2.11 Summary

Things you should know: Trading from an endowment

Endowment $(e_1,e_2)$: the “starting bundle” an agent is “endowed” with
Endowment budget line : the budget line for an agent with an endowment $(e_1,e_2)$ who can buy and sell the two goods for prices $(p_1,p_2)$; given by the equation $p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2$ with intercepts $x_1^\text{max} = e_1 + \frac{p_2}{p_1}e_2$ $x_2^\text{max} = e_2 + \frac{p_1}{p_2}e_1$
MOST IMPORTANT: the equation of the endowment budget line ONLY depends on the price ratio $p = \frac{p_1}{p_2}$, not the individual prices. If you double all prices, the budget line doesn’t change.

Offer curve: a parametric plot showing the optimal consumption bundle (gross demands) at various prices. Because an agent is not forced to trade, their offer curve must lie ON or ABOVE the indifference curve passing through their endowment.
Gross demand for good j ($x_j^\star$): the quantity of good $j$ an agent chooses to consume facing an endowment budget line
Net demand for good j ($x_j^\star - e_j$): the amount by which an agent’s gross demand for good $j$ exceeds their endowment of good $j$.
Net supply of good j ($e_j - x_j^\star$): the amount by which an agent’s endowment of good $j$ exceeds their gross demand for good $j$.
Sometimes we use net demand for everything, and allow it to be negative; sometimes we say that net demand shows only the prices where the agent wants to buy more of a good than they’re endowed with, and net supply shows only the prices where the agent wants to sell some of their endowment of the good.
MOST IMPORTANT: An agent will supply good 1 and demand good 2 if their $MRS(e_1,e_2) < \frac{p_1}{p_2}$ and demand good 1 and supply good 2 if $MRS(e_1,e_2) > \frac{p_1}{p_2}$

If the price at which you can buy things differs from the price at which you can sell them, there is a kink at the endowment point:

the price ratio to the left of the endowment (agent sells good 1 and buys good 2) is $p_1^\text{sell}/p_2^\text{buy}$
the price ratio to the right of the endowment (agent sells good 2 and buys good 1) is $p_1^\text{buy}/p_2^\text{sell}$
the corollary to the “most important” item for a single price is that an agent will supply good 1 and demand good 2 if their $MRS(e_1,e_2) < \frac{p_1^\text{sell}}{p_2^\text{buy}}$ and demand good 1 and supply good 2 if $MRS(e_1,e_2) > \frac{p_1^\text{buy}}{p_2^\text{sell}}$ and if neither of these is the case, they optimize by remaining at their endowment. Comparing the MRS at the endowment to the price ratios is the critical first step to solving these kinds of problems.

The labor supply model is an endowment model in which good 1 is leisure ($R$, measured in hours), good 2 is consumption ($C$, measured in dollars for), and the price ratio is the wage rate $w$.
The endowment is $\overline R$ hours of leisure (usually 24) and $M$ dollars of nonwage income. This implies that the equation for budget line is: $wR + C = w \overline R + M$
Labor supply curve: plots $L^*(w)$ in a diagram with $L$ on the horizontal axis and $w$ on the vertical axis; note that $L$ is the net supply of good 1, so this is just a net supply curve.
Alternative framing that we didn’t get into: if consumption is measured in a “composite good” with a price level $p$, the slope of the budget line represents the real wage $w/p$ rather than the nominal wage $w$.

Given an endowment $(e_1,e_2)$ and constant prices $p_1$ and $p_2$, draw the endowment budget line.
Illustrate how a change in endowment and/or prices affects a budget line.
Given an endowment and any utility function (not just Cobb-Douglas!), find the optimal bundle to which the agent will trade.
Given an endowment and any utility function, derive and plot the offer curve, gross demand, net demand, and net supply for either good as a function of the market prices.
Given different prices for buying and selling, draw the budget constraint and determine whether the agent will buy, sell, or remain at their endowment.

Represent an employment offer in leisure-consumption space
Determine the equation for a budget line with a wage
For any utility function, determine the optimal labor supply choice for any wage rate $w$ and nonwage income $M$
Extend the model beyond the simple case, to include situations like overtime pay or “take it or leave it” employment contracts

Ex 2.1 walks you through the canonical process of optimizing from an endowment with a Cobb-Douglas utility function.
Ex 2.2 uses the results of Ex 2.1 to derive net demand and net supply functions.
Ex 2.3 and 2.4 are old exam questions; Ex 2.4 in particular is a problem with different prices for buying and selling, so you should try that one for sure!
Ex 2.5 and 2.6 are old exam questions on labor supply; Ex 1.6 in particular is a tricky question and very much worth doing, as questions like this show up on exams all the time.

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