Using the Lagrange Method, maximize U = X1/2Y1/2 subject to X + Y = 4.

| November 24, 2016

1. Using the Lagrange Method, maximize U = X1/2Y1/2 subject to X + Y = 4.

(This is an advanced method for solving 2-variable constrained optimization problems withan equality constraint, and is especially useful for multiple and complex constraints.See Ch. 4 Appendix in Besanko & Braeutigam or Pindyck & Rubinfeld for more details.)

_2_a) Write the objective function, U(X,Y), and the constraint, L(X,Y).Write the Lagrangian function, Φ (or Λ) = U(X,Y) – λ × L(X,Y).

b) Find the first-order conditions (F.O.C.) for the Lagrangian: ∂Φ/∂X, ∂Φ/∂Y, ∂Φ/∂λ = 0.Find Y in terms of X.

c) What is MUX, MUY in terms of λ? What is MRSYX? What is the optimal condition andoptimal consumption?

d) State the dual of this consumer optimization problem. (No calculations required.)

2. Using the Lagrange Method, minimize C = wL + rK subject to Q = A Kα Lβ.

(This is a Cobb-Douglas production function, where A, α, β are constants; this is used toestimate demand functions, demand elasticities, and long-run costs.See Ch. 7 App. in Pindyck & Rubinfeld or Ch. 6 App. in Besanko & Braeutigam for details.

)a) Find the cost-minimizing levels of capital (K*), labor (L*).

b) If there are constant returns to scale (α + β = 1), what is the cost function?

c) What is the elasticity of substitution along a Cobb-Douglas production function?

d) Often it is impossible to estimate the production function (Q = A Kα Lβ ) fromengineering or process data. But economic data – costs (C), input prices (w, r) and output(Q) – is easily available. How can we estimate the production function – i.e. find α and β– from this economic data?[Hint: start with the cost function (lines 1-2 in (b) above), but do not assume constantreturns to scale (α + β = 1), as we trying to estimate α and β; simplify and take logs.]

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