An economy consists of two types of people, males (indexed by M) and females (in

Question

An economy consists of two types of people, males (indexed by M) and females (indexed by F). There are four males, all identical to each other, while there are eight females who are identical to each other. There are two goods, x and y. The price of y is normalized to $1; the price of x is p. The table below shows the type, utility, endowment, and demands for and y for each type of person. At a walrus allocation, each male will consume the same bundle, while each female will consume a different bundle. Calculate the walrus equilibrium price p and walrus consumption bundle (x^M, y^M) for a male, and the bundle (x^F, y^F) for a female. Check that the walrus allocation is Pareto efficient. Pick the market for good y and set the total demand for y given by equal to the total supply of good y of unit(s). Solving, we obtain p = $. Therefore, the walrus prices are (p_x, p_y) = Then at the walrus allocation, a male consumer consumes the bundle while a female consumer consumes From the given utility functions, the marginal rates of substitution for a male consumer is MRS^M =, while that for a female consumer is MRS^F = At the walrus allocation, MRS^m =, and MRS^F =. Therefore, the walrus allocation Pareto efficient.

Explanation / Answer

Male = 4 and Female = 8
At Walrus Allocation , each male consume same bundle i.e. Xmym
Total Demand = (mM/(2p) + mM/2)yM + [(mF/(4p) + 3mF/4)/8] yF = xp + y = 4p + 12 = 16 units
Beta= $1
Bx,By = (4,12)
Male consumes = Xmym
Female consumes = xF(yF)3
MRSM = d(xMyM)/dm = xM yM-1.m + yM . m . xm-1
Similary, MRSF = d(xF (yF)3/dF
MRSM = MRSF at Walras allocation. So, it will be pareto efficient where   MRSM = MRSF

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