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Micro Theory Pt 1.1 - Utility functions & Budget Constraints
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Terms in this set (39)
Two main principles
1. Optimization principle: people try to choose the best consumption they can afford
2. Equilibrium principle: Prices adjust until D=S
Draw a demand curve
Reservation price
Highest price that a person will accept and still purchase the good.
Comparative statics
exercise comparing two "static equilibria" without worrying how the market moves between them.
monopoly
A situation where a market is dominated by a single seller
discriminating monopolist
case where the monopolist sells at different prices to different people (ex. auctioning off apartments)
result of discriminating monopolist
exactly the same people will get the apartment as in the market solution (those valuing at at least P*)
Draw revenue of monopolist & give equation
See index card
=p*D(p)
Rent control consequences if below P*
will create excess demand (shortage)
How to notate a bundle of goods
(x,y) where x is commodity 1 & y is commodity 2.
Assumptions about preferences
1. completeness:
at least one of the following holds, a is at least as good as b, or b is at least as good as a.
2. transitive:
if a is at least as good as b, and b is at least as good as c, then a is at least as good as c.
Give notation for higher preference (more utility) for good a
u(a) ≥ u(b)
an agent is indiﬀerent between consumption bundles (x,y) and (x′,y′) if:
(x,y) is at least as good as (x′,y′), AND (x′,y′) is at least as good as (x,y).
Find the indifference curve for: u(x,y) = 3x^2y^3
--> Just choose any constant (here, 2) & set equal to utility function
--> 3x2y3 = 2
--> simplify:
y =(2/3x^2)^1/3
Assumptions of preferences
1. commodities are goods aka more is better
requirements of proper indifference curve & what it looks like if properties met
downward sloping & never cross
1. commodities are goods
2. preferences are complete
3. preferences are transitive
Draw 2 variations of complete AND transitive indifference curves
1. MUx(x,y) is ____
2. MUx(x,y) = _____
1. the ratio of the change in utility to the change in consumption of x (when this change is small) if the consumption of y remains constant.
2. MUx(x,y) = ∂u(x,y)/∂x(x,y)
If u(x,y) = 10x^1/3*y^2/3 then
Solve for MUx(x,y)
MUx(x,y) = ∂u(x,y)/∂x(x,y)
= 10/3x^-2/3*y^2/3
= (10y^2/3) / (3x^2/3)
If u(x,y) = 10x^1/3*y^2/3 then
Solve for MUy(x,y)
MUy(x,y) = ∂u(x,y)/∂y(x,y)
= 10*(2/3) x^1/3 y^-1/3
= (20x^1/3)/(3y^1/3)
Marginal rate of substitution of good x for good y is the ratio __?
How do you verbalize the numerator?
MRSxy(x,y) =∆y/∆x
∆y is the maximal amount of good 2 that the agent would sacriﬁce to obtain ∆x more of good 1
MRSxy(x,y) is the NEGATIVE of ___?
the slope of the indiﬀerence curve of u at (x,y)
MRSxy(x,y) = ___ aka ___
MRSxy(x,y) = MUx(x,y)/MUy(x,y)
aka = [∂u(x,y)/∂x(x,y)] / [∂u(x,y)/∂y(x,y)]
What does decreasing MRS mean?
the tangent to the indiﬀerence curves becomes ﬂatter when x increases. This means that it is more diﬃcult to substitute x for y when having more and more of x.
Illustrate decreasing rate of marginal substitution
Define budget line/constraint
Give equation
the line that joins all combinations (x,y) at which the agent spends exactly all his/her income Y .
x(px) + y(py) = Y
y = (Y/py) - (px/py)x
Given px = 3, py = 2 and Y = 8
Find equation of the budget line
3x + 2y = 8
aka y = 4 - (3/2)x
What is the slope of the budget line?
= ?
the marginal rate of transformation of y into x
aka MRTxy
therefore,
MRTxy = - px/py
Given px = 3 and py = 2,
Find MRTxy
MRTxy = −3/2
What are the end points of the budget constraint?
on vertical (y) axis: Y/py
on horizontal axis: Y/px
What happens to budget constraint if:
1. py increases?
2. px increases?
3. Y increases?
1. vertical intercept moves down/line flattens
2. horizontal intercept shifts left/line steepens
3. both intercepts shift out equally with line
What does indifference curve look like with smooth preferences?
concave and downward sloping
What must be the case (2) for a consumption bundle to be optimal looking at graph?
within budget line AND on utility curve
1. Draw & highlight point where indifference curve steeper than budget line
2. Give mathematical illustration
1. see index card (pg 19 pt 1 notes)
2. MRSxy(x1,y1) > px/y = −MRTxy
aka
px − MRSxy(x1,y1)py < 0
1. Draw & highlight point where indifference curve flatter than budget line
2. Give mathematical illustration
1. see index card (pg 20 pt 1 notes)
2. MRSxy(x2,y2) < px/y= −MRTxy
aka
MRSxy(x2,y2)py - px < 0
1. Illustrate where utility is maximized
2. Give mathematical illustration
1. At point e on graph below
2. MRSxy(ˆx,ˆy) =px/py= −MRTxy
What 2 equations should we solve to find where preferences are maximized, knowing the utility function?
1. (ˆx,ˆy) is in the budget line.
2. MRSxy(ˆx,ˆy) = −MRTxy.
Equivalently,
1. pxˆx + pyˆy = Y
2. MRSxy(ˆx,ˆy) = px/py
Suppose that
u(x,y) = 3x^1/3*y^2/3,
px = 1, py = 4, and Y = 300.
What is the bundle of goods (ˆx, ˆy) that maximizes the agent's preferences?
1. Find the 2 equations:
-> 1)ˆx + 4ˆy = 300
2) MRSxy(ˆx,ˆy) = 1/4
2. Calculate MRSxy(ˆx,ˆy)
-> = [MUx(ˆx,ˆy)] /[MUy(ˆx,ˆy)]
-> MUx = 3/3
ˆx^-2/3
* ˆy^2/3
= (ˆy^2/3) / (ˆx^2/3)
-> MUy = 6/3
(ˆx^1/3)
* ˆy^-1/3
= (2ˆx^1/3)/ˆy^1/3
-> Put MUx/MUy together:
[(ˆy^2/3) / (ˆx^2/3)]
/
[(2ˆx^1/3)/ˆy^1/3]
So, MRSxy(ˆx,ˆy) = ˆy/2ˆx
3. Plug back into our 2 equations & simplify #2 with goods equal to each other:
->1) ˆx + 4ˆy = 300
->2) ˆy/2ˆx=1/4
or 2ˆy= ˆx
4. Plug ˆx into equation #1 to solve for ˆy then ˆx:
-> 2ˆ y + 4ˆ y = 300
orˆy = 50!
-> Since 2ˆ y = ˆx & y = 50,
ˆx = 100!
1. Illustrate linear preferences
2. What three preference maximization situations (points of perfect bundles) are there when preferences are linear?
1. See index card (pg 24&25 pt 1 notes)
2.
1) at Y/py
2) at Y/px
3) any point on budget line
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