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Demand with endowments.doc

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McMaster University
Hannah Holmes

Demand With endowments Fred Aswani th Econ 2GG3 (Ed. June 26 2011) (Most of this material is based on Varian Ch. 9. & E.E.A 11.4) • In the previous discussions we treated income and prices as exogenous-determined outside the model. We now consider an alternative set exogenous variables as prices and endowments. We will use this in the intertemporal model also use this in studying the labor/leisure choice. • Consider the case where the consumer is endowed with x* and y* units of the two goods. If x and y on other hand represent the consumers gross demands (amount the consumer ends up consuming) then it must be the case the value of purchases must be equal to the value of goods she is endowed with. i.e xP x yP = y*P + y*x y This can be rewritten as: * * y = (1/P y( x P +xy P - xy ) andxdy/dx = -P /P x y Here, x P + yxP acts yust like income (say, M*) More of either endowment (x*P or y*P x shifts yhe constraint outward and parallel. • The endowment bundle is always on the budget line. The slope however as before only depends on the price ratio. Note too that a 10% increase in the price of x cannot be distinguished from a 10% decrease in the price of y since they pivot through the same point (the endowment point). We can rewrite the income constraint as: x*P -xP = yP -y*P x x y y Or (x*-x)P x(y-y*)P y Here the equation can be thought of as Sales of x = Purchases of y • This therefore represents the BL expressed in terms of net demands. The difference between what the consumer ends up with (gross demand) and the initial endowment. If (x-x* )> 0 then she is a net buyer/demander otherwise if less than 0 the net seller or supplier i.e. (x*-x)>0. Consumers less x that her endowment. Income and substitution effects. • In analyzing these effects it is important to consider the case where a consumer is a net seller or a net buyer. Consider the case where the consumer is a net seller of x and its price increases. The substitution effect as before is to decrease consumption of x whose price has risen. • But since price of x has risen it expands his budget set (endowment effect)-because he is a net seller of the good whose relative price has risen. The income effect is therefore to increase its consumption. These effects therefore oppose each other even though x is a normal good. We can show that for a net seller if the price of what she sells goes up, she will not switch to being a net buyer. (To be shown in class). • How about in the case where the individual is a net buyer? The substitution effect is still negative. The budget set shrinks if you are a net buyer of the good whose price has risen and therefore if the good is normal the income effect is negative too. We can however show that if the consumer is a net buyer and the price of what she buys decreases she is better off remaining a net buyer. The endowment problem-Finding demands To solve for the optimal x and y one must solve the following two equations: 1. M*=x*P + y*x =xP + yy x y 2. MRS=P /P x y We have used M* as a shorthand. M* is exogenous and the solutions are: x=x (P ,x ,y*) y=y(P ,x My, Note however that now when P changesxit has both a direct effect on x and also an indirect effect through M* Example Find the demands associated with U=xy given the consumer is endowed with x* and y*. Letting M*
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