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(See Steps) Consider a firm which produces a good, y, using two factors of production, x1 and x2. The firm’s production function y=f(x_1,x_2)=x_1^1/2x_2^1/4

Question: Consider a firm which produces a good, y, using two factors of production, x1 and x2. The firm’s production function

\[y=f\left( {{x}_{1}},{{x}_{2}} \right)=x_{1}^{1/2}x_{2}^{1/4}\] (4)

Note that (4) is a special case of the production function in Question 1, in which \(\alpha =1/2\) and \(\beta =1/4\). Consequently, any properties that the production function in Q1 has been shown to possess, must also be possessed by the production function defined in (4).

The firm faces exogenously given factor prices \({{w}_{1}},{{w}_{2}}\in {{\mathbb{R}}_{++}}\) , where w1 is the price of a unit of x1 and w2 is the price of a unit of x2. The firm’s total cost of production, TC, is

\[TC={{w}_{1}}{{x}_{1}}+{{w}_{2}}{{x}_{2}}\] (5)

where x1 denotes the number of units of factor 1 used in production and x2 denotes the number of units of factor 2 used in production, Therefore, the function TC is the objective function for this problem. For any given level of output y that the firm produced, it wishes to choose the levels of x1 and x2 which minimize the cost of producing that level of output. That is, the firm’s optimization problem is

\[\begin{aligned} & \underset{{{x}_{1}},{{x}_{2}}}{\mathop{\min }}\,\left( {{w}_{1}}{{x}_{1}}+{{w}_{2}}{{x}_{2}} \right) \\ & \text{s}\text{.t}\text{.} \\ & \left( {{x}_{1}},{{x}_{2}} \right)\in S*=\left\{ \left( {{x}_{1}},{{x}_{2}} \right)\in {{\mathbb{R}}_{++}}|\,\,x_{1}^{1/2}x_{2}^{1/4}\ge y \right\}\,\,\,\,\,(6) \\ \end{aligned}\]
  1. i) What assumptions about the properties of (5) and (6) guarantee that there is a
    global solution to the firms minimization problem’? Briefly explain.
    ii) Does (5) satisfy the required assumption(s)? Briefly explain,
    iii) Does (6) satisfy the required assumption(s)? Briefly explain. Hint: Draw a graph of S* for a given level of output, bearing in mind that \({{f}_{1}}>0,{{f}_{2}}>0\) .
    Use the graph to argue that (6) does or does not satisfy the required assumption(s).
    iv) What is the degree of homogeneity of the production function
    \(f\left( {{x}_{1}},{{x}_{2}} \right)=x_{1}^{1/2}x_{2}^{1/4}\)
    Briefly explain.
  2. It can be shown that the constraint is binding for this problem, so the constraint set effectively becomes \(S**=\left\{ \left( {{x}_{1}},{{x}_{2}} \right)\in {{\mathbb{R}}_{++}}|\,\,x_{1}^{1/2}x_{2}^{1/4}=y \right\}\)

Consequently, the firm’s minimization problem can be solved using the method of Lagrange, with

\(g\left( {{x}_{1}},{{x}_{2}} \right)=x_{1}^{1/2}x_{2}^{1/4}-y\)

  1. Is the constraint qualification satisfied for this problem? Briefly explain.
  2. What is the implication of your answer to (b) i)?
    (c) i) Write down the Lagrangean function for this problem, using the formulation
    \(L=TC+\lambda g\left( {{x}_{1}},{{x}_{2}} \right)\)
    where \(\lambda \) denotes the Lagrange multiplier.
    ii) Write down the first-order conditions for the firm’s minimization problem
    (d) Solve the first-order conditions for x1, x2 and \(\lambda \) as functions of w1, w2, and y.
    (e) i) Derive the bordered Hessian matrix for this problem.
    ii) Expand the bordered Hessian by the first row and use Theorem 4.12 to determine whether an interior minimum or interior maximum is achieved at the values of x1 and x2 which satisfy the FOC.
    Price: $2.99
    Solution: The downloadable solution consists of 6 pages
    Deliverable: Word Document


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