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1. Airbus and Boeing Competition

  • Firms: Airbus and Boeing competing in the aeroplane market.

  • Variables:

    • qA: aeroplanes produced by Airbus

    • qB: aeroplanes produced by Boeing

    • pA: price charged by Airbus

    • pB: price charged by Boeing

  • Demand functions:

    • For Airbus: qA = 10 - pA + βpB (with β ≥ 0)

    • For Boeing: qB = 10 - pB + αpA (with α ≥ 0)

  • Cost structure: Each firm incurs a cost of cqi, where c > 0.

(a) Profit Functions

  • Airbus Profit:

    • πA = pA * qA - c * qA = pA * (10 - pA + βpB) - c(10 - pA + βpB)

  • Boeing Profit:

    • πB = pB * qB - c * qB = pB * (10 - pB + αpA) - c(10 - pB + αpA)

(b) First-Order Conditions for Profit Maximization

  • Airbus: To maximize profit, set the derivative of πA with respect to pA to zero:

    • dπA/dpA = 0

  • Boeing: Set the derivative of πB with respect to pB to zero:

    • dπB/dpB = 0

(c) Concavity of Profit Functions

  • Check profit concavity:

    • Investigate the second derivatives of πA and πB with respect to their own prices.

  • Explore concavity conditions: are the second derivatives negative, or do they exhibit other behaviors?

  • Determine the nature of solutions from first-order conditions (maximum or minimum).

(d) Matrix Representation of Conditions

  • First-Order Conditions in matrix form:

    • AP = B where P is vector of prices (pA, pB)T and A, B are defined parameter matrices.

(e) Solutions for Prices

  • Given condition (αβ < 4), solve the first-order conditions system to find optimal prices (p∗A, p∗B).

(f) Comparative Statics

  • Effect of α and β change:

    • Discuss how variations in α and β influence the optimal prices (p∗A, p∗B).

    • Interpret findings from economic perspective.

2. Government Unemployment and Inflation

  • Government objectives: Minimize loss function L(u, π, α) = (u + απ)^2

  • Constraints: Phillips curve given by π = g(u, β).

(a) Lagrangian Construction

  • Write Lagrangian based on the optimization problem considering the constraint.

(b) First-Order Conditions

  • Compute conditions from Lagrangian optimization.

(c) Second Order Conditions

  • Analyze conditions to check if they indicate a minimum.

(d) Comparative Statics on Inflation Rate

  • Investigate how the increase in β affects the optimal inflation rate in terms of comparative statics.

(e) Envelope Theorem Application

  • Use the envelope theorem to compute the comparative statics effect on the loss function, ∂L/∂β.

3. Household Consumption and Savings

  • Decision factors: Household must decide on consumption and savings across two periods with income w1 and w2 and utility v(c1, c2).

  • Savings: Can save s1 at rate r.

(a) Budget Constraints

  • Write out the separate budget constraints and combine them into an intertemporal constraint.

(b) Binding Constraints Proof

  • Prove that budget constraints should bind at maximum to ensure no contradictions exist in the consumption decisions.

(c) Lagrangian for Optimization

  • Develop the Lagrangian for the constrained maximization problem.

(d) First-Order Conditions

  • List the first-order conditions for the maximization problem based on Lagrangian methods.

(e) Sufficient Conditions for Maximum

  • Indicate conditions on the utility function v ensuring maximum at first-order conditions solution.

(f) Implicit Function Theorem

  • Analyze how c2 changes as a function of c1 through the implicit function theorem.

4. Econometric Analysis of Saving and Income

  • Hypothesis: There’s a linear relationship between saving (yi) and income (xi).

  • Modeling equation: yi = α + βxi + εi.

  • Objective: Minimize sum of squared errors: Σ(εi)^2.

(a) Relation Between Minimization and Maximization

  • Show the relationship of minimizing the sum of squared errors to maximizing the negative value of the same.

(b) First-Order Conditions Derivation

  • Write first-order conditions from the minimization setup of parameters α and β.

(c) Optimal Parameters

  • Solve for the optimal values of α and β based on the established conditions.

(d) Nature of Solutions from Conditions

  • Explore if solutions from the first-order conditions indicate a minimum or maximum scenario.

5. Consumer Utility Maximization

  • Utility function: u(x) subject to budget constraint px ≤ w.

(a) Lagrangian for Constrained Maximization

  • Formulate the Lagrangian for the optimization problem.

(b) First-Order Conditions

  • Derive the first-order conditions necessary for optimization from the Lagrangian.

(c) Conditions for Maximum With Concavity

  • Show that if the utility function u is concave, it assures a maximum at the solution.

(d) Taylor Expansion Application

  • Utilize Taylor expansion to illustrate utility properties under concavity.

(e) Proving Existence of λ

  • Conclude that there exists λ > 0 such that u(y) ≤ u(x) + λp(y - x).