Week 4 Flashcards
In the standard case how is the utility maximization problem of the consumer given by?
What will a consumer choose?
a real vector x1,x2 that maxes thier utility subject to it being within budget set.
When is a constraint binding and non-binding ?
Binding when it is satisfied with equality
Non-binding when it is satisfied but not with equality
When m>0 how many of the constraints will be binding at optimal bundle x1,x2
at most two of the constraints
How to tell if budget constraint binds at the optimum?
when utility function is monotonic. This means increasing consumption of both goods always increases utility. Hence point which consumer does not spend all income cannot be optimal.
How to tell if budget constraint is binding?
optimal bundle will be on the budget line.
From the budget line solve for x2 as a function of x1
Once you get the equation of x2 in terms of x1 What does the non-negativity constraint x2>=0 do?
.
What happens when we replace the equation of x2 in terms of x1 in the objective function and the non negativity transformed equation.
How to get to (x1,x2) from
Solve the maximization problem we will get x1* which we can plug into x2=m/p2 - (p1* x1)/p2 to get x2*
How do we solve
First step is to compute the derivative of objective function. Using chain rule we get.
What are the 4 possible cases of the solutions to the objective function.
(a) Interior solution where derivative of objective function is zero at max
(b) We have a corner solution where derivative of objective function is always positive and it is optimal to consume the maximum possible of x1 which is m/p1
(c) We have a corner solution where derivative of objective function is always negative and it is optimal to consume the least possible of x1 which is 0
(d) We have a point which the derivative is zero but does not correspond to a maximum it is a minimum
What is a necessary condition to have an interior solution in the consumer’s problem?
the derivative of objective function equals zero. At the optimum derivative becomes
What does this mean?
At optimal bundle (x1,x2) the MU of good 1 must equal the MU of good 2 times the price ratio p1/p2.
What does the price ratio p1/p2 mean?
amount of good 2 consumer would afford if they sacrificed a unit of good 1.
At optimum level what happens to the MRS and price ratio?
They must be equal
MRS = - p1/p2
What does the MRS mean?
Willingness of consumers to substitute both goods and remain indifferent.
What is the graphical interpretation of MRS = - p1/p2
At interior solutions the IC curve will be tangent to the budget line.
What is MRS = - p1/p2 known as? when is it sufficient to have a maximum?
Tangency condition and it is only sufficient if preferences are strictly convex.
How to be certain if the optimal point is a maximum not a minimum?
We need to find the second derivative of the objective function and verify it is negative.
Derive the second derivative from the first derivative below.
When will the solution be unique?
if second derivative is negative at every point in budget line not just (x1, x2)
How to check for corner solutions?
we analyse the derivative of the objective function.
Looking at case b, when will the derivative be positive across all bundles on budget line?
If for all x1 and x2 such that x1>= 0 and x2>= and p1x1+p2x2=m It is then optimal to sacrifice good 2 to increase consumption of god 1 along all bundles on budget line.
Which is the optimal bundle for the case of a corner solution?
Where we spend all income on good 1, (x1, x2) = ( m/p1, 0) which is equivalent to MRS being always greater in absolute terms than price ratio
Show if IC curves are always steeper than budget line and if budget line is always steeper than IC curves. Also show what the optimal bundle will be for the both cases.
If there is a corner solution and it is optimal to spend all income on good 2 how will the MRS and price ratio be?
When will optimal bundle be on the budget line regarding preferences?
Only if they are monotonic
How to look at optimal bundles inside budget sets.
What 4 cases do we have to look at to solve problems where optimal bundles inside budget sets?
1) both non-negativity constants are binding
2) only the non-negativity contained for good 1 is binding
3) only the non-negativity contained for good 2 is binding
4) both non-negativity constants are non-binding
We then see which results in higher utility.
How do we solve the 4 cases?
1) we don’t solve anything simply compute u(0,0)
2) and 3) where we set one of the two variables to zero, we have to solve a maximization problem of just one variable max u(0,x2)
4) we have to solve the unrestricted two variable problem max u(x1,x2)
How do we solve consumer problems.
First we compute MU to see if we have a monotonic preferences and thus know if budget constraints will be binding
Second we compute the MRS and compare it with the slope of the budget line to see if we have interior solutions of corner solutions
Third if we have interior solutions, we need to check if preferences are strictly convex and compute second derivatives or plot IC curves to verify that bundle tangency condition is actually the one that reaches the highest IC curve
