Quantitative Methods In Economics Class Project Due Tuesday

Quantitative Methods In Economicsclass Projectdue Tuesday 530conside

Quantitative Methods In Economics class Project due: Tuesday 5/30 Consider a firm that has two different product lines. Let q1 and q2 represent the quantities produced and sold of goods 1 and 2, respectively. We have that q1, q2 ≥ 0. The cost of joint production is given by: c(q1, q2; c, θ) = c q1²/2 + c q2²/2 + θq1q2, where c > 0 is a cost parameter, and θ captures how joint production affects total costs. Assume that θ 0 be the prices of products 1 and 2, respectively. The firm’s profit function is given by: π(q1, q2; p1, p2, c, θ) = p1q1 + p2q2 – c q1²/2 – c q2²/2 – θq1q2

Paper For Above instruction

The analysis of a firm producing two products jointly under cost synergy provides an insightful framework into how production and market forces interact. This essay explores the firm's profit maximization condition, consumer demand behavior, and the resulting equilibrium prices, considering the influence of joint production costs.

Profit Maximization and Supply Functions

To determine the optimal output levels for products 1 and 2, we start by formulating the profit maximization problem. The profit function is given by π(q1, q2; p1, p2, c, θ) = p1q1 + p2q2 – c q1²/2 – c q2²/2 – θq1q2. Differentiating π with respect to q1 and q2, and setting the derivatives equal to zero yields the first-order conditions (FOCs):

  • ∂π/∂q1: p1 – c q1 – θ q2 = 0
  • ∂π/∂q2: p2 – c q2 – θ q1 = 0

Solving this system of equations provides the supply functions:

qS1(p1, p2, c, θ) = [p1 – θ q2] / c


qS2(p1, p2, c, θ) = [p2 – θ q1] / c

Expressed simultaneously, we obtain:

c q1 + θ q2 = p1


c q2 + θ q1 = p2

which can be solved to find explicit functions of prices and parameters. Solving these yields:

qS1(p1, p2, c, θ) = (c p1 – θ p2) / (c^2 – θ^2)


qS2(p1, p2, c, θ) = (c p2 – θ p1) / (c^2 – θ^2)

These functions illustrate how quantities depend on the prices, cost parameters, and the interaction term θ. Notice that for feasible (positive) outputs, the denominators must be positive and parameters well-behaved.

Utility Maximization and Demand Functions

Consumers’ net utilities for goods 1 and 2 are given by U1(q1; p1) = 1 – (1 – q – p1q1) and U2(q2; p2) = 1 – (1 – q – p2q2). The consumer maximization problem involves choosing q1 and q2 to maximize utility considering the respective prices, with the constraints q1, q2 ≥ 0. Setting up the utility maximization problem and taking derivatives yields:

  • ∂U1/∂q1: p1 – 1 = 0
  • ∂U2/∂q2: p2 – 1 = 0

on the interior boundary points, indicating that the demand functions for each good are essentially:

qD1(p1) = max{0, 1 – p1}


qD2(p2) = max{0, 1 – p2}

The demand functions are non-increasing, with consumers only purchasing positive quantities if prices are below 1, reflecting the utility structure. The boundary points (i.e., zero consumption) are included when prices exceed 1. These demand functions illustrate that, under this utility setup, the consumer’s willingness to pay diminishes linearly with price.

Equilibrium Conditions and Price Analysis

The general equilibrium occurs where supply equals demand for both products:

  • qS1(p1, p2, c, θ) = qD1(p1)
  • qS2(p1, p2, c, θ) = qD2(p2)

1. Equilibrium prices exceeding or equal to 1 would imply zero demand, leading to trivial or degenerate solutions. Given the demand functions, the possibility of prices ≥ 1 equating to positive quantities is null, justified by the consumption utility model.

2. In equilibrium, due to the symmetry and the structure of the demand and supply functions, prices must be equal, pE, reflecting the market-clearing condition that both goods are priced identically at equilibrium.

3. The equilibrium price pE(c, θ) lies strictly between zero and one, ensuring positive quantities demanded and supplied. To demonstrate that, substitute the supply functions into the demand functions and resolve for p.

4. Comparing the case with no cost interaction (θ=0), the equilibrium prices are higher in this benchmark because the joint cost savings (θ

5. Deriving ∂ pE/∂ θ > 0 involves differentiating the equilibrium price expression with respect to θ, revealing that as positive interaction effects increase, so does the equilibrium price, consistent with increased marginal costs or less cost-saving synergy.

In conclusion, the analysis illustrates the nuanced effects of joint production costs on firm behavior and market prices. The equilibrium dynamics depend critically on the parameters c and θ, affecting production decisions and market outcomes.

References

  • Varian, H. R. (2014). Intermediate Microeconomics: A Modern Approach. W.W. Norton & Company.
  • Pindyck, R. S., & Rubinfeld, D. L. (2018). Microeconomics (9th ed.). Pearson Education.
  • Perloff, J. M. (2016). Microeconomics: Theory & Applications with Calculus. Pearson.
  • Frank, R. H., & Bernanke, B. S. (2015). Principles of Economics. McGraw-Hill Education.
  • Tirole, J. (1988). The Theory of Industrial Organization. MIT Press.
  • Mankiw, N. G. (2020). Principles of Microeconomics. Cengage Learning.
  • Mas-Colell, A., Whinston, M. D., & Green, J. R. (1995). Microeconomic Theory. Oxford University Press.
  • Friedman, M. (1953). Essays in Positive Economics. University of Chicago Press.
  • Nash, J. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36(1), 48-49.
  • Samuelson, P. A. (1947). Foundations of Economic Analysis. Harvard University Press.

Related Assignments