Insurance and Production Function Xingze Wang, Ying Hsuan Lin, and Frederick Jao (2007)

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1 Isurace ad Productio Fuctio Xigze Wag, Yig Hsua Li, ad Frederick Jao (2007) Priciples of Microecoomics, Fall 2007 Chia-Hui Che September 28, 2007 Lecture 10 Isurace ad Productio Fuctio Outlie 1. Chap 5: Reducig Risk: Isurace 2. Chap 6: Outlie of Producer Theory 3. Chap 6: Productio Fuctio: Short Ru ad Log Ru 1 Reducig Risk: Isurace Reducig Risk: Diversificatio Isurace Example (House isurace). Assume that oe house has the probability p to catch fire, with loss l each time, i.e. the ower s wealth will reduce from y 1 to y 2 = y 1 l. If the ower pay premium k to buy a isurace which covers the loss l whe there is a fire, her wealth will be y 3 = y 1 k, for the situatios listed (see Table 1). No Isurace Isurace No Fire y 1 y 3 = y 1 k Fire y 2 = y 1 l y 3 = y 1 k Table 1: Wealth of House Ower i Differet Situatios. Assumig the ower is a risk-averse, the utility fuctio is cocave. u (y) < 0 If the expected wealth at both situatios is equal, y 3 = (1 p) y 1 + p y 2. We have k = p l. Page 1 of 1

2 Figure 1: The Utility Fuctio of Risk Averse Perso. Page 2 of 2

3 The isurace premium is equal to the expected payout by the isurace compay, ad we say the isurace is actuarially fair. Sice the perso is risk-averse, u(y 3 ) > (1 p) u(y 1 ) + p u(y 2 ). she will choose to buy isurace. If the isurace is actuarially ufair, The k > p l. y 3 < (1 p) y 1 + p y 2. We do ot kow if the perso wats to buy or ot util we get her specific utility fuctio, but it is easy to imagie she may buy isurace if k is close to pl. Now we cosider what is the maximum isurace premium that the compaies ca charge ad the costumer is still willig to buy the isurace. I this case, let y 3 be the house ower s wealth after beig charged the maximum premium. The, (Figure 1) u(y 3 ) = pu(y 2 ) + (1 p)u(y 1 ). Thus the maximum isurace premium charged is k = y 1 y 3 = y 1 E(y) + Risk P remium = p l + Risk P remium So are isurace compaies more willig to take risk? If ot, why are they willig to sell isurace? The Law of Large Numbers ca explai this. Let L be the total loss from customers, It is a radom variable. The average loss shared by each customer is L, ad E( L ) = p. The expected payout for L by the isurace compay will be Whe E(L) = p l The probability that the loss shared by each customer is equal to a fixed umber pl is almost 1. (Figure 2) L P robability( = p l) 1. Note that this argumet oly applies to the situatio whe customers fire accidet evets are idepedet. Example (Illegal parkig). Govermet has two reasoable methods to puish illegal parkig. Hire more police, get caught almost for sure but fie is low. Hire less police, get caught sometimes but the fie is high. The latter might be more effective sice people are risk averse abd are afraid to take risk of beig fied to park illegally. Page 3 of 3

4 Figure 2: Distributio of L with Differet Customer Numbers. Page 4 of 4

5 2 Outlie of Producer Theory Productio Fuctio: Iputs to Outputs Give Quatity Produced, Choose Iputs to Miimize the Cost Choose Quatity to Maximize Firm s Profit The Productio Fuctio is The two iputs: q = F (k, L). k: Capital L: Labor It is easier to chage labor level but ot to chage capital i a short time. Short ru. Period of time i which quatity of oe or more iputs caot be chaged. For example, capital is fixed ad labor is variable i the short ru. Log ru. Period of time eed to make all productio iputs variable. I the log ru, both capital ad labor are variable. Page 5 of 5

Parametric Density Estimation: Maximum Likelihood Estimation

Parametric Density Estimation: Maximum Likelihood Estimation Parametric Desity stimatio: Maimum Likelihood stimatio C6 Today Itroductio to desity estimatio Maimum Likelihood stimatio Itroducto Bayesia Decisio Theory i previous lectures tells us how to desig a optimal