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
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
point at which the individual is indifferent between the risky option and receiving a certain, fixed amount
When is Risk-Premium Negative + Pareto Improvement of Efficiency
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
Utility of Wealth v. Expected Utility Function
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
Zero Elasticity
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