Choices with endowment

In the real world, consumers not only buy and use products but they can possess and sell them too. Let’s consider a market with only two commodities: good 1 and good 2 and let’s suppose that consumers posses some quantities of both goods which represent the consumer’s endowment (ω1; ω2). The consumer’s gross demand (x1;x2) is the final quantity he ends up consuming. The consumer’s net demand (x1- ω1; x2 - ω2) is the difference between what he ends up consuming and what he already has. Obviously, gross demands must be positive numbers or 0 but net demands can be also negative numbers. This is because net demand represents the change in the quantities of the goods. Net demand is negative when the consumer wants to sell some of its goods.

THE BUDGET CONSTRAINT WITH AN ENDOWMENT

In this scenario, we are not considering an income for consumers but instead they have to exchange the goods they already possess to obtain a combination of them they like more. The consumer can consume any combination of the two goods that has the same value of its endowment. To understand why, think of a consumer that possesses 2 apples and 3 oranges: he cannot hope to consume in the end 20 apples and 50 oranges; he might consume 1 apple and 4 oranges or 5 apples and 0 oranges. This condition of the budget constrain can be expressed algebraically by:

p1x1 + p2x2 = p1ω1 + p2ω2.

The value of the goods you end up consuming (gross demand times the price of the goods) = the value of the endowment

Or we can rewrite that expression using net demands as:

p1(x1 − ω1)+p2(x2 − ω2)=0

the value of what you want to acquire (positive) must be equal to the value of what you want to sell (negative) and therefore their sum will need to be 0

If:

• the net demand for good 1 (x1 − ω1) is positive the consumer is called a net demander or net consumer of good 1.

• The net demand for good 1 (x1 − ω1) is negative the consumer is called a net seller or net supplier of good 1.

When the prices of the two goods are set, the value of the endowment is fixed and therefore you can plot the budget line. In order to represent this kind of problems we will use a graph where on the x-axis is measured the quantity of good 1 and on the y-axis the quantity of good 2. The budget line is the line with slope -p1/p2 that passes through the endowment. It has a vertical intercept on the y-axis given by the value of the endowment divided by the price of good 2.

Vertical intercept = (p1ω1 + p2ω2)/ p2 = (p1ω1/p2 + ω2)

It represents the maximum quantity of good 2 the consumer can possess without consuming good 1. The horizontal intercept on the x-axis is given by the value of the endowment divided by the price of good 1.

Horizontal intercept = (p1ω1 + p2ω2)/ p1 = (ω1 + p2ω2/p1)

It represents the maximum quantity of good 1 the consumer can possess without consuming good 2.

WHAT HAPPENS WHEN YOU CHANGE THE ENDOWMENT?

If we assume that prices are fixed in the market and we change the value of the endowment of the consumer we will se a shift in the budget line. The budget line shifts in a parallel fashion because remember that the slope of the line is -p1/p2 and we assumed both of them to be fixed. If we change the endowment in a way that the bundle is worth more so that: p1ω’1 + p2ω’2 >p1ω1 + p2ω2, the budget line will shift outward and therefore the consumer is better off because he can still buy the bundles he could have afforded with the old budget line and now can buy other bundles which are better for him. If we change the endowment in a way that the bundle is worth less so that: p1ω’’1 + p2ω’’2 < p1ω1 + p2ω2, the budget line will shift inward and therefore the consumer is worse off because he can not buy anymore the bundles he could have afforded with the old budget line and now can only buy other bundles which are worse for him. If the endowment changes but its total value stays the same the budget line will not shift.

WHAT HAPPENS WHEN YOU CHANGE PRICES?

Because the slope of the budget line depends on the prices of the two goods, a change in one or both price will change the slope of the line. Because the consumer has no income but can only buy and sell commodities, a change in the prices of the goods will affect the decision about both buying and selling the goods. If only the price of good 1 decreases, the ratio -p1/p2 will be closer to 0 and therefore the line will be flatter. Economically this means that the consumer will be able to buy more of good 1 (because it costs less) but less of good 2 (because the money the consumer gets by selling his endowment of good 1 is less and therefore can spend less on good 2). If only the price of good 1 increases, the ratio -p1/p2 will be higher (in absolute value) and therefore the line will be steeper. Economically this means that the consumer will be able to buy less of good 1 (because it costs more) but he will be able to buy more of good 2 (because the money the consumer gets by selling his endowment of good 2 is higher and therefore can spend more on the good 1). The opposite happens with good 2: if its price increase the line becomes flatter and if its price decreases the budget line becomes steeper. When the line changes in slope, it pivots around the endowment point. This is because, for each combination of prices, the endowment is yet possessed by the consumer and therefore it is always affordable no matter how the prices change.

Consider the endowment point as in the picture and suppose prices change so that the budget line becomes flatter. If the consumer after the price change remains a net seller of good 1, he will be surely worse off. To understand why, consider that all the points all the new budget line to the left of the endowment were available before the price change because they are under the old budget line. Because the consumer chose initially points higher than the ones available now, he is now settling for less (because the new available bundles are valued less). If instead the consumer is a net buyer of good 1 and that the budget line becomes flatter because of a price change once again. We can say for sure that, because the consumer was in the beginning a net buyer of good 1, he will continue to purchase that good. To understand why consider that, as said before, if the consumer will sell good 1 he will surely be worse off; furthermore notice that, as the curve becomes flatter, it becomes higher than the original one in all the points on the right of the endowment point. Therefore, the consumer can now afford bundles that are better than the original ones in that part of the graph and, because the consumer chose initially to buy, why wouldn’t him now that conditions are more favourable? The same reasoning holds when considering the opposite situation: when the budget line becomes steeper. If that happens, a buyer of good 2 will continue to buy and a seller of good 2 will be surely worse off.

HOW THE SLUTSKY METHOD CHANGES

When we considered a price change in the case in which consumers had an income to spend among two goods, we decomposed the change in the choice of the consumer in an income effect (due to the change in the purchasing power of the consumer) and the substitution effect (due to the fact that consumer may change their choice depending on which good is relatively cheaper). Now, because we are considering the case in which the consumer has no income but an endowment of the two goods, not only we need to consider the substitution and the income effect but also a third effect: the endowment income effect. It is due to the fact that, when the prices of the goods change the value of the income you possess can increase or decrease as well. Remember that the sign of the substitution effect is always opposite to the change in price (if the price of a good increases the consumer will buy less of it, while if the price decreases the demand for that good increases). The sign of the sum of the normal and the endowment income effects is ambiguous and it depends on whether the consumer is a net demander or supplier of the good. For example, suppose that we are considering normal goods and suppose that the consumer is a net buyer of the good even after its price has increased, because the good has become more expensive, the consumer will necessarily consumer less of it. If the consumer is a net supplier instead the situation is a bit more complicated because we can not predict which will be the outcome because it depends on which effect is stronger: the negative one (substitution) or the positive one (sum of the income effects). Let’s consider that graphically: consider point A as the choice of the consumer before the price change. After the price change, the budget line becomes flatter pivoting around the endowment (1). The final choice of the consumer after the price change is point C. To understand why we need to decompose the total effect into the various effects just described:

• Substitution effect: it is the movement from point A to point B. To find that point you have to construct a fictitious budget line that passes through the original choice of the consumer (2) but having as slope -p1/p2 using as prices the ones after the price change and then repeating the standard process to find the optimal choice;

• Ordinary income effect: it is the movement from point B to point D. To find that point you have to construct a fictitious budget line that passes through the endowment (3) having slope -p1/p2 using as prices the ones after the price change and then repeating the standard process to find the optimal choice;

• Endowment income effect: it is the movement from point D to point C. To find that point you have to construct a fictitious budget line that passes through the vertical intercept and having slope -p1/p2 using as prices the ones after the price change and then repeating the standard process to find the optimal choice. In this case we chose to represent that line from the vertical intercept because it is the point that keeps the total value of the bundle the same regardless of the price change (in the picture is represented a situation in which the price of good 1 increases making the budget line steeper; the price of good 2 has not changed so if you keep the value of endowment fixed you will be able to buy the same quantity of good 2).