ECON 301 Midterm Flashcards
Risk Premium + How to Calculate
Risk Premium: max amount of money a risk-averse person would pay to avoid taking a risk and get certain expected value of uncertain wealth
- Exact Method: E(U(W))=U(E(W)-F)
Approximate Method - Coefficients of Risk-Aversion rA(W)=-U”(W)U’(W) Arrow-Prate coefficient of absolute risk aversion at wealth, summary and measure of degree of risk=aversion, changes with wealth, F=rA(EW)Var(W)2
Certainty Equivalent + How to Calculate
Certainty Equivalent CE=E(W)-F or CE=U-1(E(U(W)))level of income/wealth, expected utility of random wealth, best lottery/random wealth with highest expected utility
When is Risk-Premium Negative + Pareto Improvement of Efficiency
- Risk-Loving implies negative risk-premium : they need to be compensated to avoid uncertainty
- Pareto Improvement of Efficiency: risk loving/neutral give insurance to risk averse in exchange for a fee
Risk-Attitudes v. Actuarially Fair Condition Table
See doc
How is Demand of Insurance (x) Calculated
Maximize GU(WG-px)+BU(WB-px+x)
First Order Set to 0: GU’(WG-px)(-p)+BU’(WB-px+x)(1-p)=0
Confirm Concave Maximum (not minimum): GU”(WG-px)p2+BU”(WB-px+x)(1-p)2<0
BU’(WB-px+x)(1-p)=GU’(WG-px)(p)
U’(WB-px+x)U’(WG-px)=G(p)B(1-p)
Condition C: U’(WB-px+x)U’(WG-px)=1-BB(p)(1-p) Optimal x depends on probability of accidents B and price of insurance p
Ex. U(W)=log(W),U’(W)=1x, U”(W)=-1W2<0, WG-pxWB-px+x=p1-p1-BB
Insurance
Replaces an uncertain wealth with a more certain one, allows consumer to reduce/eliminate risk
Risk Averse + Preference + Marginal Utility of Wealth + Proof
Risk-Aversion: EU(W) <U(EW) prefers expected value of income guaranteed rather than random income/lottery
* Decreasing Marginal Utility of Wealth U’‘(W)<0 concavity of utility of wealth U(W): values of an additional dollar is smaller the more money you have, not willing to risk getting another dollar when you can lose a dollar with equal probability that has higher marginal utility
* Utility Functions: U(W)=log(W), U(W)=W=W1/2, U(W)=aW-bW2
Proof of Risk-Aversion + Loving: Fundamental Theorem of Calculus
Risk Neutral + Preference + Marginal Utility of Wealth + Proof
EU(W)=U(EW)
Constant Marginal Utility of Wealth U’‘(W)=0, U(W)=a+bW,b>0 linearity of utility of wealth: values of an additional dollar is same the more money you have, indifferent b/w random income and expected value of income, risk of getting another dollar when you can only lose a dollar with equal probability that has same marginal utility
Proof of Risk-Neutrality: EU(W)=i=1niU(Wi)=i=1ni(a+bW)=i=1nia+i=1nibWi=a+bi=1niWi=a+bEW=U(EW)
Risk-Loving + Preference + Marginal Utility of Wealth + Proof
Risk-Loving: EU(W)>U(EW)
Increasing Marginal Utility of Wealth U’‘(W)>0 convex utility of wealth U(W): prefers random income rather than guaranteed expected value, value of an additional dollar is larger the more money you have, willing to risk getting another dollar when you can only lose a dollar with equal probability that has lower marginal utility
Utility Functions: U(W)=W>1,U(W)=aeaW
Expected Utility v. Utility of Expected Wealth
- Face uncertainty of receiving the random income → Expected Utility EU(W)=i=1niU(Wi), considers probability and utility, how people psychologically value uncertain outcomes
- Face certainty of receiving the expected income→ Utility of Expected WealthU(EW)=i=1nU(i=1niWi), considers expected wealth and utility, ignores probability distribution and effects of risk perception, just utility of average wealth
Utility of Wealth v. Expected Utility Function
- Utility of Wealth Function U(W)=U(I)=v(p1,…,pn,I)=u(x1(p1,…,pn,I),…,xn(p1,…,pn,I)): measures utility of money, indirect utility function with fixed prices and varying W, omitting prices from indirect utility function
- Expected Utility Function EU(W)=i=1niU(Wi):
- Conclusion: EU(W1)>EU(W2) this individual will prefer random income W1
How to evaluate utility under uncertainty?
U(W)
E(W)
E(U(W))=EU(W)
U(E(W))=U(EW)
Elasticity + Importance + Equation
Elasticity: % change in 1 variable resulting form a 1% increase in another variable
Importance: allows percentage comparison of quantity responsiveness to price, predicts effects of Demand and Supply shifts, price controls & taxation
Equation: ep=%Q%P=QQPP=QPPQ
* Elasticity of Demand use demand function → always negative
* Elasticity of supply use supply function → always positive
Infinitely Elastic Demand
- HORIZONTAL: at any P* a tiny change in price leads to huge change in quantity demanded dQdP=-infinity
Consumers will buy as much of the good at P* but any higher=0 demand, any lower=infinity demand
Zero Elasticity
- VERTICAL: quantity demanded is the same at any price dQdP=0
Consumers absolutely need to purchase Q*, would not buy any larger or smaller quantity (ex.Meds, bread in poor countries)
Elastic Demand
e(-infinity,-1):
Total expenditure decreases when price goes upd(expenditure)d(price)=Q(e+1)<0
Unit Elastic
e(-1): when price changes, total expenditure on good doesn’t change
d(expenditure)d(price)=Q(e+1)=0
Function: QD=aP e=QPPQ=PaPeaePe-1=1
Isoelastic Demand
Isoelastic Demand: elasticity is constant along the demand curve/hyperbola
Function: QD=aPe
Inelastic Demand/Supply
Inelastic Demand/Supply e(0,-1): demand is relatively unresponsive to price changes
When price goes up total expenditure on the good increases (same quantity demanded x higher price=higher total expenditure) d(expenditure)d(price)=Q(e+1)>0
Effects of Price Changes on Expenditure Depending on Elasticity of Demand Table
See doc
Cross-Price Elasticities
Cross-Price Elasticities ebDa=dQDa/dPb Pb/QDa: % change in QD in one good resulting from 1% incr. In price of another good
Income Elasticity of Demand
Income Elasticity of Demand eID=dQDdIIQD: % change in quantity demanded resulting from 1% increase in income
Price Elasticity of Linear Demand
Price Elasticity of Linear Demand: ep=QDPPQD=-bPa-bP<0, (QD=a-bP dQDdQS) always positive
Elasticity Large (absolute value) ← P Large & Quantity Small
Elasticity Small (absolute value) ← P small & Quantity Large
Elasticity & Effects of Demand Shifts
- Elastic Supply (flat): large change in Q & small change in P
- Inelastic Supply (steep): small change in Q & large change in P
- Elastic Demand (flat): large change in Q & small change in P
- Inelastic Demand (steep): small change in Q & large change in P
Knowing elasticities gives same info as Demand & Supply Curves
See doc
Individual Demand + Function + Curve + Derivation
Individual Demand: relation b/w quantity demanded and its determinants (preferences, prices & income) indicating how much an individual consumer will purchase
* Derivation: obtained by solving consumer utility maximization problem subject budget constraint
* Individual Demand Function: x(p1,p2,…,pn,I)=D(p1,p2,…,pn,I)
* Individual Demand Curve: relation b/w quantity and price of demanded good for fixed levels of prices and income x(p1,p2,…,pn)=D(p1,p2,…,pn)
Income Consumption v. Engel Curves + Graph
- Income Consumption Curve: changes in individual demand as income changes
- Engel Curves: relation b/w income & demand for the good, upward slope (normal good), downward slope (inferior good)
Market Demand Curve
Market Demand Curve: sum of all the individual demand curves in the market, relating quantity that all consumers buy in mkt to price of that good (add horizontally)
Shifts: changes in number of consumers leaving/entering market and factors that influence the demand of many consumers
Consumer Surplus
Consumer Surplus (per unit at a point on line or full): measures consumer benefit of consuming a good in units of money, answers the question how well does the economy work, how much consumer benefit produced?
* Calculation: difference b/w willingness to pay and price/actual amount paid, area under demand curve above price line
Constraint Maximization Problem + Process
Constraint Maximization Problem:
max u(x1,…,xn) such that I=p1x1+…+pnxn, where xi is quantity of good & pi is price of good & I is consumer’s income
1. Form Lagrangian Function L=u(x1,…,xn)+(I-p1x1-…-pnxn), where is Lagrange multiplier on budget constraint
1. First-Order Conditions for xi, Lxi=ui’-pi=0
1. Set Equal Marginal Principle ==u1’p1=u2’p2 or Set MRS equal Price Ratio u1’u2’=p1p2 isolate for x1 (If it cannot hold it is a corner solution, the item that gives more quantity for same income will be spent to the max 1pi units)
1. Sub xi into budget constraint I=p1x1+…+pnxn isolate for x2 gives both demand functions
1. If pi given sub in & solve for optimal bundle quantities
Constraint Minimization Problem + Process
Find I’ when prices change to (p1’,…p’n) to be at same utility level/indifference curve as with I & (p1,..,pn)
Compute level of utility attained at old I & (p1,..,pn)=u= standard Consumer Max Problem
max u(x1,…,xn) subject to p1x1+..+xnpnI to create demand functions x1(p,…,p,I),…,,xn(p,…,p,I)
Sub Demand Functions into utility function to create an Indirect Utility Function: shows highest level of utility obtained under any I & (p1,..,pn), decision-making under uncertainty
u(x1(p,…,p,I),…,xn(p,…,p,I))=v(p1,…,pn,I)
Minimal Income min p1’x1+…+pn’xn needed to achieve old level of utility uu(x1,…,xn) under new prices (p1’,…,p2’)
2 Methods
Lagrangian Method: L=p1’x1+…+pn’xn+(u-u(x1,…,xn))
Compute Expenditure Function E(p1’,…,p2’,u)=p1’x1h,…,pn’xnh: minimum income needed to achieve given utility level u under prices (p1’,…,p2’)
Need to Raise Individual Income by: E(p1’,…,p2’,u)-I so individual has enough money to attain the previous utility level u
Solution Quantities are called Hicksian/Compensated Demands: (x1h(p1’,…,pn’,u),…,xnh(p1’,…,pn’,u))
Duality Method: utility maximization + expenditure minimization
Set Equation u=v(p1,…,pn,I) & isolate for I creating expenditure function E(p1,…,pn,u)=I
Revealed Preferences Approach + Draw Graph
- observing consumer choices under different budget constraints we can infer which baskets are prefered
- Due to Law of Diminishing Marginal Rate of Substitution all consumers have convex indifference curve
Main Principle of theory Of Consumer Behavior:
Main Principle of theory Of Consumer Behavior: consumers choose combo of goods that will maximize utility (preferences) given the limited budget (income & prices) available to them.
Main Principles of Econ Behavior
Rationality: each does what is best (varies) for themselves
Self-Interest: each maximizes its objective
HH max utilities (benefits)
Firms max profits
Market + Scope + Importance
- Market: collection of buyers and sellers who, through their actual or potential interactions, determine the price of a product or set of products, and quantities transacted
- Scope/Boundaries: geographical and in terms of range of products produced/sold within it, if a product is interchangeable with another they are in the same market
- Ex. Sweeteners (is a market) v. Corn Syrup (not a market)
- Ex. Blood Pressure Drugs (is a market) v. Drugs (not a market)
- Importance of Understanding Definition: (i) firms must understand who it actual+potential competitors are for various products that it sells or might sell in the future, (ii) public policy decisions (mergers or concentrations of mkt)
*
What Shifts Demand Curve?
- Income: Normal Goods: as income incr. demand incr. at same price (luxuries), Inferior Goods: as income incr. demand decr. at same price (fast food)
- Price of Other Goods: Complements: goods consumed together, price and demand shifts opposed (ex. Incr. price → decr. demand), Substitutes: two goods interchangeably in consumption, price and demand shifts same (ex. Incr. price → incr. demand)
What Shifts Supply Curve?
- Production/input costs (wages, interest rates, raw materials, energy, technology): decr. Input costs → produce more for same P/sell same Q for lower P
- Price of jointly produced goods → symbiotic, without a competition for inputs (ex.Beef & hides (ex.price incr. → supply incr.) shift in same direction
- Article Reading Natural Gas & Oil
- Price of a good that is produced from the same inputs → competition (ex.price incr. → supply decr.) shift in opposing directions
- Military v. civilian aircraft
Market Mechanism
Moving to Equilibrium:
In a free competitive mkt P & Q change until the mkt clears, starts where there is a shortage or surplus
* Equilibrium: Q0 mkt clears at price P0 so quantity supplied equates to quantity demanded, no change
* Surplus: P1 too high → S>D → will start to discount to sell supply → P will fall until equilibrium
* Shortage: P1 too low → D>S → will start to increase price to lower demand → P will rise until equilibrium
* Assumptions:
* At any given price a given quantity will be produced & sold
* Competitive Mkt: both sellers & buyers have little mkt power, ability to individually affect mkt price, or else P & Q can be fixed or changed at will for monopolist’s interest
3 Assumptions of Consumer Preferences
- Complete: able to rank all basket of goods which is better or worse
- Transitive: must be logically sound/rational, A>B>C
- Monotonicity or Non-Satiation Consumers: always prefer more of any good to less, cuz free disposal
- “Bads”: more of it is bad, pollution
- “Goods”: more of it is good, pollution reduction
Rule for Optimal Point
MRS doesnt equal PF/PC so individuals can reallocate income to increase utility
MRS>PFPC decr. Clothing, incr. Food consumption until they equal
MRS<PFPC decr. Food, incr. Clothing consumption until they equal
Corner Solution
Corner Solution: consumer optimally spends all income on one good and none on another, highest indifference curve consumer can attain, typically MRSPFPC
* Price change can cause corner solution to change corners or change into interior solution even with same preferences/indifference curves
Interior Solution + Equal Marginal Principle
Interior Solution - Best/Optimal consumption bundle has highest utility (highest indifference curve) that is on the budget line (affordable), when indifference curve is tangent to budget line=slopes are equal (or corner solution if linear utility function), any point higher will be above the budget line, Absolute Value of Slope of Indifference Curve MRS=-dCdF=MUFMUC=PFPC equals Absolute Value of Slope Budget Line
- Located on budget line: consumer spends all income b/c more is better
- Gives most preferred combo of G+S: highest attainable indifference Curve among all points on the budget lines
Equal Marginal Principle: MUFPF=MUCPC utility is maxed when budget is allocated so that marginal utility per dollar of expenditure is the same for each good, no incentive to change behavior (equilibrium), utility of last cent spent on food equals utility of last cent spent on clothing
Logical Mathematical Proof by Contradiction (Null-Hypothesis/Assume not true): try to prove that not on equal marginal principle gives greater utility
See doc
Non Linear Budget Constraint
See doc
Budget Constraint + Derivation + Changes
Budget Line: depicts all combos of goods for which total money spent equals total income
- Rearrange equation: C=IPC-PFPCF or F=IPF-PCPFC
- Slope (Price Ratio) -PFPC=dCdF or -PCPF=dFdC units of one good traded off for increase of 1 unit of the other good, rate at which the two goods can be substituted without changing the amount of money spent, ratio of prices of 2 goods with a minus sign
- Intercept (Units if all Income spent on 1 good) IPC or IPF, simplest way to draw is identifying these, maximum units
- Income: parallel shift outward/inward from original line, intercepts move by same amount cuz more/less goods can be bought
- Price: intercept of good with price change moves accordingly IPC or IPF If ratio of prices change unevenly it will change slope of budget line If ratio of prices stays the same, but incr/decr. It will shift outward/inward
Marginal Utility of a Good v. Law of Diminishing Marginal Rate of Substitution
Marginal Utility of a Good Partial Derivative of Utility Function In Terms of Good of Interest - MU=u(x1,…,xn)x: additional satisfaction obtained from consuming 1 additional unit of good, changes with the amount of good available, non-satiation axiom implies it is positive sloping
Law of Diminishing Marginal Rate of Substitution (Decreasing Convex): more X, the more X willing to give away for the other good, slope of indifference curves decreases the more of a single good it has. Can’t be convex
Diminishing Marginal Returns Principle
Diminishing Marginal Returns Principle - Concavity of MRS: slope/utility of a good decreases as the more given
Upward Slopes UX>0: the more of a good the more utility, marginal utility is positive
Concave U2X2<0: as consumption of a good incr. it’s marginal utility decr. due to satiation a good is worth more when we’ve consumed less of it worth less when consumed lots, slope increases decreasingly
As X incr. & Y decr.=MUX decr. as MUY incr. so MRS=-dYdX=MUXMUY
Marginal Rate of Substitution
Marginal Rate of Substitution Slope of Indifference Curve=MRS=-dYdX|U=U=UXUY=MUXMUY depends on Marginal utility of Each Good = Ratio of Marginal Utility: how a person trades one good for another remaining at the same utility level/indifference curve, amount of one good given up to obtain 1 extra unit of another good
Mar
Indifference Curves
Graphical Representation using Indifference Map to describe preferences/ranking of individuals, can differ b/w people
Indifference Curves: represent all combinations of mkt baskets that one is indifferent b/w, equally satisfied with any basket on the same indifference curve
Most Preferred - Above curve>curve>below curve - Least Preferred
Along curve are equally preferred
Downward Slope=follows Non-satiation axiom
Upward Slope violates Insatiation=more is better
Northeast more preferred as more of every good
Indifference curves cannot cross to satisfy transitive and non-satiation
Prove things by contradiction, assumptions are disproved
Derive Indifference Curve using Utility Function: set U=#1,#2…, isolate for C or F, graph each new function
Every bundle (x1,…,xn) that has same utility u(x1,…,xn)=u forms a indifference curve, along curve equally preferred
Utility Function
Utility Function u(X,Y) = Preference Ordering of Bundles: describes which has bigger utility/preference, magnitude and unit meaningless can’t say it is twice as high, all relative, Ordinal: the only thing of significance is its order, intervals are meaningless
Utility Function Theorem: if individual preferences satisfy completeness, transitivity & continuity properties there exists a utility function
Still is a utility function even after applying the same increasing function f to all, making it increasing as it still shows rank preference v=f(u()) as equality b/w functions are still same
EX. U(F,C)=F+2C mkt basket with 8 food, 3 clothes utility is 14=8+2(3)
Set of different indifference curves U3>U2>U1
