Mathematical tools for intermediate economics classes
Iftekher Hossain
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Calculus of Multivariable Functions

Section 6

Use of Partial Derivatives in Economics; Some Examples

Practise questions

1. Find the marginal productivity of different inputs for each of the following functions \(Q\):

  1. \(Q = 3x^{2} + 2xy + y^{2}\)
  2. \(Q = 0.5K^{2} - 2KL + L^{2}\)
  3. \(Q = 10K^{0.5}L^{0.5}\)
  4. \(Q = 10K^{0.6}L^{0.4}\)
  5. \(Q = AK^{\alpha }L^{\beta }\)

Hint: Evaluate the second-order derivatives at the given points and check the sign.

  1. \(MP_{x} = 6x + 2y, \qquad MP_{y} = 2x + 2y\)
  2. \(MP_{K} = K - 2L, \qquad MP_{L} = -2K + 2L\)
  3. \(MP_{K} = 5K^{-0.5}L^{0.5}, \qquad MP_{L} = 5K^{0.5}L^{-0.5}\)
  4. \(MP_{K} = 6K^{-0.4}L^{0.4}, \qquad MP_{L} = 4K^{0.6}L^{-0.6}\)
  5. \(MP_{K} = A \alpha K^{\alpha - 1}L^{\beta }, \qquad MP_{L} = A \beta K^{\alpha }L^{\beta - 1}\)

2. Find the marginal utility of each good for each of the following functions \(u(x,y)\):

  1. \(u(x,y) = 3x^{2} + 2xy + y^{2}\)
  2. \(u(x,y) = 5x + 2y\)
  3. \(u(x,y) = x^{0.5}y^{0.5}\)
  4. \(u(x,y) = 5x^{0.6}y^{0.4}\)
  5. \(u(x,y) = Ax^{\alpha }y^{\beta }\)

Hint: Evaluate the second-order derivatives at the given points and check the sign.

  1. \(MU_{x} = 6x + 2y, \qquad MU_{y} = 2x + 2y\)
  2. \(MU_{x} = 5, \qquad MU_{y} = 2\)
  3. \(MU_{x} = 0.5x^{-0.5}y^{0.5}, \qquad MU_{y} = 0.5x^{0.5}y^{-0.5}\)
  4. \(MU_{x} = 3x^{-0.4}y^{0.4}, \qquad MU_{y} = 2x^{0.6}y^{-0.6}\)
  5. \(MU_{x} = A \alpha x^{\alpha - 1}y^{\beta }, \qquad MU_{y} = A \beta x^{\alpha }y^{\beta - 1}\)

3. Find the marginal rate of substitution \(MRS_{x,y}\) for each of the following functions \(u(x,y)\):

  1. \(u(x,y) = 5x + 2y\)
  2. \(u(x,y) = x^{0.5}y^{0.5}\)
  3. \(u(x,y) = 5x^{0.6}y^{0.4}\)
  4. \(u(x,y) = Ax^{\alpha }y^{\beta }\)
  1. \(MRS_{x,y} = \frac{5}{2}\)
  2. \(MRS_{x,y} = \frac{y}{x}\)
  3. \(MRS_{x,y} = \frac{3y}{2x}\)
  4. \(MRS_{x,y} = \frac{\alpha y}{\beta x}\)


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