<p>I am working on the following question but have not been able to come up with a suitable way to proceed. The setup is as follows:</p>
<p>There is a technology (for example, a vaccine) which reduces the probability of the occurrence of a bad state. The probability of the bad state before and after adoption is 'p' and 'q', respectively. The agent's utility function is given by h(x). The agent gets a utility 'u' from adoption. If the bad state occurs, irrespective of the adoption decision, there is a dis-utility of '-f'.</p>
<p>For a <strong>risk-neutral</strong> agent with h(x) = x. The expected utility from adoption is</p>
<p><span class="math-container">$$U(\text{adopt})=q(u-f)+(1-q)(u),$$</span></p>
<p>whereas the expected utility of not adopting is</p>
<p><span class="math-container">$$U(\text{not adopt})=p\cdot (-f).$$</span></p>
<p>When we compare the to utilities, the agent adopts if,</p>
<p><span class="math-container">$$q(u-f) + (1-q)\cdot u&gt;p\cdot (-f)$$</span> which implies that the agent adopts if <span class="math-container">$$u/f&gt;(q-p).$$</span> The RHS gives us a <em>threshold</em>.</p>
<p>For a <strong>risk-averse</strong> agent, with utility given by <span class="math-container">$h(x)$</span>, the utility from adoption is</p>
<p><span class="math-container">$$U(\text{adopt}) = q\cdot h(u-f)+(1-q)\cdot h(u),$$</span> whereas the utility of not adopting is,</p>
<p><span class="math-container">$$U(\text{not adopt}) = p\cdot h(-f).$$</span> This does not give us a threshold as we have not specified the function <span class="math-container">$h(x)$</span>.</p>
<p><strong>My question is as follows</strong>: how can I compare the threshold of the agent for the two preference types (even though I have not specified <span class="math-container">$h(x)$</span>)? Is there a way that I can say with certainty that the threshold for a risk-averse agent is lower/higher than that of a risk-neutral agent? If so, how do you prove that the threshold for the risk-averse agent is lower/higher than that of a risk-neutral agent?</p>

CC BY-SA 4.0
2024-04-28T07:52:47.460Z
I think in the risk averse case $U(\text{not adopt}) = p\cdot h(-f)$ should be $$U(\text{not adopt}) = p\cdot h(-f) + (1-p)\cdot h(0).$$ Also, are you familiar with the connection between concavitiy of $h$ and risk aversion?
Giskard
CC BY-SA 4.0
2024-04-28T07:57:35.233Z
Dear @Giskard, thank you for your comment. I should have mentioned that I assume that $h(0)=0$. Yes, I am familiar with the connection between the concavity of $h$ and risk aversion. A risk-averse agent has a concave function, where $$h((1-\alpha)x+\alpha y) \geq (1-\alpha)h(x) + \alpha h(y)$$
Ramandeep
<microeconomics><utility><expected-utility><risk><risk-aversion>
Comparing agent decision-making under risk-neutrality and risk-Aversion
Ramandeep
# Comparing agent decision-making under risk-neutrality and risk-Aversion

(Asked by: username_0 on 2024-04-28)

I am working on the following question but have not been able to come up with a suitable way to proceed. The setup is as follows:

There is a technology (for example, a vaccine) which reduces the probability of the occurrence of a bad state. The probability of the bad state before and after adoption is 'p' and 'q', respectively. The agent's utility function is given by h(x). The agent gets a utility 'u' from adoption. If the bad state occurs, irrespective of the adoption decision, there is a dis-utility of '-f'.

For a **risk-neutral** agent with h(x) = x. The expected utility from adoption is

$$U(\text{adopt})=q(u-f)+(1-q)(u),$$

whereas the expected utility of not adopting is

$$U(\text{not adopt})=p\cdot (-f).$$

When we compare the to utilities, the agent adopts if,

$$q(u-f) + (1-q)\cdot u>p\cdot (-f)$$ which implies that the agent adopts if $$u/f>(q-p).$$ The RHS gives us a *threshold*.

For a **risk-averse** agent, with utility given by $h(x)$, the utility from adoption is

$$U(\text{adopt}) = q\cdot h(u-f)+(1-q)\cdot h(u),$$ whereas the utility of not adopting is,

$$U(\text{not adopt}) = p\cdot h(-f).$$ This does not give us a threshold as we have not specified the function $h(x)$.

**My question is as follows**: how can I compare the threshold of the agent for the two preference types (even though I have not specified $h(x)$)? Is there a way that I can say with certainty that the threshold for a risk-averse agent is lower/higher than that of a risk-neutral agent? If so, how do you prove that the threshold for the risk-averse agent is lower/higher than that of a risk-neutral agent?

**Tags:** microeconomics, utility, expected-utility, risk, risk-aversion

> **Comments:**
> **username_1**: I think in the risk averse case $U(\text{not adopt}) = p\cdot h(-f)$ should be $$U(\text{not adopt}) = p\cdot h(-f) + (1-p)\cdot h(0).$$ Also, are you familiar with the connection between concavitiy of $h$ and risk aversion?  
> **username_0**: Dear @username_1, thank you for your comment. I should have mentioned that I assume that $h(0)=0$. Yes, I am familiar with the connection between the concavity of $h$ and risk aversion. A risk-averse agent has a concave function, where $$h((1-\alpha)x+\alpha y) \geq (1-\alpha)h(x) + \alpha h(y)$$  

---

## 0 answers

