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This question involves analyzing a simple Keynesian model of income determination, defined by the equations given in the image:

1. **Consumption Function**: \( C_t = \beta_0 + \beta_1 Y_t + u_1 \)
2. **Investment Function**: \( I_t = \beta_0 + \beta_1 Y_t + \beta_2 Y_{t-1} + u_2 \)
3. **Income Definition**: \( Y_t = C_t + I_t + G_t \)

The question has multiple parts. Let's go through each part with a structured solution.

### (a) Derive the Reduced Form Coefficients of the Behavioral Equations
The reduced form expresses each endogenous variable in terms of exogenous variables and constants.

1. Substitute \( C_t \) and \( I_t \) into the income equation \( Y_t = C_t + I_t + G_t \).
2. Solve for \( Y_t \) in terms of exogenous components \( G_t \), and possibly lagged terms (since \( Y_{t-1} \) appears in \( I_t \)).

Let me know if you would like a detailed derivation for each substitution.

### (b) Interpretation of Reduced-Form Parameters for Total Effects
To show the total effect of a change in an exogenous variable (like \( G_t \)) on the endogenous variables, we look at how \( Y_t \), \( C_t \), and \( I_t \) respond to a shift in \( G_t \), incorporating both direct and indirect effects.

### (c) & (d) Identification of Behavioral Equations and the Model
Using order and rank conditions:
1. **Order Condition**: Check if the number of excluded variables from each equation matches the required order for identification.
2. **Rank Condition**: Verify that the rank of the matrix of coefficients aligns with identification criteria.

This step can be intricate, involving matrix analysis of variables.

### (e) Appropriate Estimation Method
Based on the structure, simultaneous equation methods like **Two-Stage Least Squares (2SLS)** or **Instrumental Variables (IV)** may be suitable due to the potential endogeneity in \( Y_t \).

### (f) Conditions for Exact Identification
Exact identification typically requires that the number of independent exogenous variables matches the number of endogenous variables, aligning with both order and rank conditions.

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Would you like detailed steps or specific calculations for any of these sections? Here are additional questions to

Given the simple Keynesian model of income determination:

Ct = β0 + β1Yt + u1
It = β0 + β1Yt + β2Yt-1 + u2
Yt = Ct + It + Gt

a) Derive the reduced form coefficients of the behavioural equations.
b) Show that the reduced-form parameters measure the total effect, direct and indirect, of a change in the exogenous variable on the endogenous variables.
c) Establish the identification state of each of the behavioural equations, using the order and rank conditions.
d) What is the identification state of the entire model?
e) What estimation method is appropriate for each of the behavioural equations?
f) Show under what conditions could the system become exactly identified.

Explore the Keynesian model of income determination with equations for consumption, investment, and income. Learn to derive reduced form coefficients, analyze identification states using order and rank conditions, and apply estimation methods like 2SLS for behavioral equations.

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