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This problem involves calculating the equilibrium price of Harry's good (good 1), considering the supply and demand functions for two goods and a government subsidy for good 1.

The given functions are:

For good 1:
- Demand: $ Q_{D1} = 170 - P_1 + 4P_2 $
- Supply: $ Q_{S1} = 3P_1 - 58 $

For good 2 (substitutable product):
- Demand: $ Q_{D2} = 120 + P_1 - 5P_2 $
- Supply: $ Q_{S2} = 2P_2 - 84 $

Additionally, the government provides a unit subsidy of $38 for good 1, which affects the supply price for good 1.

### Step-by-step approach:

1. **Subsidy adjustment for good 1**: The subsidy effectively reduces the price paid by the supplier of good 1 by $38. Therefore, the supply function for good 1 needs to be adjusted accordingly:
   \[
   Q_{S1} = 3(P_1 - 38) - 58 = 3P_1 - 114 - 58 = 3P_1 - 172
   \]
   So the adjusted supply equation becomes:
   \[
   Q_{S1} = 3P_1 - 172
   \]

2. **Equilibrium for good 1**: To find the equilibrium, we set the demand equal to the supply for good 1:
   \[
   Q_{D1} = Q_{S1}
   \]
   Substituting the demand and adjusted supply equations:
   \[
   170 - P_1 + 4P_2 = 3P_1 - 172
   \]
   This gives the first equation to solve for $ P_1 $ and $ P_2 $.

3. **Equilibrium for good 2**: Similarly, for good 2, set the demand equal to the supply:
   \[
   Q_{D2} = Q_{S2}
   \]
   Substituting the demand and supply equations for good 2:
   \[
   120 + P_1 - 5P_2 = 2P_2 - 84
   \]
   This gives the second equation.

Now we have a system of two equations to solve for $ P_1 $ and $ P_2 $:

1. $ 170 - P_1 + 4P_2 = 3P_1 - 172 $
2. $ 120 + P_1 - 5P_2 = 2P_2 - 84 $

I'll solve this system for you now.The equilibrium prices for the two goods are:

- $ P_1 = \frac{535}{4} = 133.75 $ (equilibrium price for good 1, Harry's good)
- $ P_2 = \frac{193}{4} = 48.25 $ (equilibrium price for good 2, the substitutable product)

Thus, the equilibrium price of Harry's good is **133.75**.

Let me know if you need further clarification or details!

Here are 5 related questions to expand your understanding:
1. How does a government subsidy typically affect the supply curve?
2. What would happen to the equilibrium price if the subsidy were removed?
3. How would a change in the demand function for good 1 impact the equilibrium prices?
4. What is the relationship between the price of good 2 and the demand for good 1?
5. How does substitutability between two products affect their demand and supply curves?

**Tip:** When solving systems of equations, it's always helpful to simplify equations first and check if a subsidy or tax affects supply or demand directly!

Motivated by a recent government initiative for young entrepreneurs, Harry sets up a new company producing software (good 1) for the hotel industry. The supply and demand functions for his good and a substitutable product (good 2) are given. The government provides a unit subsidy of $38 for good 1. Work out the equilibrium price of Harry's good.

Learn how to calculate the equilibrium price for a product considering supply and demand functions and a government subsidy. This problem explores key economic concepts such as supply and demand functions, adjustment for subsidies, and finding the equilibrium price using algebraic methods.

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