Consumer Theory The consumer s problem 1
The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y). y MRS(x,y) = 1/2 x 2
The Marginal Rate of Substitution (MRS) Conceptually, the MRS(x,y) is the quantity of good y that will compensate the consumer if he reduces his consumption of x in one (infinitesimal) unit, so that the consumer maintains the level of welfare he has when he consumes the bundle (x,y). In other words, the MRS(x,y) is the consumer s value for one (infinitesimal) unit of good x, measured in units of good y, when he has the bundle (x,y). 3
1. u(x,y) = xy The MRS: Examples Denote by u(x,y) = xy = u* the utility level at the consumption bundle (x,y). Then Therefore u* = xy y =f(x) = u*/x. f (x) = -u*/x 2. Substituting u*=xy we obtain MRS(x,y) = -xy/x 2 = y/x. Evaluating the MRS at (2,1) yields MRS(2,1) = 1/2. 4
The MRS: Examples y MRS = slope = 1/2 x 5
2. u(x,y) = 2x + y The MRS: Examples Denote by u* = 2x + y = u* the utility level at the consumption bunde (x,y). Then Therefore u* = 2x + y y = f(x) = u* - 2x. MRS(x,y) = f (x) = 2. In this case, the MRS is a constant and equal to 2. 6
y 4 The MRS: Examples 3. The goods x and y are perfect substitutes. 2 0 1 2 x 7
3. u(x,y) = min{x,2y} The MRS: Examples This utility function is not differentiable at (x,y) when x 2y. For these points, the MRS is not defined. At points (x,y) such that x > 2y, we have MRS(x,y)=0. 8
Examples 3. MRS(x,y) = 0 if y < x/2, and MRS(x,y) is not defined if y x/2. y u(x,y)=min{x,2y} y = x/2 x 9
MRS as the ratio of marginal utilities We can find an expression for the MRS(x,y) without knowing the function y = f(x) that defines the indifference curve. In order to calculate the MRS(x0,y0), we start from the equation that defines the indifference curve at point (x0,y0) u(x,y) = u0, (*) where u(x0,y0) = u0. The Implicit Function Theorem establishes conditions that guarantee that this equation defines a function around the point (x0,y0), and under these conditions ensures that the derivative of this function can be obtanined by total differentiation. 14 10
MRS as the ratio of marginal utilities Denoting the partial derivatives of u(x,y) with respect to x and y as ux and uy, respectively, and taking the total derivative of the equation (*), we obtain dx ux + dy uy = 0. Hence the derivative of the function defined by equation (*) is dy/dx = -ux/uy Therefore MRS(x0,y0) is obtained by evaluating this expression at (x0,y0): MRS(x0,y0) = -ux(x0,y0)/uy(x0,y0) 11
MRS as the ratio of marginal utilities We apply this formula to examples 1 and 2 above. 1. u(x,y)=xy. We have u x = U/ x=y, and u y = U/ y=x. Hence MRS(x,y)= - u x /u y = y/x. 2. u(x,y)=2x+y We have u x = U/ x =2, and u y = U/ y =1. Hence MRS(x,y)= - u x /u y = 2/1= 2. 12
The consumer s problem The consumer chooses the consumption bundle that maximizes his welfare (that is, his utility) on the set of his feasible consumption bundles (that is, on his budget set). Thus, the consumer s problem (CP) is: Max x,y u(x,y) s. t. p x x + p y y I x 0, y 0. 13
The consumer s Problem Axioms A1, A2 and A4 imply that there is a utility function u: R 2 + R that represents the consumer s preferences. Moreover, the function u is continuous. When prices are positive, the consumer s budget set is compact (that is, closed and bounded). Hence, Weierstrass Theorem implies that the consumer s problem has a solution. 14
The consumer s Problem Axiom A3 implies that the function u(x,y) is non decreasing in x and non decreasing in y; furthermore, it is increasing in (x,y). Hence a solution to the CP, (x*, y*), satisfies: (1) p x x*+ p y y*= I. 15
The consumer s Problem Proof: If p x x+ p y y= I ε < I, then the bundle (x+ ε/2p x,y+ ε/2p x ) is in the budget set, and is preferred to (x, y) by A.3. 16
The consumer s Problem Axiom A5 implies that u is concave. Hence a local maximum of the function u is a global maximum; that is, second order conditions need not be checked. 17
The consumer s problem Characterizing a solution to the CP. Let (x*, y*) be a solution to the CP. Then: 2.a. If x*> 0 MRS(x*,y*) p x /p y 2.b. If y*> 0 MRS(x*,y*) p x /p y 18
The consumer s problem y I/p y B At B, the MRS p x /p y. The bundle C is preferred to B and is feasible. Therefore, B is not optimal. C I/p x x 19
The consumer s problem Interior solutions: (x*,y*) >> (0,0) (1) p x x+ p y y = I (2) MRS(x,y) = p x /p y 20
Corner solutions: The consumer s problem Only good x is consumed: x*= I/p x, y*= 0 (2) MRS(I/p x, 0) p x /p y Only good y is consumed: x*= 0, y*= I/p y (2) MRS(0, I/p y ) p x /p y 21
Examples 1. u(x,y) = xy; p x =1, p y =2, I=80. We have MRS(x,y) = y/x. Using (2) (MRS(x,y) = p x /p y ) we have y/x = 1/2 x = 2y Substituting in (1) (xp x + yp y = I) we have x+2y =80 2x=80. That is, x*= 40, y*= 20, and u* = x*y* = 800. There are no corner solutions since u(x,0)=u(0,y)=0<u*. 22
Examples y x 23
Examples 2. u(x,y) = 2x + y; p x =1, p y =2, I=80. We have MRS(x,y) = 2. Interior solutions: (1) p x x + p y y = I x + 2y = 80 (2) MRS(x,y) = p x /p y 2= 1/2?? Equation (2) is not satisfied. Hence there is no interior solution! 24
Corner solutions: y Examples MRS(0,40) = 2 > p x /p y = 1/2. 40 0 20 40 80 x The bundle (0,40) is not a solution. 25
Examples Corner solutions: y MRS(80,0) = 2 > p x /p y = 1/2. 40 0 80 x The bundle (80,0) is a solution. 26
Examples 3. u(x,y) = min{x,2y}; p x =1, p y =2, I=80. MRS(x,y) = 0 if y < x/2 (the indifference curve is horizontal at these points). The MRS(x,y) is not defined if y x/2 (at these points, the indifference curve is vertical or it has several tangent lines). The method discussed, which is based on the MRS, is not useful in solving this problem. 27
Examples Let s see that the solution is the bundle (40,20), like the graph below suggests. y y = x/2 40 0 40 80 x 28
Examples Let s suppose that (x*,y*) solves the CP. a. If y* < x*/2, since x* + 2y* = 80, we have y* = (80- x*)/2 < 40- y* y* < 20. Therefore u(x*,y*)=2y* < 40 = u(40,20). b. If y* > x*/2, since x* + 2y* = 80, we have x* = 80-2y* < 80- x* x* < 40. Therefore u(x*,y*)=x* < 40 = u(40,20). (a) and (b) imply that (x*,y*) = (40,20) is the solution to the CP. 29