BEPP 250 Unit 1

Question 0

Attachment rate vs EW-product price ratio

Answer 0

Attachment rate is the Percent of the buying population that buys extended warranties.
Ew-product price ratio is the ratio of the extended warranty to the price.
They should be equal in theory, but attachment rate is usually higher because consumers are willing to spend more because they think there’s a higher risk of product breaking than there actually is.

Question 1

Risk averse

Answer 1

Averse to risk
Doesn’t like risk
Will pay slightly more to minimize the risk.

Question 2

Suppose you think w% probability the tv needs repair and cost of repair is $L. Let $t be the price of the EW.
Get willingness to pay:

Answer 2

If you don’t buy the extended warranty, then w% of the time you lose $L. 100-w% if the time, you lose nothing
Let U be the function of the price.
wU(-L) + wU(0)is how much you pay if you don’t buy the warranty
If you do buy the warranty, then you just pay t. So U(-t)
Willingesss to pay is when prices are equal
Price t* such that
U(-t*) = wU(-L)+ wU(0)

So you’re willing to pay $w(L)
Willingness to pay is that price

Question 3

Model for willingness to pay:

Answer 3

Willingness to pay is based on aversion to risk and belief about the likelihood of repair.

Question 4

Someone’s Risk aversion relation to their estimated probability the tv needs repair

Answer 4

If they have higher aversion to risk, then they think there is a lower chance the tv needs repair.
Even if two people have same willingness to pay for a product, when you introduce a cheaper product, then the willingness to pay for the person with the lower r is lower than the other person.
Decrease in WTP is much steeper for the less risk averse person.

Question 5

Risk aversion actual relation with buying ew or not

Answer 5

Risk aversion doesn’t explain why people are willing to buy EW at observed prices.
Base decision on expected value of the warranty. Buy EW if price of EW < probability of repair * cost of repair.
People overestimate likelihood of failure by a large amount (actual is 5 percent, estimated 13-15 percent) so retailers sell EW at very high prices
If you tell someone actual failure rate is lower, their WTP goes way down.

Question 6

If told actual percent tv breaks, then what

Answer 6

Then consumers are way better of, sales and profits go way down. In real world, consumer welfare is negative. When told this, they are slightly positive.

Question 7

What happens if EW decreases

Answer 7

Consumer welfare goes down, as consumers don’t get any real value from extended warranties. Making it cheaper makes it more likely consumers get scammed

Question 8

Microeconomics

Answer 8

Body of knowledge that studies indifivudal level behavior and decision making in order to provide tools, results, and ideas to help us
Understand observed phenonomem and guide decisions

Question 9

Main principles of micro

Answer 9

1. Optimization (individual objectives + constraints —>shape decisions and observed behavior.
2. Equilibrium (observed outcomes <— interaction across individuals’ decisions, no incentive to change further, system is at rest)

Question 10

3 assumed properties of preferences (include on cheat sheet)

Answer 10

1. Completeness. When facing a choice between 2 bundles, a consumer can rank them so that either: A>B, A<B, or A~B
2. Transitivity. Consumers rankings are logically consistent in the sense that if A>B and B>C, then A>C
3. Monotonicity. More of a good is better.
   All else the same, more of a commodity/ characteristic is better than less.
   A good is different from a bad.
   We are never satisfied.
   U can’t go down as x goes up. U must increase as x goes up. Marginal utility can go down but must remain positive.

Question 11

Monotonicity

Answer 11

Think of it that we can freely dispose excess beverages in our bundle. Don’t want to drink, then throw it away, doesn’t cost anything
If you have a point C, then you draw vertical and horizontal lines. To the top right, you prefer the point, bottom left you strictly prefer C
Rest you can’t tell, so get indifference curve

Question 12

Indifference curve

Answer 12

Consider preferences such that A~B.
Graphically represent the set of all bundles that one is indifferent to by the indifference curve
Equally prefer all the bundles on this curve.

Question 13

Indifference curves and preferences

Answer 13

Because of monotonicity, you strictly prefer points (bundles) above the IC. So, you can draw IC’s next to each other. And you know that you’ll prefer the IC’s further to the top right, as more of a good is better, and more goods to the top right.

Question 14

Budget constraint

Answer 14

Can’t have unlimited goods, even though we want to, because of a budget constraint.
Budget set is the set of feasible (affordable) bundles given price and income. Set of all bundles that lie on and below the budget line.

Question 15

Budget line

Answer 15

Suppose we have two goods, x and y. With corresponding prices p1 and p2. And income I
Budget line is p1x+p2y = I
Get it in terms of Y and graph. Bundles below and on it are those that are affordable.

Question 16

Which bundle do we choose?

Answer 16

Need to be able to afford the bundle, and remember that by monotonicity, more of something is better.
So, we want to be on the highest indifference curve that is still affordable.
So this will be the indifference curve that is tangent to the budget line.
It has to be tangent as if we chose one that intersected, there’d be some point that is still affordable on a higher IC.

Question 17

Utility function

Answer 17

Utility function is an easier way to show which we prefer.
If (x,y) > (x’,y’), then u(x,y) = U > U’ = u(x’,y’)
Value of the utility function doesn’t matter, it’s an ordinal function.
Just matters relative, ie that you prefer this bundle to another bundle.

Question 18

Partial derivative
For example take partial deriv WRT x and then WRT y for
2(x-2)^2 -4xy +3y^2

Answer 18

Need partial derivatives for maximization problems with multiple variables.
Partial deriv WRT x —> treat y as a constant
4(x-2) -4y
And for WRT y—> treat x as a constant
-4x +6y

If we want the minimum of this function, equate the partial deriv to 0 and then to each other.

Question 19

Relationship between IC and utility function

Answer 19

Assign a number, U1 = u(x1,y1) and another U2=u(x2,y2)
We prefer the bundle on the higher IC.
Always will prefer bundle on higher IC, and we represent the IC with the utility function. So the top IC is U2 and bottom is U1 and we strictly prefer bundles on U2

Question 20

Utility maximization problem in words

Answer 20

Predicting which bundle is chosen (most preferred affordable bundle)is equivalent to finding (x,y) such that:

1. u(x,y) = U is the largest AND
2. (x,y) satisfies the budget constraint

Question 21

Utility maximization problem in math

Answer 21

Objective:
Choose bundle (x,y) to maximize utility.
u(x,y)
Subject to budget constraint:
Chosen bundle (x,y)should be affordable
px X + pyY <= I
X and Y are always greater than or equal to 0, as can’t have negative quantity of a good.

Question 22

Utility maximization graphical solution

Answer 22

The optimal solution is the point where the IC is tangent to the budget line, as if the IC hits the budget line twice, then there’s some point on a higher IC that we would prefer.
Mathematically, this point is captured by equality of slopes of budget line and the IC at the optimal bundle.
Budget line slope is -px/py because we have the EQ px x + py y =I
See next card for slope of IC

Question 23

Slope of indifference curve

Answer 23

Consider an increase in x given by delta x. To be on the same IC, y must decrease by delta y.
The ratio delta y / delta x tells us how much we need to give up in terms of y if we want to have an incremental increase in x and have utility remain constant.
For a very small increase in x, the ratio delta y /delta x is the first deririvate dy/dx, which is the slope of the IC at (x,y)
This slope, dy/dx measures the trade off between y and x and has a special name, marginal rate of substitution or MRS
So, slope of ic is the deririvite of y / dx

Question 24

MRS and marginal utility

Answer 24

Change in x given by delta x changed our utility by delta x (U) Same thing for y Same IC, so delta U = 0 So, delta x U * delta X + delta y U * delta Y =delta U =0 So, MUx * dx + MUy * dy =dU =0 Where MUx is the marginal utility of x which measures the change in utility for a small change in x

Question 25

Marginal utility

Answer 25

Additional benefit you get from consuming the marginal unit, which is the very last unit you consume or an additional unit. MU x =the partial derirative of the utility function with respect to x = partial deriv u(x,y) / partial deriv x Assuming monotonicity, utility increases in consumption. MUx must be positive for all x. However the additional benefit, may decrease as x becomes large (MU still positive though) This is diminishing marginal utility, MUx is decreasing, meaning the partial deriv of MUx with respect to x which is the second partial deriv of the utility function wrt x is negative.

Question 26

Bang for buck

Answer 26

We get for the slope of the IC =-MUx/MUy Rearrange this, and we get equality of bang for buck—> MUx/px = MUy/py Bang for buck is this the benefit cost ratio of the marginal unit. The benefit of the last unit / the cost of the last unit. Want to increase consumption by a tiny bit of the good that has the highest B4B until the B4B’s are equal. Once equal or you hit a constraint (can’t consume more or can’t consume negative quantities), then you stop.

Question 27

Utility max problem basic structure and solution

Answer 27

``` Remember we are trying to maximize utility subject to our budget constraint. Max u (x,y)where x and y are greater than or equal to 0 We also have budget function px X + py Y = I Need to find two unknowns, x* and y*. Equate the bang for bucks for interior solutions and for a corner solution, you consume more of the good if their B4B is strictly greater than the other B4B And the budget constraint. ```

Question 28

Interior solution utility max problem

Answer 28

If optimal solution is strictly positive, then B4Bs should all be equal when evaluated at optimal solution. Use equality of B4B to get x* and y* Make sure you show that x and y are greater than 0

Question 29

``` Example of útil Max problem:interior solution: Max sqrt (xy) such that Px X + Py Y = I ```

Answer 29

Step 1. Examine the bang for bucks. MU x = partial deriv with respect to x = SQRT (y) / 2SQRT (x) Divide by px to get bang for buck MUy =SQRT (x)/2SQRT (y). Divide by py to get B4By X and Y must be strictly positive, as if x or y were zeros then B4B would go to infinity, which can’t happen. So, we equate the B4B. Cross multiply and get y = px/py * x Step 2. Plug this into the budget constraint PxX + py (px/py *x) = I Now solve for X* = I/2px Then solve for y* by using x* and what we got from step one. Y*=px/py * x* = px/py * I/2px. So, X*=I/2px and Y*=I/2py

Question 30

Corner solution útil Max

Answer 30

Either zero consumption of one of the goods or hitting some constraint such as a maximum consumption constraint. B4B won’t be equal. Goods you won’t consume will have lowest B4B

Question 31

Útil Max corner solution ex. | Max x + y such that 2x+y =1

Answer 31

B4Bx = 1/2 and B4By = 1 So B4B y is strictly greater, so we consume all y and no x. X*=0. Plug this into the budget constraint, y*=I

Question 32

Demand function

Answer 32

Gives us the optimal quantity demanded as a function of prices and income.

Question 33

Law of demand

Answer 33

If the demand for a good is decreasing as the price goes up | Price and quantity have an inverse relationship.

Question 34

Normal and inferior good

Answer 34

Normal good is something where demand increases with income. Inferior good is when demand decreases with income. Consume more of something (McDonald’s)when you’re poorer

Question 35

Complement and substitute goods

Answer 35

Substitute is when your demand is increases as the price of something else increases. McDonald’s and Burger King are substitutes. If one price goes up, consume other Complement is of demand decreases as price of something else goes up. If price of milk goes up, less coffee demanded.

Question 36

Demand elasticities

Answer 36

Measures the percent change in quantity demanded with respect to a percent change in one factor such as goods own price or ones income. Units free.

Question 37

Own price elasticity

Answer 37

If we increase our own price by 1%, how much with the demand of that good increase by? (dq / q) / (dp/p) Rearrange to get dq/ dp *p/q In our example where q1(p1)=I/2p1, Our own elasticity = -I/2p1^2 * p1 / (I/2p1) = -1 So, as we increase our own price by 1%, demand for that good decreases by 1%

Question 38

Cross price elasticity

Answer 38

As price of Pepsi goes up by 1%, what happens to quantity demanded of coke? See cheat sheet (dq1/q1) / (dp2/p2)

Question 39

Income elasticity

Answer 39

If we increase income by 1%, what happens to quantity demanded of our good? Put on cheat sheet (dq1/ dI) * I/q1

Question 40

Inverse demand willingness to pay

Answer 40

Inverse demand function gives us info about ones willingness to pay for the marginal unit of a given quantity of the good. Recall that the demand function tells us that at a given price, say $1, if you demand, say 5 units of Kombucha, the 5th units B4B no longer dominates the other goods B4B (coke) Bottles 0-4 of Kombucha, higher B4B than coke. So keep consuming, but bottle 5 B4B =B4B coke, so stop consuming more. This is the optimal demand for K=5 bottles Inverse demand asks what should the price or cost of the last bottle be if you optimally demand 5 bottles and this Kombuchas B4B no longer dominates cokes? How much are you willing to pay for the 5th bottle if you optimally consume 5 bottles P1(q1) = the function (1/2q1)in our case

Question 41

Consumer surplus

Answer 41

How do we measure the net value of consuming a total of q1 quantities of good 1? Since each point on the inverse demand curve (p1 on y and q1 on x), gives us the WTP or value of that marginal unit, we add up these values from 0 to q1. Value of consuming total of q1 quantities is the area below the inverse demand curve from 0 to q1. Consumer surplus =total value -total purchase cost Top left triangle. Willing to pay much more, got it at a cheaper price.

Question 42

Be sure to know how to take derivatives

Answer 42

Ok

Question 43

Uber surge pricing

Answer 43

Dynamic pricing takes advantage of supply and demand. High demand and low supply, surge pricing discourages low willingness to pay riders from requesting rides while encouraging drivers to accept requests. Balances out supply and demand Low willingness to pay people out and higher price results in more drivers

Question 44

Production functions

Answer 44

A firm transforms inputs (factors of production)such as capital (K)and labor (L)into output (q) The production function, q=f(K,L) describes the technology for transforming inputs into output.

Question 45

Short run vs long run

Answer 45

In the short run, at least one input (fixed input)can’t be adjusted. If suppose K is fixed at K bar, then q =f(K bar, L) In long run all inputs can be adjusted Depends on horizon

Question 46

Suppose gas is purchased at start of day, divers hired by week, cars leased by month Horizon = month, then which inputs fixed? Week?

Answer 46

If month, then car is fixed because lease car by month Can change gas and driver If week, then car and driver are fixed, can change gas.

Question 47

Returns to scale

Answer 47

For some m>1, Constant returns to scale (CRTS) if: f(mK,mL)= mf(K,L) If double both inputs, same as doubling original output. Increasing returns to scale (IRTS)if: f(mK, mL) > mf(K,L) Output from doubling inputs > double of regular inputs Decreasing returns to scale (DRTS)if: f(mK,mL) < mf(K,L) Double inputs produces less than double what you actually produce. For example, too many people, can’t do as much.

Question 48

Returns to scale Example: Consider function: f(K,L) = K^a * L^b What happens to output if we double both inputs?

Answer 48

``` f(2K,2L)= (2K)^a * (2L)^b = 2^(a+b) * K^a * L^b = 2^(a+b) f(K,L) Compare that to 2f(K,L) If a plus b = 1, then crts If a+b is bigger than 1, then IRTS if less than 1, DRTS ```

Question 49

Marginal product of labor

Answer 49

The additional output produced by an additional unit of labor (marginal unit), holding all other factors constant. Partial deririvatve of q wrt L =partial deriv f(K,L) / partial deriv L

Question 50

Average product of labor

Answer 50

Ratio of output to the amount of labor used | = q/L = f(K,L) / L

Question 51

Marginal and average product relation

Answer 51

Marginal product drives average product If MPl> APl, the average goes up If MPl < APl the average goes down Very last unit pulls the average in that direction Prove this using calculus —> partial deriv of APL WRT L. We get derivative is positive if MPL is bigger than APL, meaning apl goes up when MPl is bigger

Question 52

Diminishing marginal returns

Answer 52

Production function is increasing in its inputs (marginal productivity is positive) But we also would think that is a firm increases its use of an input (holding other inputs fixed),the marginal productivity of this input goes down. Partial deriv of MPL / partial deriv of L = second partial deriv of f(K,L)/ partial deriv L squared This should be negative to have diminishing marginal returns But, we see with an example where q =f(K,L)= K^a * L^b, MPL = bK^aL^(b-1) because partial deriv with respect to L. Partial deriv of the MPL = b(b-1)K^a * L^(b-2) When will this be positive? Remember b>0 If b=1,then partial deriv of MPL wrt L =0. If b is bigger than 1, then increasing marginal returns and less than 1 then diminishing marginal returns.

Question 53

Isoquant

Answer 53

Similar to indifference curve. A collection of input bundles that produce the same amount of output. q= f(K,L)where K and L are the input bundle.

Question 54

Car isoquant example where we have red car and blue car

Answer 54

q= f(R,B) | Trade off between the two. Black car is one axes and red car the other. Draw the curve (concave up)

Question 55

Isoquant slope

Answer 55

Slope measures tradeoff between inputs Marginal rate of technical substitution How many black car hours can we give up in exchange for an increase in red car hours and still produce 240 miles? Slope = MRTS = dB/dR = -MPR / MPB Where MP R Is marginal product of R

Question 56

Isocost line

Answer 56

Collection of input bundles that have the same cost Px (x) + Py(y) = C y = -(px/py) (x) + C/Py All the bundles on the line cost C

Question 57

Isocost trends

Answer 57

Bundles on isocost lines toward bottom left are cheaper. Also will produce less output. Bundles on each line cost the same.

Question 58

Cost minimization problem in words

Answer 58

We want to find the cheapest input bundle that will produce at least q units of output, on or above the isoquant. Optimal bundle is where isoquant is tangent to isocost, slope of isoquant = slope of isocost Need at least q units of output, want cheapest bundle such that we get q. So, cost minimization. We want to find the cheapest input bundle (x*,y*) that produces at least q units of output.

Question 59

Cost minimization mathematical setup

Answer 59

Isocost slope = -px/py Slope of isoquant is -MPx/MPy So equate and rearrange and we get equality of B4B MPx/px =MPy/py Min (where x,y>=0) of Px X + Py Y Subject to production constraint f(x,y) >= q

Question 60

Interior solution cost minimization

Answer 60

If optimal solution is strictly positive, then B4B should be equal when evaluated at optimal solution. Use equality of B4B MPx/ Px = MPy / py

Question 61

Corner solution cost min

Answer 61

Either zero demand of one of the inputs or hitting some constraint such as a maximum usage constraint. Here, B4Bs aren’t equal

Question 62

Cost function

Answer 62

Let (x*,y*) be the solution to our cost min problem for a given quantity, q>0. Lowest cost of producing at least q is C(q) =Px X* + Py Y* C(q) is what we refer to as the cost function We get the cost function by: 1. solving for optimal input demands given q 2. Plug in optimal input demands to the cost eq. Multiply these input demands by their respective prices

Question 63

Marginal cost

Answer 63

Cost of producing the marginal unit of output | MC(q) = dC(q) / dq

Question 64

Average cost

Answer 64

Costs divided by output produced | AC (q) = C(q)/q

Question 65

Marginal and average cost relation

Answer 65

If MC is bigger then AC is increasing in q | If MC is smaller, then AC decreasing in q

Question 66

Economics and diseconomies of scale

Answer 66

Economies of scale is if the average cost goes down as output increases. dAC(q)/dq is negative Diseconomies of scale is if average cost goes up as output increases. dAC(q)/dq is positive. Gets more expensive for later units of q

Question 67

Main difference cost minimization and utility max

Answer 67

We do the production constraint for cost minimization q= f(x,y) As opposed to budget constraint for utility max

Question 68

LR cost function interior example C(q) = min (x and y greater than or equal to 0) Px X + Py Y q <= f(x,y)= xy

Answer 68

``` Step 1 X * and y* > 0 because if either were zero, then q would =0. That can’t happen, as q must be positive. So, we can use equality of B4B. B4Bx = y/px B4By = x/py Equate them, we get y= (px/py) (x) ``` Step 2. Remember the constraint is q, not the budget, so we plug this into q=xy Q = x (px/py)(x) X* = SQRT(py/px *q) Plus back in where we have Y in terms of x, and we get Y* Step 3. Get the cost function Plus x* and y* into the original cost equation. C(q)=px X* + Py Y*

Question 69

LR cost min corner ex C(q) = min (x and y greater than or equal to 0) Px X + Py Y q<= f(x,y) =5x + 7y Also, q>0 and assume 7/py < 5/px

Answer 69

Step 1. B4Bx = 5/px and B4By =7/py We are given B4Bx is strictly greater, so therefore, y*=0 Step 2. q= 5x+7y, y*=0,so q=5x, x*=q/5 Step 3. Plug into cost function, C(q) = px (q/5)

Question 70

Short run cost function

Answer 70

In the short run, we have a fixed input. | So we are just finding one unknown

Question 71

Short run cost min interior example C(q) = min (x and y greater than or equal to 0) Px X + Py Y q= f(x,y) = xy Y=y bar > 0 where q>0

Answer 71

Step 1. X star must be positive, as need to produce q>0 q=x(y bar) X*= q/y bar Step 2. Plug into cost function, C(q) = px/ y bar (q) + Py (y bar)

Question 72

Short run cost min corner ex C(q) = min (x and y greater than or equal to 0) Px X + Py Y q<= f(x,y)= 5x+7y Y=y bar >0 and q>0. Suppose 7/py < 5/px

Answer 72

Step 1. B4B x is bigger than B4B y, but we are forced to use some y bar greater than zero. We see that y bar alone produces q= 7y bad. So, only use x when q>7 y bar. So, optimal input demand for x =0 if q<=7y bar and x* = (q-7 y bar)/5 if q>7 y bar Step 2. Cost function depends on q. For q > 7 y bar, we have C(q)= px (q-7y bar)/5 + Py (y bar) If q is less than or equal to 7y bar, C(q)= py Times y bar