Derivative Intuition
Describe the following derivative in your own words:
dU/dx = 2
As my consumption of x increases by one unit, my Utility increases by 2.
(Think in context of coffee. As my consumption of coffee increases, my utility increases by one util per unit)
Imagine that you only get utility from a bag of chips if you also have 2 cups of salsa to consume with it. The reverse is also true (a cup of salsa only has value to you if you’ve got half a bag of chips to pair with it). Which utility function over bags of chips, C, and cups of salsa S, represents your preference?
a. 𝑈(𝐶, 𝑆) = min (𝐶, 2𝑆)
b. 𝑈(𝐶, 𝑆) = min (2𝐶, 𝑆)
c. 𝑈(𝐶, 𝑆) = 𝐶 + 2𝑆
d. 𝑈(𝐶, 𝑆) = 2𝐶 + 𝑆
a. U(C,S) = min (2C,S)
because we know that Salsa is double the units of Chips...
2(Units Chips) = (Units Salsa)
Think of them as the same measurement
Extra Credit: How do we find the engel curve?
Positive -> Normal good
Negative -> Inferior Good
What is elasticity at levels of:
1
0
and .5
Unit Inelastic
Inelastic
Graphically, what represents the convex nature of Cobb - Douglas ICs?
Their slope gets flatter as you move from left to right
What is a partial derivative in context of U(X,Y) (100 pts) AND...what is their role in MRS?(100 pts)
Partial Derivative: They are widely used to analyze functions involving multiple variables, such as utility, production, and cost functions. They allow us to isolate the impact of one variable (on Utility) while controlling for the other(s).
Their Role in MRS: Imagine you have a fixed budget for food, and your utility depends on both pizza (X) and burgers (Y). The MRS tells you how many burgers you’d give up to have one more slice of pizza while still maintaining the same satisfaction level.
More: Me use partial derivatives as a ratio. Holding Utility constant, we determine the rate at which we are willing to give up units of Y for additional units of X
Find X* and Y* from the following utility function and budget constraint:
(x,y) = xy^1/2
X = $1
Y = $2
M = $24
x = 16
y = 4
y* = (b/a+b)*(M/Py)
x* = (a/a+b)*(M/Px)
Can two goods be inferior over income levels?
No, budget must be exhausted
Under what condition will an income increase have no effect on your consumption of Y?
You have [Which utility function] AND what is true about MRS and PR
You have perfect subs and the MRS is greater than the price ratio
What does the second derivative show us in a two good utility function?
The most common interpretation of the second derivative in utility functions is related to the law of diminishing marginal utility. If it is less than zero, it implies that as you consume more x, utility diminishes. If it is greater than zero, your marginal utility is increasing.
There is no optimality condition for perfect subs. We COMPARE price ration to MRS and then decide which good to consume all of. This is because the slopes are constant and the IC will be tangent with the price ratio at either M/Px or M/Py. We don't balance our consumpiton of subs.
What is the equation for elasticity and what are the three steps associated with finding it?
E(x,M) = dx/dM * M/x
1. Take the partial derivative of your max value with respect to M.
2. Multiply that partial derivative by the ratio of the input variable to the response variable
3. Substitute the original equation in for the response variable and simplify.
Interpretation: Ex,y = a
If y goes up by 10%, x goes up by a*10%
What is the difference between Ordinary and Giffen Goods? And what is an example of each?
Ordinary: when price goes down, consumption goes up
If my demand for Y is given by the equation y* = M/py^2.
What is the elasticity of demand of Y? Is it inelastic, unit elastic, or elastic?
Elastic -> -2
How do we determine if X is a "bad" instead of a "good"
Hint: if you remember from last session, I incorrectly said that the derivative of the marginal utility of X with respect to X is negative...
The marginal utility (first derivative) of X is negative
The second derivative would indicate diminishing marginal returns or increasing marginal returns.
What is the super-secret way to check if you got MRS for Cobb Douglass correct?
MRS = (a/b)(Y/X) where U = X^aY^b
What is the difference between the Engel Curve and Income expansion path
Engel Curve is X,M and Income expansion is X,Y
If I have the demand function:
y* = (b/a+b)(M/Py)
is my good y ordinary or Giffen
Ordinary
As the price of X gets larger and larger, to what point in the (x,y) place does the price-offer curve converge, assuming you have perfect complements utility?
The origin
from the equation: Use derivative and inequality
x* = M - px^2 + py^1/2
Is X a normal or inferior good? (167 pts)
Is X a ordinary or giffen good? (167 pts)
Is X a sub or compliment for Y? (167 pts)
X is a normal good as 1> 0
X is an ordinary good as -2Px < 0
X is a substitute for Y as 1/2Py^-1/2 > 0
What are the utility maximizing demands, B* and P* for U(B,P) = min(2B,P)
Pb = $1
Ps = $2
M = $10
P = 4
Can a good always be giffen?
No, budget restricts it
What two circumstance make price changes matter?
1. You are currently consuming the good that has a change in price.
2. The price of the good you are NOT consuming drops so substantially that it changes which of the two goods you consume.
Solve U(A,B) = A^.4 * B^.6
Pa = 2
Pb = 3
M = 120
A* = 24 B* = 24