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Frist mid term review

# Frist mid term review - 1 Drawing the demand functions and...

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1. Drawing the demand functions and supply functions and &ning the equi- librium price and quantity by solving the two equations together Q D = a bP;Q S = c + dP: 2. Finding the price elasticity of demand and supply dQ D dP P Q D = bP Q D ; dQ S dP P Q S = dP Q S 3. Fitting linear equations to demand data: demand elasticity (at equilibrium): & 0 : 5 equilibrium price: P = \$2 equilibrium quantity: Q = 12 million metric tons per year demand curve: Q = a bP Finding a;b & 0 : 5 = b ± P Q = b ± 2 12 ; hence b = 3 : Substitute this value and values of P ;Q into the demand equation Q = a bP ; we get the value for a = 18 : Therefore demand is given by Q = 18 3 P 4. Drawing indi/erence curves for di/erent utility functions through a point ( x 0 ;y 0 ) u ( x;y ) = ax + by;u ( x;y ) = x a y b ;u ( x;y ) = a ln x + b ln y;u ( x;y ) = min( ax;by ) ;u ( x;y ) = ln x + y 5. Drawing a budget constraint p x x + p y y = I: 6. Finding optimal consumption by solving MRS = p x p y ;p x x + p y y = I: (This applies to Cobb-Douglas, logarithm and quasi-linear case with interior solutions) (a) Cobb-Douglas case: u ( x;y ) = x 2 y MRS = 2 xy x 2 = 2 y x and more generally when u ( x;y ) = x a y b MRS = ax a ± 1 y b bx a y b ± 1 = ay bx 1

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Therefore, this MRS depends only on the ratio y x : This makes the Cobb-Douglas functions popular with economists. A useful formula to remember x = a a + b I p x ;y = b a + b I p y : Examples: (a1) suppose u ( x;y ) = xy; and p x = 2 ;p y = 3 ;I = 10 : optimal consumption choice. Set y x = 2 3 Combine with the budget equation 2 x + 3 y = 10 The two equations are solved simultaneously, and we get a unique solution. For instance, substitute y = 2 3 x into the budget equation, we get 2 x + 3 2 3 x = 10 and we get x = 2 : 5 ;y = 5 3 : Therefore the optimal consumption choice is the bundle (2 : 5 ; 5 3 ) : (a2) Finding the Engel curve: Suppose u ( x;y ) = xy; and p x = 2 ;p y = 3 ; let income be denoted by I: From the optimal consumption, we get the two equations y x = 2 3 ; 2 x + 3 y = I Substitute y = 2 3 x into the budget equation, we have 2 x + 2 x = I hence x = I 4 ;y = I 6 : In other words, half of the income is spent on X to purchase I 4 quantities of the good, and half of the income is spent on Y to purchase I 6 quantities of the good. The two solutions
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Frist mid term review - 1 Drawing the demand functions and...

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