Notes 10: Risk and Uncertainty

Transcription

1 Economics 335 April 19, 1999 A. Introduction Notes 10: Risk and Uncertainty 1. Basic Types of Uncertainty in Agriculture a. production b. prices 2. Examples of Uncertainty in Agriculture a. crop yields due to a variety of factors b. number of days available for field work c. frost dates d. death loss in livestock e. egg breakage f. output prices g. input prices h. fulfillment of contracts i. accidents j. health problems 3. Causes of Uncertainty a. random forces (1) nature (2) human error (3) technology and obsolescence (4) machine failure (5) labor strikes (6) world events (7) institutional changes

2 2 b. timing of decisions time [input outputs] Because most agricultural production processes take significant time between the application of inputs and the realization of outputs, firms must often make decisions without ful information about the value of the product produced at the time they commit the inputs. 4. Some Strategies to Manage Uncertainty a. strategies for production uncertainty (1) crop insurance (2) lower risk enterprises (3) lower risk techniques in a given enterprise 4) diversification across enterprises (5) lease arrangements on inputs such as land or equipment (6) share cropping (7) health/liability insurance b. price or market uncertainty (1) forward contracts (2) hedging in a futures market (3) options markets (4) government programs (5) spreading of sales

3 3 B. The Decision Problem 1. Definition of a Decision Problem The decision problem is concerned with how a decision maker chooses among alternative courses of action when the consequences of each action are not know at the time the choice is made. It is assumed that Nature chooses the state of the world, which then determines which actions actually result in which consequences. 2. Components of the Decision Problem a. consequences or outcomes (c i ) These are the things which matter to a decision maker. We denote these consequences c 1, c 2, c 3,..., etc. The set of all possible consequences is denoted C. The consequence or outcome of a decision must be measured in some manner, such as profit, cost, or more generally, utility. For most agricultural problems, dollar profits can be determined by budgeting techniques, assuming the necessary data are available. (Note: Dollar profits will not be the appropriate measure of individual utility if the marginal utility of a dollar of income or wealth is not constant. Normally, marginal utility is not constant, so we must distinguish between dollar payoffs and actual utility values.) As an example, consider the possible consequences of applying a herbicide to a given field of corn. The direct consequences might be denoted c 1 = all weeds killed c 2 = 50 % or more but less than 100% of the weeds killed c 3 = less than 50% of the weeds killed c 4 = no weeds killed. We might measure these in terms of the change in profits per acre from a reduction in weed growth. For example, if all weeds are killed, the dollar benefit might be $20.00 per acre. We then assign a utility value to these benefits of $20. If utility and dollars are the same, then u(all weeds killed) = $20. b. feasible actions (a j ) Risk becomes important when people make decisions, when they must choose among actions which result in different possible consequences. The set of all actions available to the decision-maker is denoted A, where a is a typical action in A. For our purposes, the set of actions will be defined so that they are mutually exclusive and exhaustive, in the sense that they include all feasible acts. Actions are regarded as discrete and may be denoted a 1, a 2..., etc. The set of action sfor the herbicide problem might be a 1 = apply herbicide from company A. a 2 = apply herbicide from company B. a 3 = apply no herbicide.

4 4 c. mapping from actions to consequences When a person makes a choice, he does not know the consequence which will actually occur. Each action may lead to a number of possible consequences, so we have a mapping from actions to subsets of all the possible outcomes. For the above herbicide problem the mapping for the first action might be: a 1 < {50 % or more but less than 100% of weeds killed, less than 50% weeds killed} while the second and third the mappings might be: a 2 < {all weeds killed, 50 % or more but less than 100% of weeds killed, less than 50% weeds killed}, a 3 < {less than 50% weeds killed, no weeds killed} d. states of nature (s i ) The state of nature determines which consequence will actually result from an action. The states of nature to be considered in agricultural problems are those that can affect the production or pricing process. Definition of states of nature usually requires judgment about such variables as rainfall, prices, governmental actions regarding farm programs, grain exports to China, etc. States of nature are regarded as independent of actions; the decision maker s choice of an action is assumed to have no effect on the occurrence of the states of nature. States of nature are regarded as discrete and can be denoted s 1, s 2, s 3,... etc. The set of all states is denoted S. For the problem above the states of nature might be defined as {wet year, normal year, and dry year). Thus, the consequences of action 2 could vary with the state according to the following rule: c(a 2 s 1 ) < less than 50% weeds killed c(a 2 s 2 ) < all weeds killed c(a 2 s 3 ) < 50 % or more but less than 100% of weeds killed When consequence c occurs, the resulting utility is u(c). So, when a person chooses action a and state s occurs, the resulting utility is u(c(a s)), which we can write in short-hand as u(a s). Thus, if utility is the same as the dollar payoff (which is normally not the case), then u(a 2 s 2 ) = u(all weeds killed) = $ e. probabilities Each state of nature s occurs with some probability p. For example, there may be two states which are equally likely, in which case p(s 1 ) = p(s 2 ) = ½. Alternatively, we could have p(s 1 ) = 1/3 and p(s 2 ) = 2/3. More precisely, we suppose that the decision-maker has some beliefs about what these probabilities are. The function p(s) is called the subjective probability function. These probabilities represent the decision maker s beliefs about how likely the different states are to occur. They have the axiomatic properties ascribed to probability and must sum to one when added over all states. These probabilities will depend somewhat on the information available to a decision-maker, and thus they may change when the decision maker acquires new information.

5 5 3. Summary a j ö the j th act or risky prospect s i ö the i th state of nature p(s i ) ö the probability that s i will occur (or the belief about this probability) c(a j -s i ) ö the consequence of action a j when state s i occurs u(a j -s i ) ö u(c(a j -s i )) ö the utility that results if a j is chosen and s i occurs D. Making the Choice: Expected Utility Theory 1. Preferences over actions The decision maker must have a basis for selecting among alternatives actions: a choice indicator or objective function. He must have some preferences over actions in order to choose some action which he prefers over all others. We represent these preferences in terms of a relation. If the decision maker likes action a as much as he likes action b, then we write a b. This relation has the following properties: a. All actions are comparable: For all a, b 0 A either a b or b a (completeness) b. Preferences must be consistent with other preferences. If a b and b c, then it must also be the case that a c (transitivity) We can express these preferences in terms of a utility function U, where U(a) represents utility a decision maker assigns to an action a. This function is consistent with preferences if we have a b iff U(a) U(b) So we must ask, what should the function U look like? How should a decision maker choose between actions? 2. Some possible decision criteria to compute U(a) a. Maximin for the faint of heart. Choose the action which gives the best worst case scenario. In other words, take each action a and look at the lowest wealth level which might result. Let U(a) be this wealth level. For example, if action 1 leads to a 99% chance of being a millionaire and a 1% chance of being a pauper, you will prefer to take action 2, which offers a 50% chance of being moderately rich and a 50% chance of being middle class. You don t want to consider even the possibility of becoming a pauper, so you take the safe route, even though you are giving up an almost certain chance of becoming a millionaire.

6 6 b. Maximax for the true gambler. Choose the action wich gives the best case scenario. In other words, take each action a and look at the highest wealth level which could result. Let U(a) be this wealth level. With these preferences, you prefer action 1 to action 2, even if the probability of becoming a millionaire drops to 1% and the probability of becoming a pauper rises to 99%. c. Most Likely State What state is most likely? Let U(a) be the wealth which results in that state. In this case, with the probabilities in (a) above, action 1 is better than action 2, since it will probably give higher wealth than action 2. You do this even if the probability of becoming a millionaire drops to 51%. 3. Expected wealth or expected value The rules above do not seem plausible if applied under all circumstances. One should take into account all the possible consequences of an action. One way of doing this, is to look at the expected wealth, or expected value, which results. EV a ( ) = p( s) v( a s) s S The expected value which results from action a is given by where v(a s) is the monetary return to the action a in state s.. Example: Suppose that action a involves rolling a die. The consequences are the numbers 1-6. Let the monetary return to each consequence be a dollar payment equal to the number on the die. EV(a) = (1/6)($1) +(1/6) ($2) (1/6)($6) = $21/6 = $ Expected Utility Expected value takes into account all possibilities, but it ignores risk. For example, suppose action 1 gives a 50% chance of getting $1,000,001 and a 50% chance of getting $0. Suppose action 2 gives a 100% chance of getting $500,000. You will prefer action 1 to action 2 because the expected value you get with action 1 is $500, which is higher than $500,000. But most people would prefer action 2, since the risk involved is much lower. So, we need something more general. We suppose that, instead of maximizing expected value, the decision maker maximizes expected utility. There is a preference scaling function u(v) which represents the utility that results from wealth v. Thus, the utility function over actions is ( ) U a E U a p s u v a s ( ) = ( ) = ( ) ( ) s S Alternatively, if we use the utility function u(a s) we defined before, we could say that

7 7 Example: Suppose again that we roll a die. Let ( ) = p( s) u( a s) U a s S u(v) ö v Then the utility of rolling a die is 1 6 1ø 2ø 3ø 4ø 5ø 6 ö This is the same as the utility of a 100% chance of getting $ (1.8053) 2 = $ Note that this is lower than the expected value of $3.50. The gamble is not worth as much as $3.50 because the decision-maker does not like to take risks. He prefers a 100% chance of $3.50 to this gamble with an expected value of $3.50. We call this risk-aversion. E. An Example of Cattle Marketing Decisions 1. Assumptions a. Assume cattle bought on contract are purchased one month prior to delivery with payment at the time of delivery (at a price determined at the time of the contract). b. Assume cattle bought live are purchased and paid for on the day of delivery. c. Assume that live steers weigh 1200 lbs. d. Assume that all of the cattle will grade choice. 2. The Example with Certainty Consider a farmer who has ten steers to sell and must decide now whether to sell them on contract or wait a month and sell them for cash. Action a 1 is to sell on contract and action a 2 is to wait. The states of the world are summarized by the prices today and next month while the payoffs are the returns from the strategies given the prices. The farmer knows both contract and live prices with certainty. His information can be summarized in a table as follows: Contract Price $.60 Live Price $.65 The monetary returns to selling the cattle on contract are

8 8 10 x.60 x 1200 = 7,200 Similarly, live monetary returns are given by 10 x.65 x 1200 = 7,800 In this case the farmer should wait and sell the animals live, since the return is higher and there is no uncertainty. 3. Cattle Marketing with Uncertainty Assume contract prices are $.60 for choice steers. Contract monetary returns = 10 x 1200 x $.60 = $7200 a. Assumptions About States i. State I: Cash prices are $.60 for choice Live returns = 10 x 1200 x $.60 = $7200 ii. State II: Cash prices are $.55 for choice steers. Live returns = 10 x 1200 x $.55 = $6600 iii. State III: Cash prices are $.65 for choice. Live returns = 10 x 1200 x $.65 = $7800 b. Probabilities Assume the different states have different probabilities as outlined below. State 1.1 State 2.4 State 3.5 c. Expected Value The expected value of selling on the contract is $7200 since the outcome is the same under all states. The expected value of selling the cattle live is (.1)$ (.4)$ (.5)$7800 = $7260. The information can be summarized in a table. State I State II State III EV Likelihood Sell on a contract $7200 $7200 $7200 $7200 Wait and sell the cattle live $7200 $6600 $7800 $7260

9 9 If the producer cares only about expected value, he should sell the cattle live. The actual optimal decision depends on risk preferences. d. Expected Utility Assume the scaling function is given by u(v) = 1-e -.001v The expected utility of selling the cattle under a contract is then expected utility is u(7200) = 1-e = The expected utility of selling the cattle live is (.1)u(7200) + (.4)u(6600) + (.5)u(7800) = (.1)( ) + (.4)( ) + (.5)( ) = The information can be summarized in the following table. Expected Value and Expected Utility Probability Contract Returns Contract Utility Live Returns Live Utility State State State Expectation The contract provides a higher expected utility, even though the expected value is lower. So the producer will sell on the contract. The expected utility of the contract is higher because the risk is lower. Using a scaling function allows us to incorporate risk preferences into the decision.

10 10 F. The Shape of Preference Scaling Functions 1. Linear Scaling Functions A decision maker who is indifferent to risk will have a scaling function which is linear in wealth. Each additional dollar of wealth has the same value. Graphically this gives: Risk Neutral Utility Utility U(Wealth) Risk Neutral Wealth In this case, the individual basically maximizes expected value. He does not care at all about how much risk is involved. Therefore, this person will prefer to sell the cattle live. Suppose, for example, that in the example above we have u(v) = 2v + 3. Then the expected utility of selling the cattle live is: EU = (.1)u(7200) + (.4)u(6600) + (.5)u(7800) = (.1)( ) + (.4)( ) + (.5)( ) = 2 [(.1) (.4) (.5)7800] + 3 = = 2 EV + 3 = Notice that the expected utility is directly proportional to the expected value. We would get the same behavior if we simply supposed that u(v)=v. The expected utility of selling under the contract is u(7200) = = < The expected utility of selling the cattle live is higher, because the expected value is higher. A risk-neutral person will sell the cattle live in order to get higher expected returns and will not care about the risk of getting lower returns under some states of the world. 2. Risk Averse Scaling Functions A decision maker who is averse to risk will have a scaling function which is concave in wealth. Each additional dollar of wealth has less value than the previous dollar acquired.

11 An individual who is risk averse will not take a fair bet in the sense that she will prefer a certain amount to a bet which has the same expected value. For example consider two states of nature with p 1 = p 2 =.5 and a scaling function given by u( ) = x x 2. For this case u(100) = 10, u(150) = 16, u (200) = 21, u(250) = 25, u(300) = 28 A graph of the scaling function is as follows: 11 Notice that the slope of the scaling function is decreasing as wealth rises. Now consider a lottery or bet where the outcomes are 100 and 300 each with probability.5. Then the expected utility of this lottery is u(100)(.5) + u(300)(.5) = 10(.5) + 28(.5) = 19 < u(200) = 21. Graphically we can see this as follows: The line represents a linear combination of the values of u(#) associated with wealth levels of 100 and 300. The value of the line at the midpoint 200 is exactly (.5)u(100) + (.5)u(300) = (.5)(10) + (.5)(28) = 19, which is the same as the expected utility of the gamble. Note that, because the function is concave, the line segment falls underneath the curve between 100 and

12 The utility of getting wealth 200 for sure is higher than the utility of a gamble which might get you 100 and might get you 300. Therefore, whenever a person has a concave scaling function, that person is risk-averse. She will prefer certainty to risk. A risk-averse person will still take risks (risk is unavoidable), but she will insist on being compensated for risk in terms of a higher expected value. Consider again the cattle market problem discussed above. In this problem, the scaling function u is concave, and the person is risk-averse. She prefers to sell the cattle under a contract, rather than face the uncertain price of live cattle. But suppose that we change the probabilities, so that now the probability of state III is 70%, and the probability of state II is only 20%. Now the expected utility of selling the cattle live is: (.1)u(7200) + (.2)u(6600) + (.7)u(7800) = (.1)( ) + (.2)( ) + (.7)( ) = This is now higher than the expected utility of selling under the contract ( ). Even a risk-averse person is willing to take some risks, if the probability of a good outcome is sufficiently high. If the expected value is sufficiently large, and the risk is sufficiently small, then the risky action may yield a higher expected utility than the risk-free action. 3. Risk Loving Scaling Function We normally assume that decision-makers do not like risk. But the theory of expected utility allows for people who actually like to take risks. A decision maker who prefers risk will have a scaling function which is convex in wealth. Graphically this gives Each additional dollar of wealth has greater value than the previous dollar acquired. The decision-maker might take an action which involves risk, even if the expected value is actually lower than that which she could obtain with a riskless action.