# How to calculate the maximum willingness to pay? (Reservation price)

I have to solve the following exercise. I cant figure out what I have to do know. I hope i can get some help.

Suppose you have the following utility: $$U (x_1, x_2) = 4x_1 ^ .5+2x_2 ^ .5$$ and you have the income m = 20. If you stay in Aarhus over the weekend you can buy these two goods at prices $$p_1 = 4$$ and $$p_2 = 2$$.

Alternatively you can borrow a car and drive down to Mutter Poetz in Flensburg and buy the goods at prices $$p_1 = 1$$ and $$p_2 = 2$$.

Suppose you get neither the benefit nor the dis-benefit of anything other than these two benefits (ie you don't care that you spend time on the drive for example).

What is your maximum willingness to pay to borrow the car? (Fuel cost is included in the amount you have to pay to borrow it)

I have tried to solve a maximization problem in both situations.

The first situation, where i stay in Aarhus and buy the goods to the prices $$p_1=4$$ and $$p_2=2$$ with the following budgetconstraint: $$4x_1+2x_2=20$$

$$x_1^\star=3.33$$ and $$x_2^\star=3.33$$

in the second situation, where i drive to Flensburg and buy the goods to the prices $$p_1=1$$ and $$p_2=2$$. With the following budgetconstraint: $$x_1+2x_2=20$$

$$x_1^\star=17.78$$ and $$x_2^\star=1.11$$

The optimal $$x_1$$ and $$x_2$$ is given by the following

$$x_1^\star=\frac{m}{p_1+\frac{p_1^2}{p_2\cdot4}}$$

$$x_2^\star=\frac{m}{p_2+\frac{p_2^2\cdot4}{p_1}}$$

I dont know if im doing it right and whether i should use it to solve the exercise.

• In the second situation where you drive to Flensburg, your budget constraint needs to include the cost of renting the car. So your income would be $20-r$ or something. Then you'd find a bundle that gets you the same utility as in the first situation where you stay in Aarhus, while also satisfying the marginal rate of substitution for the second problem. – Kitsune Cavalry Mar 16 at 0:26
• Okay, thank you for the answer. I will try that. But what about the reservation price? – Tarek Badr Mar 16 at 0:48
• I got the following when I solved the problem with the cost $x_1^\star=17.78-\frac{8r}{9}$ and $x_2^\star=1.11-\frac{1}{18}\cdot{r}$. – Tarek Badr Mar 16 at 1:01
• The value of $r$ that gets you the conditions I described above is the reservation price. Keep in mind you are not really looking to get the exact same bundles. You are looking for the same utility. – Kitsune Cavalry Mar 16 at 2:35
• I understand now. Thank you :). – Tarek Badr Mar 16 at 3:16