Ch12Uncertainty(中级微观经济学-清华大学,钟笑寒)

State-Contingent Budget Constraints
Buy
$K of accident insurance. Cna = m - K. Ca = m - L - K + K = m - L + (1- )K. So K = (Ca - m + L)/(1- ) And Cna = m - (Ca - m + L)/(1- )
Contingent Consumption
Q:
What is the consumption bundle that is being chosen under uncertainty? A: Contingent consumption plan. A specification of what will be consumed in each different state of nature. – Contingent: dependent on something uncertain or in the future.
State-Contingent Budget Constraints
Buy
$K of accident insurance. Cna = m - K. Ca = m - L - K + K = m - L + (1- )K. So K = (Ca - m + L)/(1- ) And Cna = m - (Ca - m + L)/(1- ) m L I.e. Cna Ca 1 1
Expected Utility
Q:
What is the utility when state 1 or 2 is certain? A: u(c1) or u(c2). Implications.The expected utility is a probability-weighted sum of utility in each state. (often be denoted as Eu.)
Uncertainty is Pervasive
What
are rational responses to uncertainty? – buying insurance (health, life, auto) – a portfolio of contingent consumption goods.
Expected Utility


Q: If U(c1,c2,1,2)=1ln(c1)+2ln(c2) is a utility function for a consumer under uncertainty, which of the following is still his utility function? Among them, which still has the expected utility form? – U‟(c1,c2,1,2)=exp(U)= c11c22 – U‟‟(c1,c2,1,2)=3U+2=1[3ln(x1)+2]+2[3ln(x2)+2] A: Both are his utility functions, but only the second reserves the expected utility form. Implications. An expected utility function is unique up to a positive affine transformation, i.e., v(u)=au+b,with a>0.
Expected Utility
One
particular convenient form that the utility function might take is, U(c1,c2,1,2)=1u(c1)+2u(c2) Which is called an expected utility function, or, a von NeumannMorgenstern utility function. The expected utility function has some very interesting and useful properties.
Contingent Consumption
(State-)contingent
consumption plan – State of nature: Different realizations (events) of some random variables. – The specification of a contingent consumption plan tells us the probability distribution of different states of nature. – What will be consumed (in different states). Usually a consumption bundle (in our previous sense). A simplified form is the monetary gamble.
State-Contingent Budget Constraints
Accident
causes a loss of $L. Without insurance, Ca = m - L Cna = m.
State-Contingent Budget Constraints
Cna
U((x1,x2),(y1,y2),1,2)=1u (x1,x2) +2u (y1,y2).
What is the MRS between x1 and x2? Does it depend on y1 or y2? A: MRS(x1,x2)=MUx1/MUx2= ux1(x1,x2)/ux2(x1,x2), is independent of y1,2! Notice it‟s usually not true for arbitrary form of utility function. E.g., U=x1+x2y.
m L Cna Ca 1 1
slope
mL

1
Where is the most preferred state-contingent consumption plan?
m L Ca

Preferences Under Uncertainty
In
general, how a person values consumption in one state as compared to another will depend on the probability that the state in question will actually occur. Suppose that we are considering two mutually exclusive states. Let c1 and c2 represent consumption in state 1 and 2, and let 1 and 2 be the probabilities that state 1 or 2 actually occurs. Thus 1+2=1. The utility function can be written as U(c1,c2,1,2)
State-Contingent Budget Constraints
Buy
$K of accident insurance. Cna = m - K. Ca = m - L - K + K = m - L + (1- )K. So K = (Ca - m + L)/(1- )
State-Contingent Budget Constraints
m L 1
Cna
m L Cna Ca 1 1
m
The endowment bundle.
mL
m L Ca

State-Contingent Budget Constraints
m L 1
Contingencies
A
state-contingent consumption plan is implemented only when a particular state of Nature occurs. E.g. Suffer from loss only if there is an accident.
Expected Utility

Q: Suppose in state 1, the consumer consumes x1 unit of good 1 and x2 unit of good 2; in state 2, he consumes y1 and y2 unit of good 1 and 2. The expected utility function is,
States of Nature
Possible
states of Nature: – “car accident” (a) – “no car accident” (na). Accident occurs with probability a, does not with probability na ; a + na = 1.
Cna
m L Cna Ca 1 1
m
The endowment bundle.
slope