I need help answering 1-4

2 Theory Consider a monopolist retailer selling a single good. The firm sets the tax-inclusive price of the good p and also a single product 'potency' characteristic (i.e. THC or alcohol content) r. The firm faces a per-unit potency tax 72 (in units of dollars per unit of potency per unit of the good). Consumers pay an ad valorem tax Tp (expressed as a percentage of the tax- exclusive price) and the firm sets the tax-inclusive price p (so that the firm earns p/(1 + 7p) per unit sold). Consumer demand q(p, x) is a function of the price and the potency and is assumed to feature constant demand elasticities (Ep, Ex) with respect to both characteristics. Marginal costs vary with the potency via mc(I) = c+ (y+TI)I. (1) In this equation, c represents marginal costs which are independent of potency and y is the marginal cost of an additional unit of potency. The marginal costs of additional potency are either costs associated with growing more potent marijuana (e.g. allowing plants to grow for a longer period of time) or with manipulation during the testing process (e.g. sending in samples that are more likely to give a particular THC reading). Putting these pieces together, the firm's profit function is n (p, I) = T - C - (y+12)1). q(p,I) (2) where q(p, I) = Apprez. 1. Write the first order conditions for profit maximization. You may write these with respect to partial derivatives of q, instead of plugging in the demand function. 2. Use the fact that the price and potency elasticities of demand are constant to find a closed form solution for x and p. Hint: If you used partial derivatives for q in the previous problem, you should be able to multiply and divide to turn those into clasticities. 3. What is the relationship between taxes on price and the potency of the product? Define the potency-adjusted price as p/r. What is the relationship between taxes on price and this object? 4. Suppose you had estimates of demand elasticities, and knew the prices, potency, and current tax regime from the data. How could you use the model to estimate c and y