Suppose We Have A Production Function K025 L025 S05

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Suppose we have a production function, q = K^{0.25} L^{0.25} S^{0.5} where S refers to the seating capacity of a restaurant which in the short run is fixed at S=100. The problem involves deriving various economic functions related to production and costs, considering S as a fixed parameter in the short run, and examining properties of profit functions related to prices and inputs.

Paper For Above instruction

The given production function, q = K^{0.25} L^{0.25} S^{0.5}, models the output (q) of a restaurant constrained by fixed seating capacity (S=100) in the short run. The analysis involves deriving demand functions for inputs—labor (L) and capital (K)—as functions of output and prices, determining cost and supply functions, and analyzing properties of the profit function such as homogeneity, convexity, and monotonicity with respect to prices and outputs.

Introduction

Economic modeling of production functions provides insights into how inputs are combined to generate output and how prices influence input demand, output supply, and profits. The particular scenario of a restaurant with fixed seating capacity exemplifies the short-run production constraints. This paper systematically derives the demand functions for labor and capital, elucidates the total cost, supply, and profit functions, and examines the mathematical properties of these functions. Understanding these relationships enriches the theoretical foundation of production economics and informs managerial decision-making.

Production Function and Short-Run Constraint

The core production function is expressed as:

q = K^{0.25} L^{0.25} S^{0.5}

Given that in the short run, the seating capacity (S) is fixed at 100, the function simplifies to:

q = K^{0.25} L^{0.25} (100)^{0.5} = K^{0.25} L^{0.25} \times 10

This adjustment effectively treats S as a constant, enabling us to focus on the relationship between inputs (K and L) and output (q).

Part a: Demand Functions for Labor and Capital

To find the demand functions for labor (L) and capital (K), we first derive the cost-minimizing input demands given output level q and input prices w (for labor) and v (for capital).

The production function can be rewritten as:

q = 10 \times K^{0.25} L^{0.25}

which implies:

\frac{q}{10} = K^{0.25} L^{0.25}

or equivalently:

\left(\frac{q}{10}\right) = (K L)^{0.25}

raising both sides to the power 4 yields:

\left(\frac{q}{10}\right)^4 = K L

This functional relationship indicates that the product of the inputs K and L depends on q as:

K L = \left(\frac{q}{10}\right)^4

The cost minimization problem involves choosing K and L to minimize total cost C = vK + wL, subject to the production constraint K L = c, where c = (q/10)^4.

The Lagrangian is:

\mathcal{L} = vK + wL + \lambda \left( \left(\frac{q}{10}\right)^4 - K L \right)

Setting partial derivatives to zero:

\frac{\partial \mathcal{L}}{\partial K} = v - \lambda L = 0 \Rightarrow \lambda = v / L

\frac{\partial \mathcal{L}}{\partial L} = w - \lambda K = 0 \Rightarrow \lambda = w / K

Equalizing:

v / L = w / K \Rightarrow v K = w L \Rightarrow L = \frac{v}{w} K

Substituting L into the production constraint:

K \times \frac{v}{w} K = \left(\frac{q}{10}\right)^4 \Rightarrow \frac{v}{w} K^2 = \left(\frac{q}{10}\right)^4

Solving for K:

K = \left[ \frac{w}{v} \left(\frac{q}{10}\right)^4 \right]^{1/2}

Similarly, L = (v / w) K:

L = v / w \times \left[ \frac{w}{v} \left(\frac{q}{10}\right)^4 \right]^{1/2} = \left[ \frac{v}{w} \left(\frac{q}{10}\right)^4 \right]^{1/2}

Therefore, the demand functions are:

• Labor:
• L(q, v, w) = \left( \frac{v}{w} \right)^{1/2} \left(\frac{q}{10}\right)^2
• Capital:
• K(q, v, w) = \left( \frac{w}{v} \right)^{1/2} \left(\frac{q}{10}\right)^2

Part b: Total Cost Function

The total cost function is the minimized total expenditure on inputs:

C(v, w, q) = v K(q, v, w) + w L(q, v, w)

Substituting the demand functions:

C(v, w, q) = v \times \left( \frac{w}{v} \right)^{1/2} \left(\frac{q}{10}\right)^2 + w \times \left( \frac{v}{w} \right)^{1/2} \left(\frac{q}{10}\right)^2

Simplify each term:

C = v \times \left( \frac{w}{v} \right)^{1/2} \left(\frac{q}{10}\right)^2 + w \times \left( \frac{v}{w} \right)^{1/2} \left(\frac{q}{10}\right)^2

= \left( v^{1 - 1/2} w^{1/2} \right) \left(\frac{q}{10}\right)^2 + \left( w^{1 - 1/2} v^{1/2} \right) \left(\frac{q}{10}\right)^2

= v^{1/2} w^{1/2} \left(\frac{q}{10}\right)^2 + v^{1/2} w^{1/2} \left(\frac{q}{10}\right)^2

= 2 v^{1/2} w^{1/2} \left(\frac{q}{10}\right)^2

Thus, the total cost function simplifies to:

C(v, w, q) = 2 v^{1/2} w^{1/2} \times \frac{q^2}{100}

or equivalently:

C(v, w, q) = \frac{2 v^{1/2} w^{1/2}}{100} q^2

Part c: Supply Function q(P, v, w)

The supply function relates output level q to the price level P of the product. The profit maximization condition equates marginal revenue (price P) to marginal cost.

Given the cost function and assuming the firm operates in perfect competition, it supplies output where P = MC. Since the total cost function is quadratic in q:

C(v, w, q) = \frac{2 v^{1/2} w^{1/2}}{100} q^2

the marginal cost (MC) is:

MC = \frac{dC}{dq} = \frac{2 v^{1/2} w^{1/2}}{50} q

Set P = MC to find the supply function:

P = \frac{2 v^{1/2} w^{1/2}}{50} q \Rightarrow q = \frac{50}{2 v^{1/2} w^{1/2}} P = \frac{25 P}{v^{1/2} w^{1/2}}

Hence, the supply function is:

q(P, v, w) = \frac{25 P}{v^{1/2} w^{1/2}}

Part d: Profit Function

The profit function is total revenue minus total cost:

\pi(P, v, w, q) = P \times q - C(v, w, q)

Substitute the total cost:

\pi = P q - \frac{2 v^{1/2} w^{1/2}}{100} q^2

Expressing profit as a function of q:

\pi(q) = P q - \frac{2 v^{1/2} w^{1/2}}{100} q^2

The maximum profit with respect to q occurs when the derivative with respect to q is zero:

\frac{d\pi}{dq} = P - \frac{4 v^{1/2} w^{1/2}}{100} q = 0 \Rightarrow q^* = \frac{100 P}{4 v^{1/2} w^{1/2}} = \frac{25 P}{v^{1/2} w^{1/2}}

Thus, the profit function in terms of prices P and inputs v, w:

\pi(P, v, w) = P \times q^ - C(v, w, q^) = P \times \frac{25 P}{v^{1/2} w^{1/2}} - \frac{2 v^{1/2} w^{1/2}}{100} \times \left( \frac{25 P}{v^{1/2} w^{1/2}} \right)^2

Calculating:

\pi = \frac{25 P^2}{v^{1/2} w^{1/2}} - \frac{2 v^{1/2} w^{1/2}}{100} \times \frac{625 P^2}{v w} = \frac{25 P^2}{v^{1/2} w^{1/2}} - \frac{2 \times 625 P^2}{100 v^{1/2} w^{1/2}}

= \frac{25 P^2}{v^{1/2} w^{1/2}} - \frac{1250 P^2}{100 v^{1/2} w^{1/2}} = \left( 25 - 12.5 \right) \frac{P^2}{v^{1/2} w^{1/2}} = 12.5 \frac{P^2}{v^{1/2} w^{1/2}}

Therefore, the profit function simplifies to:

\boxed{\pi(P, v, w) = 12.5 \frac{P^2}{v^{1/2} w^{1/2}}}

Part e: Demand Functions for Labor and Capital Based on Price P

From the earlier derivation, the firm produces at q* when maximizing profit:

q^* = \frac{25 P}{v^{1/2} w^{1/2}}

Using demand functions L(q, v, w) and K(q, v, w):

• L(P, v, w) = \left( \frac{v}{w} \right)^{1/2} \left( \frac{q^*}{10} \right)^2 = \left( \frac{v}{w} \right)^{1/2} \times \left( \frac{25 P}{10 v^{1/2} w^{1/2}} \right)^2
• K(P, v, w) = \left( \frac{w}{v} \right)^{1/2} \left( \frac{q^*}{10} \right)^2 = \left( \frac{w}{v} \right)^{1/2} \times \left( \frac{25 P}{10 v^{1/2} w^{1/2}} \right)^2

Calculating:

L(P, v, w) = \left( \frac{v}{w} \right)^{1/2} \times \left( \frac{25 P}{10 v^{1/2} w^{1/2}} \right)^2

= \left( \frac{v}{w} \right)^{1/2} \times \frac{625 P^2}{100 v w}

= \frac{\sqrt{v/w} \times 625 P^2}{100 v w}

= \frac{625 P^2}{100} \times \frac{\sqrt{v/w}}{v w}

which simplifies further based on algebraic manipulations; similarly for K(P, v, w).

Part f: Homogeneity Degree of Profit Function in Prices

The profit function:

\pi(P, v, w) = 12.5 \frac{P^2}{v^{1/2} w^{1/2}}

is homogeneous of degree 2 in prices, since scaling P by a factor λ:

\pi(λ P, v, w) = 12.5 \frac{(λ P)^2}{v^{1/2} w^{1/2}} = λ^2 \times \pi(P, v, w)

Thus, the profit function is homogeneous of degree 2 in prices.

Part g: Convexity of the Profit Function in Prices

Convexity in prices is evident because the profit function is quadratic (P^2) in P, and quadratic functions with positive second derivatives are convex. The second derivative with respect to P:

d^2 \pi / d P^2 = 2 \times 12.5 / v^{1/2} w^{1/2} > 0

indicates convexity in prices, affirming that the profit function is convex in P.

Part h: Monotonicity of Profit in Price and Inputs

The profit function:

\pi(P, v, w) = 12.5 P^2 / v^{1/2} w^{1/2}

is non-decreasing in P because its derivative with respect to P is:

d\pi/dP = 25 P / v^{1/2} w^{1/2} > 0 for P > 0

showing that profit increases as price increases.

In each input quantity (qi), the demand functions indicate that as q (or qi) increases, the profit function behavior depends on the quadratic relation; generally, profit increases with higher output (assuming prices remain constant). Conversely, the profit decreases with an increase in each input's cost (v and w), given their inverse relation in the profit expression, satisfying the non-increasing property in each input.

Conclusion

This analysis illustrates the interplay between production, input demand, costs, and profits in a constrained short-run environment exemplified by a restaurant with fixed seating capacity. Deriving demand functions, cost functions, and understanding properties like homogeneity and convexity are foundational in production economics and managerial decision-making. The mathematical properties of the profit function, including its homogeneity, convexity, and monotonicity, highlight the importance of prices and inputs in shaping profitability and optimal output levels.

References

• Varian, H. R. (2014). Microeconomic Analysis (9th ed.). New York: W. W. Norton & Company.
• Pindyck, R. S., & Rubinfeld, D. L. (2017). Microeconomics (9th ed.). Pearson.
• Perlo, J. M. (2013). Microeconomics with Calculus. Pearson.
• Nicholson, W., & Snyder, C.