How to find the equilibrium amount of $p_{2}$ in terms of $p_{1}$?

Question

There are $100$ tons of crops remaining to supply for the two months. The crop holders consider whether to sell crops now or one month later. Holders face the demand curve of each period as below:

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$q_{1}=150−p_{1}$, $q_{2}=160−p_{2}$

where $p_{t}$ and $q_{t}$ denote the price and quantity of the crop where 1 and 2 stand for this month and the next month, respectively. The monthly interest rate is 10%. Also, Here is assumed that compounding is counted once a month.

The condition of the monthly interest rate baffled me, can someone gives me some ideas to start with this question: find the equilibrium amount of $p_{2}$ in terms of $p_{1}$?

Answer 1

we need to solve: $$\begin{align} \max \quad & \pi= \pi_1+\frac{\pi_2}{1+r}=p_1q_1+p_2q_2 \\ \textrm{s.t.} \quad & q_1=150-p_1, \quad q_2=160-p_2 \\ & 0\leq q_1+q_2 \leq 100 \end{align}$$ rewriting in terms of prices $$\begin{align} \max_{p_1,p_2} \quad &\pi=p_1(150-p_1)+\frac{p_2(160-p_2)}{1.1} \\ \textrm{s.t.} \quad & 210\leq p_1+p_2 \leq 310 \end{align}$$

you can check that $\pi(p_1,p_2)=p_1(150-p_1)+\frac{p_2(160-p_2)}{1.1}$ is maximum at $(p_1,p_2)=(75,80)$ but $p_1+p_2=75+80=155\not \geq 210$, then it must be the case that at the optimum we have $p_1+p_2=210$

so, the problem we need to solve is: $$\begin{align} \max_{p_1,p_2\geq 0} \quad & \pi(p_1,p_2)=p_1(150-p_1)+\frac{p_2(160-p_2)}{1.1}\\ \textrm{s.t.} \quad & p_1+p_2=210 \end{align}$$

we can use Lagrangian method to solve the above problem, so the optimal price must satisfy the first order conditions: $$\begin{eqnarray} \frac{\partial \pi / \partial p_1}{\partial \pi / \partial p_2}=1 \tag{1}\\ p_1+p_2=210 \tag{2} \end{eqnarray}$$

from $(1): \quad 150-2p_1=\frac{160-2p_2}{1.1} \implies \boxed{p_2=\frac{11}{10}p_1-\frac{5}{2}}$

Answer 2

hint: hopefully you learned by now that if there are two options, lets say $x$ and $y$ rational people will optimally choose $x$ and $y$ such that marginal benefit of $x$ is equal to marginal benefit of $y$.

Now think what is the farmer's benefit of selling today? What is farmer's benefit of selling tomorrow? How does interest rate connect present value to the future?