Question

Write a two-equation system in "supply and demand form," that is, with the same variable \(y_{t}\) (typically, "quantity") appearing on the left-hand side: $$\begin{aligned} &y_{1}=\alpha_{1} y_{2}+\beta_{1} z_{1}+u_{1}\\\ &y_{1}=\alpha_{2} y_{2}+\beta_{2} z_{2}+u_{2} \end{aligned}$$ i. If \(\alpha_{1}=0\) or \(\alpha_{2}=0,\) explain why a reduced form exists for \(y_{1}\). (Remember, a reduced form expresses \(y_{t}\) as a linear function of the exogenous variables and the structural errors.) If \(\alpha_{1} \neq 0\) and \(\alpha_{2}=0,\) find the reduced form for \(y_{2}\) ii. If \(\alpha_{1} \neq 0, \alpha_{2} \neq 0,\) and \(\alpha_{1} \neq \alpha_{2}\), find the reduced form for \(y_{1}\). Does \(y_{2}\) have a reduced form in this case? iii. Is the condition \(\alpha_{1} \neq \alpha_{2}\) likely to be met in supply and demand examples? Explain.

Short answer

A reduced form for \(y_1\) exists if \(\alpha_1 = 0\) or \(\alpha_2 = 0\). If \(\alpha_1 \neq \alpha_2\), \(y_1\) has a reduced form, but \(y_2\) does not. Distinct \(\alpha\) values are typical in supply and demand.

Step-by-step solution

Identify the Conditions for Reduced Form

A reduced form expresses a variable as a function of exogenous variables and errors. The variable must not depend on endogenous variables. If either \(\alpha_1=0\) or \(\alpha_2=0\), the equations simplify such that \(y_1\) is only dependent on exogenous variables \(z_1\) or \(z_2\), leading to a reduced form.

Reduced Form for \(y_2\) with \(\alpha_1 \neq 0\) and \(\alpha_2 = 0\)

Given \(\alpha_2 = 0\), the second equation simplifies to \(y_1 = \beta_2 z_2 + u_2\). From this, we can write \(y_2\) in terms of \(y_1\): \(y_2 = \frac{y_1 - \beta_1 z_1 - u_1}{\alpha_1}\), thus providing a reduced form for \(y_2\).

Reduced Form for \(y_1\) When \(\alpha_1 \neq \alpha_2\)

If \(\alpha_1 eq 0\) and \(\alpha_2 eq 0\), and \(\alpha_1 eq \alpha_2\), solve the system. Subtract the second equation from the first to eliminate \(y_1\): \(0 = (\alpha_1 - \alpha_2)y_2 + (\beta_1 z_1 - \beta_2 z_2) + (u_1 - u_2)\). Solve for \(y_2\), then substitute into either equation to find \(y_1\).

Existence of Reduced Form for \(y_2\)

In the system with \(\alpha_1 eq \alpha_2\) and both non-zero, \(y_2\) is derived from \(y_1\) as shown in Step 3. However, it does depend on \(y_1\), making it not readily expressible as only a function of exogenous variables and errors, so \(y_2\) doesn't have a reduced form in this scenario.

Likelihood of Condition \(\alpha_1 \neq \alpha_2\) in Supply and Demand

In supply and demand models, \(\alpha_1\) and \(\alpha_2\) represent distinct relationships of supply and demand curves. It is common for these to differ unless the supply and demand curves are perfectly parallel, which is unlikely in real-world scenarios.

Key concepts

Supply and Demand Analysis

Supply and demand analysis is a fundamental aspect of economics that examines the relationship between the quantity of goods offered by producers (supply) and the quantity of goods consumers are willing to purchase (demand). In a two-equation system, like the one in the exercise, the quantity demanded and supplied can both be expressed as a function of various factors such as price, consumer income, and other exogenous variables.

This analysis helps in understanding how changes in these factors affect equilibrium prices and quantities in the market. For instance, a rise in consumer income typically increases demand, which can lead to higher prices and greater quantity sold, assuming supply remains constant. Conversely, if production technology improves, supply may increase, potentially lowering the price if demand stays unchanged.

Each relationship, whether it's supply or demand, is often described using equations where quantity is dependent on other variables. These equations reflect the underlying economic forces driving market behavior, illustrating how sensitive the market is to changes (elasticity). Understanding these dynamics is crucial for predicting economic outcomes and making informed business or policy decisions.

Simultaneous Equation Models

Simultaneous equation models (SEMs) allow economists to study multiple interdependent variables, like supply and demand interactions, within an economic system, all at once. In essence, such models capture the idea that variables in an economic model can simultaneously affect and be affected by each other. This contrasts with single-equation models that assume variables depend only on predetermined factors.

For the given exercise, the presence of simultaneous equations indicates that changes in demand may lead to immediate changes in supply and vice versa, creating a system of interdependencies that must be addressed together rather than in isolation. This complexity requires special techniques for solution, as traditional methods used with simpler models are insufficient.

• Steps typically involve the identification of endogenous variables (those affected within the model) and exogenous variables (outside influences on the system).
• We then apply mathematical solutions to isolate each variable involved, aiming to derive meaningful relationships or predictions between them.

Simultaneous equation models are heavily used for policy simulation, allowing stakeholders to forecast the impact of changes in policy measures on a complex economy.

Reduced Form Equations

Reduced form equations are derived expressions where endogenous variables in a simultaneous equation model are solved explicitly in terms of exogenous variables and random errors. These are particularly useful as they simplify models into more straightforward interpretations.In the context of the exercise, understanding when a reduced form equation exists is key. The reduced form exists when it is possible to express a variable solely based on exogenous variables, without interference from other endogenous variables. This situation arises under certain conditions related to model parameters such as the \(\alpha_1\) and \(\alpha_2\) coefficients.Reduced form equations have practical applications because they:

• Enable economists to predict how changes in external variables impact the system.
• Simplify the estimation process by focusing on observable factors.
• Aid in econometric analysis by providing clear functional relationships that aid statistical testing.

Thus, they play a crucial role in making complex models practically manageable and interpretable, helping both in academic research and policy-making.