Department of Economics
Problem Set 4, 2025
ECON0010:
Introduction to Mathematics for Economics
Question 1: Matrix Algebra
Consider the following three matrices, sometimes called the Pauli Matrices,
and the vector-space of 2 ×2 matrices of the form.
. a] Show that the set of matrices like A is a real vector space, and that the Pauli matrices are elements of this set.
. b] Show that the three Pauli matrices form. a linearly independent triplet.
We define a new ‘Transpose’, the so-called Hermitian transpose Mt as
Mt = (MT)* ,
and we use this to define a “double-dot” product on the vector space of matrices
A : B = Tr[A . Bt] .
. c] Show that matrices of the form of A satisfy At = A, and for two matrices
calculate A : A, B : B and A : B.
. d] Find the characteristic equations for the eigenvalues of the Pauli matrices, determine their eigenvalues and their eigenvectors, and comment on your result.
Question 2: Constrained Optimization
Consider a community of agents j = 1, ..., n, and suppose each agent has a private opinion sj on some issue I, but also a public opinion σj. We assume that the preferences of the community are given by a utility function
and because we are mainly interested in the relative scale & orientation of the two opinion-distributions we fix their shared scale by a constraint , the positive sign of this ensures that, on average, agents prefer their private and public opinions to align.
. a] Write down a Lagrangian for this constrained optimization problem.
. b] Find the first-order conditions and comment on the nature of the equations and their possible solutions if we assume A is symmetric.
Suppose it is a system with 3 agents and the matrix A is
. c] Find the solution with the highest utility, sketch how the opinions of each of the agents changes with θ and comment on your results.
Now suppose that the matrix A is not symmetric, but
. d] Find the solution with the highest utility, sketch how the opinions of each of the agents changes with θ and comment on your results.
Question 3: Difference Equations
Consider a firm with a production function f[x, y] = 10 (x y)1/3 and facing wage costs C = 2 x + 5 y. Assume that the firm is ‘learning’ where its optimal usage of resources x andy is in a process described by gradient-learning with respect to the profit they make.
. a] Find the profit-function π[x, y] and the system of difference equations that determine
(→)[t] = {x[t], y[t]} if the learning-rate is γ.
. b] Find the stationary state
Suppose the firm has reached the stationary state some time ago, and suddenly there is a small
shock in the price of one of the inputs px → px + δpx . As a result the firm’s response will lead to a small change in their use of resources, i.e.
. c] Find a linear approximation to the equations for the firm’s response , find the stationary state for , and check for stability.
. d] Now an economist argues that a better model would be to assume that changing resource usage comes at an additional administrative and logistical cost equal to They argue that the firm would not be gradient-learning, but would follow an optimal path to the equilibrium. Find the Euler equations for this scenario, also their linearized form, and comment on the difference this makes compared to the gradient- learning case.
Question 4: Optimization
Consider the following function f[] of n variables = {x1, ..., xn} and with a symmetric n × n, symmetric, matrix M with strictly positive eigenvalues,
We are interested in whether this function has any local optima.
. a] Find the first-order conditions for the local optima.
. b] Suppose we rewrite = x , where x is the length of and a unit vector in the same
direction. Show that whenever is parallel to an eigenvector of M there is a solution for x as long as the corresponding eigenvalue is positive.
. c] Suppose
Find the eigenvalues of M, the normalised eigenvectors, the corresponding solutions to the FoCs and determine any constraints on the value of a for all solutions to be real-valued.
. d] Suppose a = 1/2, find the (local) maxima , calculate f[] and determine at a = 1/2.
Question 5: Differential equations
Consider an agent with an opinion σ[t] and preferences described by
. a] If we assume that this agent learns what the optimal opinion is by a process described by gradient-learning, with a learning-rate γ, find the differential equation satisfied by σ[t] and establish wether there is a stationary state σs?
. b] Solve this equation assuming that initially the agent is ‘neutral’, i.e. σ[0] = 0.
A researcher now proposes a different model. They suggest that the evolution of the agent’s opinion is described by the equation
η - R σ[t] + μ σ,, [t] = 0 with μ > 0 .
. c] Find a solution for this equation for which σ[0] = 0 and , and comment on the
difference between this solution and the one from the model used in a] and b].
. d] Show that the new model-equation can be found as the result of the agent choosing an optimal path, assuming preferences given by
under suitable initial- and final-conditions and comment on why this model is actually more general by finding an equation describing the case where the agent is discounting future utility at time t by e- r t, and discuss the effect of future-discounting on the solution.