General equilibrium - exchange economy with time and perishable goods

Question

I am attempting to solve a general equilibrium problem from the textbook Jehle and Reny. However, I am slightly confused if my approach is correct. The question is as follows.

Consider an exchange economy with one good and two time periods. Two consumers have the following utility functions:

$U_A (x_{A1}, x_{A2}) = \sqrt{x_{A1}} + \sqrt{x_{A2}}$

$U_B (x_{B1}, x_{B2}) = \sqrt{x_{B1}} + \sqrt{x_{B2}}$

where $A1$ corresponds to time period 1 and $A2$ corresponds to time period 2.

Each consumer is endowed with one unit of the good at time period 1, BUT only B is endowed with 1 at time period 2 (A is endowed 0 at time period 2). Note that storage of the good is not possible, but consumers can engage in trade in period 1. Calculate the competitive equilibrium allocations.

I am not sure if I am overthinking the problem, but since the good is perishable. Will both agents just consume their endowments (e.g., agent 1 consumes 1 in the first period, and agent 2 consumes one in the first and one in the second).

Is there a scenario where agent 1 would want to sell some of his endowment in period 1 to purchase some in period two? Given his utility function I believe this would be the case, but showing it seems much different than the typical GE problems I have been solving.

Thanks

Answer 1

Of course, note that in general, consumer B is willing to increase her consumption in period 1. it is a matter of simply solving the problem in the regular way. i.e, MRS=p and find the $p$ which "clears" the market.

Normalize the price of the current period good to $1$ and let $p$ be the price of the second period good, then the optimum for player $A$ is: $\max_{x_{A1},x_{A2}} \sqrt{x_{A1}}+\sqrt{x_{A2}}$ s.t $x_{A1}+px_{A2}\le 1$ Clearly the constraint is binding and thus the problem becomes: $\max_{x_{A1}} \sqrt{x_{A1}}+\sqrt{\frac{1-x_{A1}}{p}}$ Calculate the first order condition, repeat the process for consumer B etc...

Answer 2

Think of the problem as a standard 2x2 pure exchange economy, because in a sense the same good in a different period can be thought of as a different good. In that case, the problem is simply to find the competitive equilibrium in an exchange economy.

let $p_1$ and $p_2$ denote the prices of periods 1 and 2 respectively.

Given, $U_A=\sqrt{x_{A1}}+\sqrt{x_{A2}}$ with endowments $\omega_A=(1,0)$

To obtain the demand functions for individual A, solve: $$\begin{align} \max_{x_{A1},x_{A2}\geq0} \quad & \sqrt{x_{A1}}+\sqrt{x_{A2}} \\ \textrm{s.t.} \quad &p_1x_{A1}+p_2x_{A2}\leq p_1\cdot1+p_2\cdot0=p_1 \\\text{gives: } \quad &(x_{A1},x_{A2})^d(p_1,p_2)=\left(\frac{p_2}{p_1+p_2},\frac{p_1^2}{p_2(p_1+p_2)}\right) \end{align}$$

Similarly, $U_B=\sqrt{x_{B1}}+\sqrt{x_{B2}}$ with endowments $\omega_B=(1,1)$ $$\begin{align} \max_{x_{B1},x_{B2}\geq0} \quad & \sqrt{x_{B1}}+\sqrt{x_{B2}} \\ \textrm{s.t.} \quad &p_1x_{B1}+p_2x_{B2}\leq p_1\cdot1+p_2\cdot1=p_1+p_2 \\\text{gives: } \quad &(x_{B1},x_{B2})^d(p_1,p_2)=\left(\frac{p_2^2+p_1p_2}{p_1^2+p_1p_2},\frac{p_1}{p_2}\right) \end{align}$$

In the same fashion as we do in typical exchange economies, normalize the price of period 1 i.e., $p_1\overset{set}=1$

So, the demand function reduces to: $$\begin{align} (x_{A1},x_{A2})^d(p_2)=\left(\frac{p_2}{1+p_2},\frac{1}{p_2(1+p_2)}\right) \quad & \& \quad (x_{B1},x_{B2})^d(p_2)=\left(p_2,\frac{1}{p_2}\right) \end{align}$$

Market clearing condition for period 2: $$\begin{eqnarray} & x_{A2}^d(p_2)+x_{B2}^d(p_2)=\omega_{A2}+\omega_{B2} \\ & \frac{1}{p_2(p_2+1)}+\frac{1}{p_2}=1 \\ & p_2=\sqrt2 \end{eqnarray}$$

Therefore, $\boxed{((x_{A1}^*,x_{A2}^*),(x_{B1}^*,x_{B1}^*))=\left(\left(\frac{\sqrt2}{\sqrt2+1},\frac{1}{2+\sqrt2}\right),\left(\sqrt2,\frac{1}{\sqrt2}\right)\right)}$ is the competitive equilibrium allocation supported by the price vector $\boxed{(p_1^*,p_2^*)=(1,\sqrt{2})}$