# Solving an exponential utility function

I have a utility function $$u(x) = a - b e^{-x/20\,000}$$ and it is given that $$u(0)=0$$ and $$u(100\,000)=1$$.

I am trying to show that $$a = 1.0067837$$ and $$b = 1.0067837$$. Here is what I tried:

\begin{align}u(0) &= a - b e^{-0/20\,000} = 0 &\implies a-be^1=0 &\implies a=be\\u(100\,000) &= a - b e^{100\,000/20\,000} = 1 &\implies a-be^5=1 &\implies a=1+be^5\end{align}

When $$a=be=1+be^5$$, taking $$e = 2.72$$, it doesn't get $$b = 1.0067837$$.

Where did I do wrong, and how can I correct it? Thank you.

The utility function $$u(x)=a-be^{-x/20\,000}$$ with the conditions $$u(0)=0$$ and $$u(100\,000)=1$$ gives $$u(0)=a-be^{-0/20\,000}\implies 0=a-be^0=a-b\implies a=b\tag1$$ and $$u(100\,000)=a-be^{-100\,000/20\,000}\implies 1=a-ae^{-5}=a(1-e^{-5})\tag2$$ so that $$a=b=(1-e^{-5})^{-1}=1.0067837$$ as desired.

Where you went wrong:

• $$u(0) = a - b e^{-0/20\,000} = 0 \implies a-be^1=0$$ should be $$a-be^0=0$$.

• $$u(100\,000) = a - b e^{100\,000/20\,000} = 1$$ should be $$a-be^{-100\,000/20\,000}$$; you just forgot the minus sign.

• Thank you so much for the teaching in details and so well thoughtful. wish you a brilliant weekend! Oct 26 '19 at 10:32
• No problem, I appreciate your trying the problem before asking. Have a nice weekend too. Oct 26 '19 at 10:33

set k=1 in a=be=1+be^5. therefore a=k+be^5. solve for k=a-be^5.

• Welcome to OR.SE! Please use MathJax to typeset the math in your posts. Your post is a bit hard to follow as written. Nov 18 '20 at 17:51