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Excise equation contract

hello this video will provide the mathematics behind the Edgeworth box we're going to have an exchange economy with two consumers consumer 1 and consumer 2 we're going to have two goods in this exchange economy good acts and good wide consumer ones utility function is given by the following ten times the natural log of good-x plus a natural log of good-y and consumer twos utility function is likewise defined here consumer one has initial endowment of 55 units of good-x and zero units of good-y consumer two has an initial endowment of zero units of good-x but a hundred and ten units of good-y his subscripts just represent consumer 1 or consumer two so what we're going to try to find here is the equilibrium the wall raisi in equilibrium so we're going to start this way we're going to first start by constructing a little constraint the budget constraint the budget constraint for consumer 1 well let me just write the budget constraint in general format I is income we got the price of good ax times units of good ax for consumer 1 plus the price of good-y times units of good-y for consumer 1 so consumer 1 has 55 units of good-x so consumer 1 maximum income is those 55 units times the price of good-x what the consumer could sell those units for that would be the consumers maximum income and I'm just going to rewrite everything this way let me put a 1 on the Y here 1 subscript on the Y and one thing I'm going to do here I'm going to divide everything through by py by the price of good-y you'll see why in a minute it's an X so I divided this top expression right here through by the price of good-y now let's let the price of good-x divided by the price of good-y dis equal to P okay we're going to simplify that so if we do that then let me go into a clean screen if we do that we're going to get 55 P equals P times good X plus good why this is consumer ones budget constraint let's do the same thing for consumer to consumer two's budget constraint I'll go over here so this consumer - and this is consumer two's budget constraint or just constraint for short so from consumer twos perspective the consumers income I can be written like this this consumer has 110 units of good-y some from the previous slide we said the initial endowment for consumer to is 110 units so that's upper level limit on consumer two's income consumer has 100 to 110 units of good-y it could sell those units @py raising this amount of income and then we just rewrite and let's do what we did before let's divide through but this time let's divide through by the price of good acts and since we define the price of good acts divided by the price...