To analyze the effects of taxes and subsidies on supply and demand, let's break down the problem step by step. We start with the demand and supply equations and then incorporate the tax and subsidy effects.

Given:
- Demand: \( D = 200 - 22p \)
- Supply: \( S = -500 + 7p \)
- Tax: \( t = 4 \)
- Subsidy: \( s = 4 \)

### 1. Equilibrium without Tax or Subsidy

First, we find the equilibrium price and quantity without considering tax or subsidy. The equilibrium occurs where demand equals supply:

\[ D = S \]

So,

\[ 200 - 22p = -500 + 7p \]

Solve for \( p \):

\[ 200 + 500 = 22p + 7p \]

\[ 700 = 29p \]

\[ p = \frac{700}{29} \approx 24.14 \]

Substitute \( p \) back into either the demand or supply equation to find the equilibrium quantity. Using the demand equation:

\[ Q = 200 - 22 \left( \frac{700}{29} \right) \]

\[ Q \approx 200 - 22 \times 24.14 \]

\[ Q \approx 200 - 531.08 \]

\[ Q \approx -331.08 \]

This negative quantity doesn't make sense economically, suggesting the need for checking the original problem's parameters or interpreting them in a realistic market scenario.

### 2. Equilibrium with Tax

With a tax of 4, the price consumers pay is \( p + t \), while producers receive \( p \):

\[ 200 - 22(p + 4) = -500 + 7p \]

\[ 200 - 22p - 88 = -500 + 7p \]

\[ 112 - 22p = -500 + 7p \]

\[ 112 + 500 = 29p \]

\[ 612 = 29p \]

\[ p = \frac{612}{29} \approx 21.10 \]

So, the price consumers pay is:

\[ p + t = 21.10 + 4 = 25.10 \]

And the quantity is:

\[ Q = -500 + 7 \times 21.10 \approx -500 + 147.7 = -352.3 \]

Again, a negative quantity indicates a recheck is necessary for realistic scenarios.

### 3. Equilibrium with Subsidy

With a subsidy of 4, the price consumers pay is \( p \), but producers receive \( p + s \):

\[ 200 - 22p = -500 + 7(p + 4) \]

\[ 200 - 22p = -500 + 7p + 28 \]

\[ 200 - 22p = -472 + 7p \]

\[ 200 + 472 = 29p \]

\[ 672 = 29p \]

\[ p = \frac{672}{29} \approx 23.17 \]

And the price producers receive is:

\[ p + s = 23.17 + 4 = 27.17 \]

And the quantity is:

\[ Q = -500 + 7 \times 23.17 \approx -500 + 162.19 = -337.81 \]

Similar to the previous scenarios, a negative quantity highlights a need to reevaluate the settings or consider if any logical errors are in the parameters provided.

### Summary and Possible Issues

The given parameters and the results suggest that the original equations might not fit typical supply and demand scenarios, or the market structure implied isn't straightforward.

Would you like further analysis on possible adjustments or detailed steps for each scenario? Here are some questions to deepen understanding:

1. What do the negative quantities imply about the market scenario provided?
2. How might we adjust the supply and demand equations for more realistic outcomes?
3. What are the practical interpretations of taxes and subsidies on supply and demand curves?
4. How do price floors and ceilings relate to taxes and subsidies?
5. What are potential economic implications of a tax versus a subsidy in this market?
6. How can we model more complex supply and demand scenarios with different forms of taxes or subsidies?
7. Can you provide real-world examples where such supply and demand scenarios might occur?
8. How would shifts in external factors (e.g., production costs, consumer preferences) affect these equilibria?

### Tip:
When analyzing economic models, always cross-check the logical consistency and practical applicability of the parameters used in the equations to ensure realistic results.

Explore how taxes and subsidies impact supply and demand using detailed equations and step-by-step analysis. Learn about equilibrium prices and quantities in economic models. Suitable for undergraduate economics students.

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