#1 Answer

a) The set of preferences provided in the question violate Transitivity.

Since, (13,21) is preferred to (16,8), and (13,21) and (12,17) are indifferent. Note that, and (12,7) and (16,8) are indifferent. This means that, (13,21) should be indifferent with (16,8), which it is not.

b) The set of preferences also violate non-satiation.

Non-satiation, in other words, also means 'more' is better. However, in the preferences above, we can clearly see (12,7) is indifferent to (13,21).

#2 Answer

The consumer demand is obtained through the maximization of utility subject to the budget constraint of the consumer. Supposing the utility function is U(x, y) and the budget constraint is w = p(x)*x + p(y)* y.

Therefore, for maximizing U(x,y) subject to the budget constraint, the langrange equation is given by,

L = U(x,y) + (w - (p(x)*x + p(y)* y))

dL/dx = d(U(x,y))/dx - p(x) = 0   (1)

dL/ dy = d(U(x,y)/dy - p(y) = 0   (2)

Dividing (1) by (2), we get,

(dU/dx)/(dU/dy) = p(x)/p(y)   (3)

The LHS of the above equation is the ratio between the marginal utility of x and y respectively and is called the Marginal Rate of Substitution(MRS). So, in order to have the same demand for two utility functions, the MRS obtained from them must be same.

a) Differentiating u1 wrt to x and y, we get,

du1/dx = 1 and du1/dy = 1

Therefore, MRS1 = 1/1 =1

Differentiating u2 wrt to x and y, we get,

du2/dx = beta (1+x+y)beta-1

du2/dy = beta(1+x+y)beta-1

MRS2 = 1

Here, MRS1 = MRS2 are equal for all values of beta. Therefore, the demand is same for both utility functions u1 and u2 for all possible values of beta.

b) Differentiating u1 and u2 wrt to x and y,

du1/dx = 0.5 y0.5 / x0.5

du1/dy = 0.5 x0.5 / y0.5

Therefore MRS1 = y/x

du2/dx = beta/x

du2/dy = (1- beta)/y

MRS2 = y/x * beta/ (1 - beta)

For the demand to be same, we set MRS1 = MRS2,

1 - beta = beta

beta = 0.5

Therefore, the the deamnds are equal for u1 and u2, when the value of beta is 0.5.