The only difference between endowment questions and normal constrained optimization questions is in the budget constraints.
Formulation and Intuition
Normally the consumer's budget constraint looks like , where are prices of the two goods, and , and m is the income. With endowment , the budget constraint becomes . With endowment of and , we think of the consumer selling her endowment for cash (of value ) and then use the cash to purchase the consumption she wants, like she did with income m in the first equation.
Note that with , when p1 increases, we expect the consumer to purchase less x1. However, this is not necessarily the case with endowment. When p1 increases, x1 becomes more expensive, but so does w1, proportionally. What that means is that the consumer can sell her endowment of w1 at a higher price. Whether she ends up consuming more or less x1 depends on her preference, ie, whether she is a net buyer or a net seller of good 1 in the first place.
Graphically, the budget line with the endowment is also slightly different from the normal budget constraint. With , the slope of the budget constraint is and when p1 increases, for example, then the budget line becomes steeper and the x-intercept of the budget line decreases. For , the slope of the budget line is still , but when p1 increases, the budget line becomes steeper by rotating through the endowment point. The reason is that for , the endowment bundle is always feasible (the consumer can always decide not to trade) so the budget line (which denotes the maximum feasible set) always passes through the endowment.
Consider the following statements with budget constraint , and some utility functions.
1, if price of a good consumer is selling goes down and consumer remains seller, welfare goes down.
This is a revealed preference argument. Let's call the new price p1' and we know p1' < p1. Also we know that x1* and x1' (the new optimal bundle) are less than w1. First we know that the new bundle (x1', x2') satisfies the new budget constraint: . Then we can show that the new bundle was affordable under the old budget constraint: . Given that the new optimal bundle was affordable but not chosen, it must be that it gives lower welfare to the consumer.
Intuitively, if a consumer chooses to sell a product and that the price of the product goes down, she is getting less value from selling (and the decrease in price is not big enough to induce her to buy the product), resulting in less welfare.
Graphically, being a seller of a good (good 1) means that the optimal points (new and old) are on the left side of the endowment point. When the price of the good goes down, the budget line rotates in the counter-clockwise direction. After the rotation, the feasible area on the left of the endowment point decreases. As the consumer is having fewer choices than before, her welfare goes down.
2, if consumer is a net buyer and price goes down, consumer will remain a net buyer.
From the previous problem we learn that when a consumer is selling a good and the price of the good goes down, the feasible set on the selling side becomes smaller. As a result any bundle that involves selling of the good cannot give better welfare than before. Moreover, the original optimal bundle (x1*, x2*) is affordable under the old budget constraint: . It is still available under the new budget constraint (with p1' being the decreased price for good 1): (because x1* - w1) is positive. As the consumer chose not to sell the good under the original budget, she would not choose to do so under the new budget.
Graphically, being a buyer of a good (good 1) means that the optimal point is on the right side of the endowment point. When the price of the good goes down, the budget line rotates in the counter-clockwise direction. After the rotation, the feasible area on the right of the endowment point increases, while that on the left side decreases. In the original budget constraint the consumer chose a point on the right side of the endowment point, revealing that she prefers that point over any bundle on the left. A decrease in the feasible set on the left, with the original bundle still affordable, can never give a better welfare than the original bundle and induce the consumer to switch to the selling side.
Budget constraints with endowment are often used in general equilibrium setting, where agents are assumed to be endowed with goods which they bring to the market to trade.
Another application is labour supply, where agents are assumed to be endowed with a certain amount of time T (think of it as T = 24 hours a day, or T = number of awake hours in a year), and they choose to allocate T between work (h) and leisure (l). For every unit of time the agent works, the agent earns a wage w per unit of time. The wage can then be used for consumption. Thus the budget constraint for the agent is c = w(T - l) + n, where n is some non-labour income.
In the intertemporal choice setting, agents are assumed to earn an income I1 when she is young, and earn an income I2 when she is old. She can decide on her consumption in the two periods of her life. If she uses more than her income I1 when she is young, she has to borrow against her future income and pays an interest r; on the other hand, if she spends less than her income when she is young, she can save the extra money for the future. The money will earn an interest rate r. The intertemporal budget constraint is thus
Consider an example with Labour Supply. Suppose the agent has a utility function of where l = number of hours of leisure and c = consumption. The agent receives a wage of w, a non-labour income of n, and T hours of endowment per day. She has to allocate her T hours between work (h) and leisure (l), what's her optimal choice?
First we form her utility function . We can find first order conditions (by the Lagrange method perhaps) and found the solution to be . Observe that the price of leisure, w, appears in the optimal choice of c. This is a key difference between optimal solutions with endowment and those with a fixed amount of income.
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