Numerical Methods for Expected Utility
Assume have Bt and non-Bt corn and have to choose which to plant
Model:
Yno = PYpf(1 – ) and YBt = Ypf
so net returns are
no = PYpf(1 – ) – K andBt = PYpf – T – K
Decision rule if no risk: Set Bt ≥ no and solve for  at which it’s economical to buy Bt
PYpf(1 – ) – K ≤ PYpf – T – K
PYpf – Ypf ≤ PYpf – T
– PYpf ≤ – T
 ≥ T/(PYpf)
Pretty simple model
Problem: yield loss is not known or certain when make decision
What’s the decision rule when have random loss?
Assume know average loss: 
Risk neutral: maximize expected profit, which for discrete case here means choose the one with
the highest expected profit, or find mean loss such that E[Bt] ≥ E[no]
First: find E[]
no] = PYpf(1 – E[]) – K = PYpf(1 – ) – K
Bt] = PYpf – T – K = PYpf – T – K
(nothing random)
Do the math and will get:  ≥ T/(PY)
Again: pretty simple idea
Main problem:
Model assumes farmer’s do not respond to risk, that the elimination of risk has no value to
farmers (notice the risk neutral model assumes have risky losses without Bt and with Bt, no risk).
Want to get at how the risk reduction benefit also has value and adjusts the threshold as to when
plant Bt; i.e., will a farmer be willing to plant Bt for expected losses less than T/(PY)?
How do we answer this question? Need to assume risk preferences, or a specific utility function.
Simple place to start: CARA utility: Constant Absolute Risk Aversion U ( )   exp( RA )
Decision rule: Maximize expected utility, which for discrete case here means choose the one
with the highest expected utility, or find loss such that E[U(Bt)] ≥ E[U(no)]
Question: how do you describe the loss distribution? Can’t just use the mean loss anymore, need
more, need to know the distribution of the loss .
For now: assume loss has a beta distribution, with mean of  and st dev  0.15 and min and max
of 0 and 1.0. [later discuss how to estimate pdf’s from data]
Now: need a decision rule: set of  and  such that E[U(Bt)] ≥ E[U(no)], or more generally: the
distribution of  such that E[U(Bt)] ≥ E[U(no)].
First: Look at E[U(Bt)] and note nothing random: PYpf – T – K, so
E[U(Bt)] = U(Bt) = –exp(–Ra(PYpf – T – K))
Decision rule: E[U(Bt)] ≥ E[U(no)] now –exp(–Ra(PYpf – T – K)) ≥ E[U(no)]
Need E[U(no)]:
E[U(PYpf(1 – ) – K)] = E[–exp(–Ra(PYpf(1 – ) – K))]
E[–exp(–RaPYpf(1 – ) + RaK)] = E[–exp(–RaPYpf + RaPYpf + RaK)]
E[–exp(–RaPYpf)exp(RaPYpf)exp(RaK)]
–exp(–RaPYpf)exp(RaK)]E[exp(RaPYpf)]
–exp(–RaPYpf + RaK)]E[exp(RaPYpf)]
Main point: what’s the E[] of a beta random variable multiplied by Z and raised to exp()?
Problem: cannot calculate a general analytical solution for expected utility, so now cannot get a
nice simple decision rule that you can write down like  ≥ T/(PY)
Can only solve this numerically: for specific distribution of , or a specific set so parameters 
and  for loss .
Monte Carlo Simulation
Suppose have variable x with pdf f(x) and want to know E[x] =
 xf ( x)dx
Suppose you cannot solve the integral
However, you can obtain many random draws from the pdf f(x)
1 K
Monte Carlo Approximation: E[x]   xk , where xk is the kth random draw from the pdf f(x)
K k 1
Key is to use enough random draws so that estimated mean “stabilizes” or converges to the
“correct” mean. Also need a way to get the random draws: pseudo-random number generators
Our problem here: loss  is random, but we already know its mean (and st deviation).
More Common problem: know x ~ f(x), but want to know E[g(x)] =
Monte Carlo Approximation: E[g(x)] 
1
K
K
 g ( x) f ( x)dx
 g ( x ) , where xk is kth random draw from pdf f(x)
k 1
k
Implementation in Excel
1) Draw random variable x
2) Calculate g(x)
3) Calculate average value of g(x)
How do you draw random variables
1) Draw Uniform (0 to 1)
2) Transform it to beta using the inverse of the CDF
3) Calculate profit and utility with and without Bt
4) Calculate average to get E[] and E[U]
5) Calculate CE and RP and RP as % mean
Can now set Ra parameter, then do analysis
Issues: how find  and ?
1) Set econ parameters (yield, price, cost, tech fee, risk aversion)
2) Set sigma
3) Vary mu until get CE = E[]