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
The MRS: Examples
1. u(x,y) = xy
Denote by u(x,y) = xy = u* the utility level at the consumption bundle (x,y). Then
u* = xy → y =f(x) = u*/x.
Therefore f’(x) = -u*/x2.
Substituting u*=xy we obtain
MRS(x,y) = |-xy/x2| = y/x.
Evaluating the MRS at (2,1) yields
MRS(2,1) = 1/2.
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The MRS: Examples
y
MRS = |slope| = 1/2
x
5
The MRS: Examples
2. u(x,y) = 2x + y
Denote by u* = 2x + y = u* the utility level at the consumption bunde (x,y). Then
u* = 2x + y → y = f(x) = u* - 2x.
Therefore MRS(x,y) = |f’(x)| = 2.
In this case, the MRS is a constant and equal to 2.
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The MRS: Examples
3. The goods x and y are perfect substitutes.
y
4
2
0
1
2
x
7
The MRS: Examples
3. u(x,y) = min{x,2y}
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.
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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)|
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MRS as the ratio of marginal utilities
We apply this formula to examples 1 and 2 above.
1. u(x,y)=xy.
We have ux= ∂U/∂x=y, and uy= ∂U/∂y=x. Hence
MRS(x,y)= |- ux/uy| = y/x.
2. u(x,y)=2x+y
We have ux= ∂U/∂x =2, and uy= ∂U/∂y =1. Hence MRS(x,y)= |- ux/uy| = 2/1= 2.
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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. px x + py y ≤ I
x ≥ 0, y ≥ 0.
13
The consumer’s Problem
•  Axioms A1, A2 and A4 imply that there is a
utility function u: ℜ2+ → 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) px x*+ py y*= I. 15
The consumer’s Problem
Proof: If px x+ py y= I – ε < I, then the bundle (x+ ε/2px,y+ ε/2px) 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.
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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*) ≥ px/py
2.b. If y*> 0 → MRS(x*,y*) ≤ px/py
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The consumer’s problem
y
I/py
At B, the MRS ≥ px/py. The bundle C
is preferred to B and is feasible.
Therefore, B is not optimal.
B
C
I/px
x
19
The consumer’s problem
Interior solutions: (x*,y*) >> (0,0)
(1) px x+ py y = I (2) MRS(x,y) = px/py
20
The consumer’s problem
Corner solutions:
  Only good x is consumed: x*= I/px, y*= 0
(2) MRS(I/px, 0) ≥ px/py
 Only good y is consumed: x*= 0, y*= I/py
(2) MRS(0, I/py) ≤ px/py
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Examples
1. u(x,y) = xy; px=1, py=2, I=80.
We have MRS(x,y) = y/x.
Using (2) (MRS(x,y) = px/py) we have
y/x = 1/2 → x = 2y
Substituting in (1) (xpx + ypy = 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*.
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Examples
y
x
23
Examples
2. u(x,y) = 2x + y; px=1, py=2, I=80. We have MRS(x,y) = 2. Interior solutions:
(1) px x + py y = I
(2) MRS(x,y) = px/py
x + 2y = 80
2= 1/2 ??
Equation (2) is not satisfied. Hence there is no interior solution!
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Examples
Corner solutions: y
MRS(0,40) = 2 > px/py = 1/2.
40
0
20
40
80
The bundle (0,40) is not a solution.
x
25
Examples
Corner solutions: y MRS(80,0) = 2 > px/py = 1/2.
40
0
80
x
The bundle (80,0) is a solution.
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Examples
3. u(x,y) = min{x,2y}; px=1, py=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.
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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
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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