MHB What is the Optimality Condition for a Firm's Profit Maximization Problem?

  • Thread starter Thread starter Ginnee
  • Start date Start date
  • Tags Tags
    Condition
AI Thread Summary
The discussion centers on deriving the optimality condition for a firm's profit maximization problem using the production function Y=zK^αN^(1-α). The problem is framed as maximizing profit by adjusting labor input N while keeping capital K fixed and cost-free. The optimality condition is established by setting the marginal product of labor (MP_N) equal to the wage rate (w), leading to the equation zF_N = w. The participant seeks clarification on how to express the optimality condition as α = 1 - (wN/Y) and is confused about the notation, specifically the meanings of z, F, and F with a subscript N. Understanding these concepts is crucial for solving the problem effectively.
Ginnee
Messages
1
Reaction score
0
If I'm given a firm's production function of

$$Y=zK^{\alpha}{N}^{1-\alpha}$$

Then assuming $$K$$ is fixed and cost free, we can get our profit maximzation problem of

$$\max_{{N}}zF(K^{\alpha}{N}^{1-\alpha})-wN$$

To find the optimality condition, $${MP}_{N}=w$$ , I take the partial derivative and find

$$z{F}_{N}=z(1-\alpha){K}^{\alpha}{N}^{-\alpha}=w$$

Here is where I'm stuck.

I need to show that the optimality condition can be written as $$\alpha=1-\frac{wN}{Y}$$

Any help would be appreciated.

Thank you,

Gin
 
Mathematics news on Phys.org
Ginnee said:
If I'm given a firm's production function of

$$Y=zK^{\alpha}{N}^{1-\alpha}$$

Then assuming $$K$$ is fixed and cost free, we can get our profit maximzation problem of

$$\max_{{N}}zF(K^{\alpha}{N}^{1-\alpha})-wN$$

To find the optimality condition, $${MP}_{N}=w$$ , I take the partial derivative and find

$$z{F}_{N}=z(1-\alpha){K}^{\alpha}{N}^{-\alpha}=w$$

Here is where I'm stuck.

I need to show that the optimality condition can be written as $$\alpha=1-\frac{wN}{Y}$$

Any help would be appreciated.

Thank you,

Gin

Dont get the notation. What is z and F and also F with a subscript N
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top