Econ 507 Problem Set 1 In Mathematical Economics

Econ 507 Ps 1econ 507 Mathematical Economicsproblem Set 1partial De

Differentiate the following functions with respect to x and y (i.e., calculate ∂f/∂x and ∂f/∂y):

1. f(x, y) = 6x + 8y – xy
2. f(x, y) = x² y³
3. f(x, y) = (xy)² + (2x³ – 7y)(ln y – eˣ)
4. f(x, y) = xy + 5 ln(x)
5. f(x, y) = √(e²ˣʸ) – ln(xy)
6. f(x, y) = ln(e²ˣ + 5y² + 10x)
7. f(x, y) = (x^{1/2} + y) (Note: Clarify if intended as product or sum; assuming product)
8. f(x, y) = ln(x² y³)
9. f(x, y) = yˣ

Paper For Above instruction

Partial derivatives play a crucial role in understanding the behavior of functions in multivariable calculus, which is foundational in mathematical economics. This paper aims to differentiate given functions with respect to variables x and y, interpret their second order derivatives, and explore the significance of these derivatives in economic modeling.

Differentiation of Functions with Respect to x and y

The first function, f(x, y) = 6x + 8y – xy, involves straightforward partial derivatives. Differentiating with respect to x, treating y as constant, yields ∂f/∂x = 6 – y. Differentiation with respect to y, treating x as constant, results in ∂f/∂y = 8 – x. These derivatives highlight how changes in each variable affect the function, illustrating marginal effects in economic contexts such as cost or utility functions.

Second function, f(x, y) = x² y³, exemplifies product rule application in partial derivatives. With respect to x, ∂f/∂x = 2x y³, since y is treated as a constant. With respect to y, ∂f/∂y = 3x² y². These derivatives are useful in optimizing production functions or analyzing responding variables in economic models.

The third function, f(x, y) = (xy)² + (2x³ – 7y)(ln y – eˣ), combines power and product rules. Differentiating with respect to x:

∂f/∂x = 2xy² + (6x²)(ln y – eˣ) + (2x³ – 7y)(–eˣ).

Similarly, for y:

∂f/∂y = 2x² y + (2x³ – 7y)(1/y) + (–7)(ln y – eˣ).

These derivatives reveal the complex interplay between variables in nonlinear models, pertinent in economic analysis involving non-linear utility or cost functions.

The fourth function, f(x, y) = xy + 5 ln(x), yields:

∂f/∂x = y + 5/x,

∂f/∂y = x.

These derivatives are essential in understanding marginal contributions to overall functions such as revenue or utility.

For the fifth function, f(x, y) = √(e²ˣʸ) – ln(xy), derivatives involve chain and product rules:

∂f/∂x = (1/2) * (e²ʸ) / √(e²ˣʸ) – (1/x),

∂f/∂y = (1/2) * (e²ˣ) / √(e²ˣʸ).

Understanding these derivatives supports the analysis of functions with exponential and logarithmic components, common in economic growth models.

Sixth function, f(x, y) = ln(e²ˣ + 5y² + 10x), differentiates to:

∂f/∂x = (2e²ˣ + 10) / (e²ˣ + 5y² + 10x),

∂f/∂y = (10y) / (e²ˣ + 5y² + 10x).

These derivatives help analyze the marginal effects of variables in macroeconomic models.

The seventh function, assuming it is the product of (x^{1/2} + y), involves:

∂f/∂x = (1/2)x^{-1/2} * y,

∂f/∂y = x^{1/2}.

Similarly, for the eighth function, ln(x² y³), derivatives are:

∂f/∂x = (2x) / (x² y³),

∂f/∂y = (3 y²) / (x² y³) = 3 / y.

The ninth function, yˣ, is differentiated as:

∂f/∂x = yˣ ln y,

∂f/∂y = x y^{x-1}.

These derivatives are critical in modeling consumer preferences and substitution effects.

Second Order Partial Derivatives and Their Significance

Second derivatives provide information on the curvature and stability of functions. For example, considering f(x, y) = (x + y)²:

• f_{xx} = ∂²f/∂x² = 2
• f_{yy} = ∂²f/∂y² = 2
• f_{xy} = f_{yx} = 2

For f(x, y) = x^{1/2} y, derivatives are:

• f_{xx} = –(1/4) x^{-3/2} y
• f_{yy} = 0
• f_{xy} = f_{yx} = x^{-1/2}

These derivatives assist in analyzing the convexity, concavity, and saddle points in economic models, critical for optimization.

Conclusion

Understanding partial derivatives and second order derivatives is fundamental in mathematical economics. They enable economists to analyze marginal effects, stability, and curvature of functions representing economic phenomena such as utility, costs, or production. Mastery over differentiation techniques enhances the accuracy of economic modeling, forecasting, and decision-making processes, thus underscoring their importance in advanced economic analysis.

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