Utility and Demand

Extract

[...] results that were derived from the previous question. Hence: From the previous questions, we know that demand for x1, x2 and y1, y2 write: x1=x2=x/2 and and . Market clearing conditions imply that demand = supply = initial endowments. We know that (from y = y1+y2) it is therefore possible to write p as a function of endowments x and as well as parameter such that: Since we know the expression for price we can compute the equilibrium allocations We know that x1=x2=x/2 and therefore x1=x2=1/2. [...]

[...] Utility and Demand Exercise 3 2 y1 has a positive or negative effect on U2 depending on coefficient a's sign. If an increase in y1 hurts the consumer as it lowers its utility. The opposite is true for a>0. A Pareto optimisation problem seeks to maximise overall utility, so we could assume that a Pareto optimum would write as but this assumes that both agents are given weights. A more general formula would allocate specific weights ? such that: where Given that x and y are in fixed supply, the Pareto optimum can be computed on the basis of x1, y1 alone, hence: There is a Pareto-optimal allocation when both indifference curve tangents are equated, hence: hence which can be re-arranged such that: When backed into the MRS for each agent, we get the following results: y1=1, and x2=x-?. [...]

[...] Assuming some price q for good a competitive equilibrium assuming that both utilities are maximised. It also implies that each consumer sets their marginal utility equal to the cost associated with each good, hence and meaning that the marginal utilities of x1, x2 are equal to their unit price and those of goods y1, y2 are equal to q. Hence: y1=p, x2=q. The social planner seeks to maximise aggregate weighted utilities subject to resource constraints: and the FOC yield the following: and while and Since we know that there are prices equated to quantities, we ca substitute those into the optimal, efficient allocation, and those write: prices p and q can achieve efficiency as a result. [...]

[...] Recall that the MRS is the ratio of marginal utilities, and writes: hence: MRS1=y1/x1 and MRS2=(ay1+y2)/x2 MRS refers to the number of x units the consumer needs to give up if she wants to increase its consumption of y by one unit. We know that the optimality condition equates MRS and the price ratio, so: and on the basis of the initial allocations - which makes up their income, demand for agent 1 would write: and for y1, which yield y1=px/2. We know that demand for x1 for agent 1 is so x2 has to make up the second half, hence x2=x/2. We use the optimality condition in order to compute the demad for y2. [...]