Exercise 1: Find the first and the second-order derivatives of the following functions:

1. v(x) = 2/x + 4x

2. k(x) = 10 (x³ )² + 5 ⁴ x

3. g(x) = ln(x⁵ )

4. f(x) = 7^log7(x)

5. u(x) = x⁴ + x³ 1/x²

Exercise 2: Consider the following system of two non-linear equations with two unknowns:

y ln(x) = 0 (I)

y 5x² + 5x = c (II)

where y and x are the unknowns and c R is some exogenous variable.

1. Assume that c = 0 and answer the following questions:

(a) How many solutions does the system have? You do not need to calculate the solution(s), but please explain your answer. (Hint: Try to sketch the equations in a two dimensional coordinate system.)

(b) Verify that (x* , y^* ) = (1, 0) is one solution.

2. Assume now that c decreases by the small amount c < 0 from 0 to c. ¹ Use first-order approximation to linearise the system and find the new solution in the neighborhood of (1, 0). That is,

(a) Write each equation as a function, y(x), with x being the argument. Don't forget to replace c with c in equation (II).

(b) Approximate each function around the point (x, y) = (1, y(1)) by using the first-order approximation method from the lecture.

(c) Use the linear functions you derived in part (b) to set up a system of two linear equations in two unknowns. Solve this system to obtain an approximate solution, say (x, y). This solution will depend on c.

(d) Let (x, y) be the exact solution to the system given that c = c. ² Without calculating it, is x larger or lower than x? Please explain your answer. (Hint: A graphical answer (or explanation) is fine.) 1 If you struggle, try this approach: Assume first a specific value for c, for example 0.1. For this value, try to answer questions (a),(b),(c) and (d). Then just replace this specific value with c and repeat your steps. 2 If there is more than one solution, let (x, y) be the solution that has the largest value of x.

Exercise 3Consider a monopolist firm that faces an inverse demand P = Q/2 + 5, and has a total cost function C(Q) = 2/3Q³ 2Q² + 6Q. The domain of the cost function, profit function, and revenue function is [0, 10].

1. Find the monopolist's profit function, (Q).

2. Find and sketch the marginal revenue function MR(Q), the marginal cost function MC(Q), and the marginal profit function (Q) in one graph. If there are intersections with the vertical axis and with the horizontal axis, please highlight them clearly.

3. In your graph, highlight the part of the domain where the profit function (Q) is convex and that where (Q) is concave. Moreover, highlight the part of the domain where the cost function C(Q) is convex and that where C(Q) is concave. Also highlight the part of the domain where the revenue function R(Q) is convex and that where R(Q) is concave.

An unexpected pandemic changes the market demand for the monopolist's product and also forced the monopolist to change its production technology. Due to these changes, the profit function of the firm becomes (Q) = 2Q² + 8Q, with the domain of the profit function still being [0, 10].

4. Using the first-order condition to find the critical points, i.e., all candidates for local extrema of the new profit function.

5. Are the critical points you found above local maximizers? Explain your answer. As the status of the pandemic improves, the market condition faced by the monopolist improves, and the profit function of the firm changes to ( Q), which is a concave function on the domain [0, 10]. Suppose that the function is increasing at 10, i.e., (10) > 0.

6. How many critical points does ( Q) have? Explain your answer.

7. Find the profit-maximizing quantity Q for the profit function ( Q).

Exercise 4: Consider the two functions below h(x, y) = x y y² + x² , g(x, y) = (x + 2y)².

1. Compute the gradients of h(x, y) and g(x, y).

2. Compute the gradient of dg(x,y)/dx .

3. Compute the Hessian matrices of h(x, y) and g(x, y).

Exercise 5 A firm's production function is Q(K, L) = [bK^+ (1 b)L^ ]^/ , with 1.

1. Show that the function Q(K, L) is homogeneous of degree .

2. Compute the marginal rate of technical substitution, and show that it is independent of .

3. Show that the marginal products of Q(K, L) satisfies the relation K Q/K + L Q/L = Q(K, L). Now suppose that = 1, b = 1/2, = 1, and = 2. Consider point (K₀, L₀) = (0, 2).

4. Estimate the overall effect on Q when K increases by 0.15 from K₀ and L increases by 0.1 from L0.

5. Now the firm would like to decrease L from L₀ by 0.2, to keep the total output unchanged, the firm wants to know how much K should be increased from K₀. Can we use the implicit function theorem to estimate the change of K in this case? Explain your answer.