2.3.
03. Problem: Consider a market with two firms that produce a homogeneous good. The inverse demand function is given by p = 100 - Q, where p is the price and Q is the total quantity produced by the two firms. The cost function for each firm is given by C(q) = 10q, where q is the quantity produced by the firm. Find the Cournot equilibrium and calculate the profits of each firm.
Solution:
The Cournot equilibrium is found by solving for the quantity that each firm produces such that each firm maximizes its profit, given the quantity produced by the other firm. Let q1 and q2 be the quantity produced by firms 1 and 2, respectively. Then, the profits of each firm are given by:
π1 = p(q1 + q2)q1 - 10q1 π2 = p(q1 + q2)q2 - 10q2
Taking the first-order condition for each firm, we have:
∂π1/∂q1 = p(2q1 + q2) - 10 = 0 ∂π2/∂q2 = p(2q2 + q1) - 10 = 0
Solving for q1 and q2, we get:
q1 = q2 = 20
Therefore, the Cournot equilibrium quantity is Q = 40, and the price is p = 60. The profits of each firm are:
π1 = (60)(20) - (10)(20) = 1000 π2 = (60)(20) - (10)(20) = 1000
04. Problem : Consider the following linear regression model: Y = β0 + β1X1 + β2X2 + ε where Y is the dependent variable, X1 and X2 are independent variables, β0, β1, and β2 are unknown parameters, and ε is the error term.
a) Show that the OLS estimator of β1 is unbiased. b) Under what conditions is the OLS estimator of β2 consistent? c) Suppose that the errors are normally distributed with zero mean and constant variance. Derive the formula for the standard error of the OLS estimator of β1.
Solution: a) To show that the OLS estimator of β1 is unbiased, we need to show that its expected value is equal to β1. Using the OLS equation, we have: β1 = ∑(Xi - X)(Yi - Ȳ) / ∑(Xi - X)² Taking the expected value of both sides, we get: E(β1) = E[ ∑(Xi - X)(Yi - Ȳ) / ∑(Xi - X)² ] = ∑(Xi - X) E[ (Yi - Ȳ) / ∑(Xi - X)² ] = β1 + ∑(Xi - X) E[ εi / ∑(Xi - X)² ] Since the error term ε has a zero mean, we have: E[ εi / ∑(Xi - X)² ] = 0 Therefore, we have: E(β1) = β1 This shows that the OLS estimator of β1 is unbiased.
b) The OLS estimator of β2 is consistent if the following conditions are satisfied:
•The true model is linear and correctly specified.
•The errors are independent and identically distributed with zero mean and finite variance.
•The independent variables are not perfectly correlated with each other.
•The sample size is large enough.
c) Suppose that the errors are normally distributed with zero mean and constant variance σ². The standard error of the OLS estimator of β1 is given by: SE(β1) = sqrt[ σ² / ∑(Xi - X)² ] where σ² is the estimated variance of the error term ε, which is given by: σ² = SSE / (n - k - 1) where SSE is the sum of squared errors, n is the sample size, and k is the number of independent variables (excluding the constant term).
Problem 2: Consider the following simultaneous equations model: Y1 = β0 + β1Y2 + β2X1 + ε1 Y2 = β3 + β4Y1 + β5X2 + ε2 where Y1 and Y2 are endogenous variables, X1 and X2 are exogenous variables, β0, β1, β2, β3, β4, and β5 are unknown parameters, and ε1 and ε2 are the error terms.
a) Write down the structural form and reduced form equations of the model. b) Derive the expressions for the OLS estimators of the parameters.
c) Under what conditions are the OLS estimators of the parameters consistent and unbiased?
Solution: a) The structural form equations of the model are: Y1 = β0 + β1Y2 + β2X1 + ε1 Y2 = β3 + β4Y1 + β5X2 + ε2 The reduced form equations
05. Question: What is the difference between cost-effectiveness analysis and cost-benefit analysis in healthcare?
Answer: Cost-effectiveness analysis (CEA) and cost-benefit analysis (CBA) are two methods of evaluating healthcare interventions
. CEA compares the costs and health outcomes of different interventions to determine which is the most effective in achieving a particular health goal. For example, CEA might compare the cost and effectiveness of two different drugs for treating a particular condition. CEA typically uses a measure such as quality-adjusted life years (QALYs) to measure health outcomes.
CBA, on the other hand, takes a broader view and considers not only health outcomes but also other benefits and costs that may be associated with an intervention. For example, CBA might consider the economic benefits of reducing absenteeism due to illness, as well as the direct medical costs of treatment. CBA typically uses monetary measures such as net present value (NPV) or benefit-cost ratio (BCR) to compare the costs and benefits of different interventions. In short, CEA focuses on comparing the health outcomes of different interventions, while CBA considers a broader range of costs and benefits, including health outcomes.
06. Question: How does moral hazard affect healthcare markets?
Answer: Moral hazard refers to the phenomenon in which individuals behave differently when they have insurance than they would if they were paying out of pocket. In healthcare markets, moral hazard can lead to an increase in demand for healthcare services and a
corresponding increase in healthcare costs. When individuals have insurance, they may be more likely to seek medical care for minor conditions or to undergo expensive treatments that may not be necessary. This is because they perceive the cost of healthcare services to be lower than it actually is. As a result, healthcare providers may over-treat or over-diagnose patients, leading to higher healthcare costs.
To mitigate the effects of moral hazard, healthcare systems may use various strategies, such as cost-sharing arrangements (where patients pay a portion of the cost of care), utilization review programs (where healthcare providers are required to justify the necessity of certain treatments), and managed care (where healthcare providers are incentivized to provide cost-effective care). These strategies aim to encourage individuals to use healthcare services only when necessary and to encourage healthcare providers to provide appropriate, costeffective care.