Economics

Question 1 (5 points)
Consider the function y = a1 + (a2/x), where a1 and a2 are constants.
a. Find the first and second derivatives of the function. (2 points)
b. If a1 > 0, draw rough graphs to show the relationship between y and x when (i) a2 > 0, and (ii) a2 < 0. (3 points)
Question 2 (6 points)
A firm’s profit is given by: π= p.y – wL, where π is profit, L is the quantity of labour, p is the product price, w is
the wage per unit of labour, and y is output. Price (p), and wages (w) are given. The firm’s cost is C= wL.
a. If y=0.5L0.5, determine the profit maximizing demand for labour (L). Then find the profit-maximizing level of output (y). (2 points)
b. Obtain the following derivatives: (∂L/∂w), (∂L/∂p), (∂y/∂w), and (∂y/∂p). Then determine the sign of each
derivative, and briefly explain what this says about the impact of w and p on L. (4 points) Question 3 (5 points) A decision maker must choose the values of variables x and y so as to minimize (ax + by) subject to the constraint that xy=c. Here a, b, and c are positive constants. a. Set up the Lagrangian function. (2 points) b. Use the FOCs to solve for the optimal values x and y*. (3 points)
Question 4 (14 points)
A consumer has the following utility function u= x1
0.5 + x2.
a. Does marginal utility diminish for both goods? Are marginal utilities independent? Answer this by finding the
appropriate partial derivatives. (4 points)
b. Find the expression for MRS for this utility function. (2 points)
c. Only one of the following bundles is the consumer’s optimum given that M=24, p1= 1, and p2=4. Explain why. This
means you must show why it represents the optimum and the other bundles do not. (4 points)