recall our simple linear demand function: Qd = a + b P where a = intercept on the Qdaxis = quantity demanded if P = 0 The value of a will depend on many factors Changes in these factors will be shown to shift the demand function. The Qaxis intercept Factors that shift the demand curve will changes in the Qaxis intercept of the linear demand curve These factors include: Changes in income Changes in prices of related goods (substitutes and complements) Changes in population Changes in environmental factors Changes in tastes Advertising expenditures A more general model of demand Linear demand function: Qd= b0 + b1P + b2Pother+ b3Y + b4Pop + ...
Where: P = own price of the good Psub = price of some other good Y = income (e.g. average household income) Pop = population etc A more general model of demand Rearranging the same expression: Qd = [b0 + b2Pother+ b3Y + b4Pop +...] + b1P or Qd = a + b1 P where a = [b0 + b2Pother + b3Y + b4Pop + ...] i.e. changes in demand determinants other than own price will simply change the Qaxis intercept Call these "demand shifters" or "shift variables" Interpreting the Coefficients Price of some other good If the coefficient for Pother < 0, then the other good is a complement Increase in Pother shifts demand curve to the left If the coefficient for Pother > 0, then the other good is a substitute Increase in Pother shifts demand curve to the right Income If the coefficient for Y > 0, then the good is normal Increase in Y shifts demand curve right Increase in Y shifts demand curve left If the coefficient for Y < 0, then the good is inferior Estimating the Demand Curve Suppose we gather some data: Pricequantity observations (Pi, Qi) Maybe the same market in different periods i.e. at price Pi, the quantity demanded was Qi Time series data Maybe different (but similar) markets at the same time Maybe different (but similar) markets at various times Crosssectional data Panel data Can we use these data to infer what the demand function is like? 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 But we don't know the deterministic portion of the model We must estimate the demand curve, using the information available i.e. observed pricequantity pairs Estimated function: ^ ^ ^ Q( P) = a + b P Where ( and ) ^ b <0 ^ a >0 P e1 Residuals
e3 P1 P2 P3 P4 e4 P5 Slope = 1 ^ b Q ^ a Residuals Residual: ^ ei = Qi  Q( Pi ) Measure of how inaccurate the estimate is We want to choose an estimated function that gives small residuals What does that mean? In previous diagram, making the curve steeper will make some residuals bigger and some smaller. Is this good? P Sum of Residuals
e3 = 50
3 30 20 2 10 1 e1 =  50 50 100 150 200 300 Q Ordinary Least Squares 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: For each observation of demand, we know: Qd i.e. the dependent variable; and P, x2..., xn i.e. 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 ] 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 Slope = 1/3 2 e = 2 e1 = 1 Qd = 20 3 P 4 6 1 2 3 8 20 Q Why Linear Estimators? Demand functions are unlikely to be linear But linear estimators... And nonlinear estimators... are simple are often not bad approximations for reality can sometimes be nonlinear estimation in disguise Linear estimators are certainly a good place to start analysis are complicated often require some knowledge of the precise nature of non linearity 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 as P 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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