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Add Allocation Agreement template initials

we're gonna do some general equilibrium analysis to consumer exchange economy where consumers have cobb-douglas utility functions so here's the set up consumer ones utility equals x times y consumer one has an initial endowment of three units of good-x and two units of good-y the subscript one just represents consumer one consumer two has the same utility function consumer two has a slightly different endowment consumer two has one unit of X and six units of good-y so we want to find the in equilibrium how many units a good accent could Y do each consumer does each consumer consume so the first thing I want to do since we're dealing with cobb-douglas utility functions and this will help simplify the the problem is that we can normalize the exponents on the cobb-douglas utility functions to sum to one so the way you do that in general say we got this utility function good ax is raised to the power a good y raised to the power of b we want to normalize exponents to sum the one you're going to just do the following calculation it's gonna be X raised to a divided by a plus B Y is going to be raised to B divided by a plus B why do we do that because the a divided by a plus B will equal the share of income spent on good x okay this is a nice property of cobb-douglas utility functions and then B divided by a plus B will represent the share of income spent on good y so doing that for consumer one we're gonna normalize exponents to sum the one so you know just following that rule on the last slide it's gonna be one divided by one plus one or just one half okay and then for the y term the exponent you're gonna have what is B B is one and it's going to be divided by a plus B where a is one so it's also 1/2 so again just following that rule so notice exponents um the 1 and again reminding us consumer ones initial endowment so what can we do with that information we know that consumer will spend half his or her income on good acts and half his or her income on good Y so in other words the price of good acts times the units of good-x represents the spending on good acts and because of this property the consumer will spend half their income on good acts so it's going to be 0.5 times M okay so the amount of spending on good acts will equal half the consumers income where m represents the consumers income so just dividing through by the price of good acts here we have basically the demand for good acts so let me just go to the next step right here what is consumer income the consumers income is going to be the amount of good acts that the consumer currently has times the price of good acts what it could be sold for okay so this is the the three is the initial endowment times the price of good acts the consumers income will also equal the number of units of Y that the consumer has times the price of good-y so for M we have this now in parentheses just the initial endowments multiplied by their respective prices added together we're going to normalize the price of good acts to equal one so where I see a price of good acts I'm just gonna plug in one so price of get ox is one price of good ox is one simplifying we have consumer ones demand for good acts 1.5 plus the price a good y let's do a similar thing now for consumer consumer ones demand for good y the consumer will spend half her income okay then the exponent on Y here's one half a consumer will spend half her come on good why so the expenditures on y will equal point five times M dividing through by the price of good-y and then once again making the substitution for what is the consumers income the initial Domino Vox multiplied by the price of X plus the initial endowment of Y times the price of Y and just simplifying here multiplying this point five through by parentheses we have consumer Ones demand for good why we're gonna do the exact same thing for consumer two since consumer two has the same utility function we will normal as a sync cobb-douglas utility function and in fact it is the same we'll just go through and normalize the exponents to sum the one so half the expenditures or half the the expenditures or half the consumers income will be on good acts the other half on good y basically following the same steps here here could the consumers income once again will be the consumers initial Adama two good acts times the price of X plus the initial endowment of Y times a price of Y so these initial diamond amounts are different than consumer one normalizing the price of good-x equal one and making that substitution we have consumer twos demand for good acts right here doing the same thing for the the good-y for consumer to making our substitutions for income and then simplifying we have consumer twos demand for good y consumer one and consumer twos demand for good x what we want to do is we're going to add up to get an aggregate demand or total demand so we're going to add consumer ones demand plus consumers twos demand together so just making the substitutions here for x subscript 1 and x subscript two simplifying we have the total demand for good acts equals two plus four times the price of good I the next step is we're just going to basically set the total demand equal to the total endowment of good ax the in total endowment of good ax was for consumer 1 let me go back at the beginning here sorry consumer 1 had 3 units of AX consumer 2 had one unit of AK so our total endowment of axes for okay so setting the total demand for good ax equal tour and total endowment for good ox and then solving for the price of good-y we get the price of good-y of 0.5 the next step is to take this price of good-y and plug it into all the consumers demand equation so for consumer 1 we plug in the price of good-y 0.5 consumer 1 will consume 2 units of X we plug point 5 into consumer twos demand for good ox consume 2 likewise will consume 2 units of good-x and then finally we take our individual demands for good y and we evaluate them at a price of good-y equal to 0.5 and we get the following results so we get our equilibrium condition here all right I hope you found this video helpful