Instructor Ram Sewak Dubey Econ 317 Problem Set 5 November 1

Instructor Ram Sewak Dubey Econ 317 Problem Set 5 November 1 20181

Find the marginal and average functions for each of the following total functions and evaluate them at Q = 3 and Q = 5:

  • (a) Total Cost Function (TC): TC = 3Q² + 7Q + 12
  • (b) Total Profit Function (π): π = Q² - 13Q + 78
  • (c) Total Revenue Function (TR): TR = 12Q - Q²
  • (d) Total Cost Function (TC): TC = 35 + 5Q - 2Q² + 2Q³

Find the marginal expenditure (ME) functions for each of the following supply functions and evaluate them at Q = 4 and Q = 10:

  • (a) Supply Function: P = Q² + 2Q + 1
  • (b) Supply Function: P = Q² + 0.5Q + 3

Find the marginal revenue (MR) functions for each of the following demand functions and evaluate them at Q = 4 and Q = 10:

  • (a) Demand Function: Q = 36 - 2P
  • (b) Demand Function: 44 - 4P - Q = 0

Find the marginal propensity to consume (MPC = dC/dY) for each of the following consumption functions:

  • (a) C = C₀ + bY
  • (b) C = 1200 + 0.8Yd, where Yd = Y - T, and T = 100
  • (c) C = 2000 + 0.9Yd, where Yd = Y - T, and T = 300 + 0.2Y

Find the marginal cost (MC) functions for each of the following average cost functions:

  • (a) AC = 1.5Q + 4 + 46 / Q
  • (b) AC = 160 / Q + 5 - 3Q + 2Q²
  • (c) AC = Q² - 4Q + 174

Find the average product (APL) and marginal product (MPL) functions for the following total product functions:

  • (a) TP = 90L² - L³
  • (b) TP = aL + bL² - cL³, with a > 0, b > 0, c > 0

Analyze the linear demand curve and related revenue functions:

  • (a) Plot the inverse demand curve P = 60 - 3Q
  • (b) Find the total revenue and marginal revenue functions mathematically
  • (c) Compare the slope of the inverse demand curve and the marginal revenue curve
  • Bonus Provide a mathematical proof that, given a linear inverse demand curve, the marginal revenue curve must have the same vertical intercept but be twice as steep.

Paper For Above instruction

This problem set encompasses fundamental concepts in microeconomics related to marginal and average functions arising from total, revenue, cost, and production functions, as well as the analysis of demand and supply curves. It aims to develop skills in calculus applications within economic contexts, including differentiation and evaluation at specific quantities, as well as graphical interpretations and proofs.

Part 1: Marginal and Average Functions

The initial task involves deriving marginal and average functions from total functions such as total cost, profit, and revenue. Marginal functions are the derivatives of total functions concerning quantity (Q), representing the rate of change, while average functions are obtained by dividing the total functions by Q, indicating per-unit measures. For example, from TC = 3Q² + 7Q + 12, the marginal cost (MC) is the derivative d(TC)/dQ = 6Q + 7, and the average cost (AC) is TC/Q = 3Q + 7 + 12/Q. Evaluations at Q=3 and Q=5 involve substituting these values into the derived functions, illustrating how marginal and average costs/profits/revenues change with quantity.

Part 2: Marginal Expenditure Function

Given the supply functions, the total expenditure (TE) can be formulated as TE = P Q. Since P is a function of Q, TE becomes a function of Q. Differentiating TE with respect to Q yields the marginal expenditure (ME). For example, with P = Q² + 2Q + 1, TE = (Q² + 2Q + 1) Q = Q³ + 2Q² + Q. The derivative d(TE)/dQ provides ME, which can then be evaluated at specified quantities.

Part 3: Marginal Revenue Function

The demand functions relate Q to P, from which the inverse demand curve P(Q) can be derived. Total revenue TR = P Q, expressed as a function of Q by substituting P(Q). Differentiating TR yields the marginal revenue MR. For instance, for Q = 36 - 2P, rearranged to P = (36 - Q)/2, TR = P Q. Differentiating TR with respect to Q helps to understand how revenue changes as Q varies.

Part 4: Marginal Propensity to Consume

The MPC measures the change in consumption with respect to income, derived as the derivative dC/dY from the consumption functions. For functions involving disposable income Yd, which depends on Y and taxes T, the differentiation reflects the marginal response of consumption to income changes.

Part 5: Marginal Cost from Average Cost

Marginal cost (MC) is related to average cost (AC) via the relationship MC = d( total cost )/dQ, or directly from AC functions as MC = AC + Q * d(AC)/dQ. Differentiating the average cost functions yields their slopes, which are essential for understanding cost behavior at different output levels.

Part 6: Marginal Product and Average Product

The total product functions describe output (TP) as a function of labor input (L). The average product (APL) is TP / L, and the marginal product (MPL) is the derivative d(TP)/dL. These functions illustrate concepts of productivity, showing how additional units of labor influence total output and average productivity.

Part 7: Revenue Analysis for Linear Demand

Plotting the inverse demand curve P = 60 - 3Q provides graphical insight. The total revenue function TR = P * Q = (60 - 3Q)Q = 60Q - 3Q² is derived, and the marginal revenue is its derivative: MR = d(TR)/dQ = 60 - 6Q. Comparing the slopes reveals that the MR curve is twice as steep as the inverse demand curve, a fundamental relationship in microeconomics. The proof involves algebraic manipulation showing that MR is derived from P(Q) and differs in slope by a factor of two, with the same intercept.

This comprehensive analysis integrates calculus and economic theory, fostering a deep understanding of the relationships between costs, revenues, outputs, and demand. Mastery of these concepts is essential for analyzing firm behavior, market dynamics, and policy impacts logically and mathematically.

References

  • Intermediate Microeconomics: A Modern Approach. W.W. Norton & Company.
  • Microeconomics. Pearson Education.
  • Principles of Economics. Cengage Learning.
  • Microeconomic Theory: Basic Principles and Extensions. Cengage Learning.
  • Microeconomics. Pearson Education.
  • Economics. McGraw-Hill Education.
  • The Journal of International Economics, 15(3-4), 471-481.
  • The Theory of Industrial Organization. MIT Press.
  • Revista de Economía Industrial, (117), 25-38.
  • Economics Letters, 128, 63-66.

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