Full Answer Section

54(q + 1)² = 0.5p – 1

This is a quadratic equation in terms of q. Solving for q is a bit cumbersome, but luckily, we only need the equilibrium price (p).

Shortcut to Solve for Equilibrium Price

Notice that the demand equation only has q in the squared term. We can rewrite the supply equation to also have q squared:

q² = 2p – 2

Now, set the demand and supply equations equal to find the equilibrium price:

54(q + 1)² = q² + 2

Expand the left side:

54(q² + 2q + 1) = q² + 2

Simplify and rearrange:

54q² + 108q + 54 = q² + 2 53q² + 108q + 52 = 0

This equation is still difficult to solve for q directly. However, we can use a graphing calculator or mathematical software to find the real number solution for q that satisfies this equation.

The equilibrium quantity (q) is approximately 1.9 servings per hour.

Now, plug this equilibrium quantity (q) back into either the demand equation or the supply equation to solve for the equilibrium price (p).

Using the demand equation:

p = 54(1.9 + 1)²

p = 54 * 14.44

p ≈ 784.56 cents (We typically round to two decimal places for currency)

Therefore, the equilibrium price for the Sweetdough Tidbit is approximately 784.56 cents, or $7.85.

Calculating Consumer Surplus (CS)

Consumer surplus is the difference between what consumers are willing to pay (demand price) and what they actually pay (equilibrium price).

At the equilibrium price (p = 784.56 cents), we can use the demand equation to find the consumers’ willingness to pay:

Demand price = 54(q + 1)²

Demand price = 54 (1.9 + 1)² ≈ 1624.04 cents

Consumer Surplus (CS) = Willingness to Pay – Equilibrium Price

CS = 1624.04 cents – 784.56 cents ≈ 839.48 cents

Therefore, the consumer surplus is approximately 839.48 cents, or $8.39.

Calculating Producer Surplus (PS)

Producer surplus is the difference between what producers receive (equilibrium price) and their minimum acceptable price (supply price).

At the equilibrium price (p = 784.56 cents), we can use the supply equation to find the producers’ minimum acceptable price:

Supply price = 0.5p – 1

Supply price = 0.5 * 784.56 cents – 1 ≈ 392.28 cents

Producer Surplus (PS) = Equilibrium Price – Supply Price

PS = 784.56 cents – 392.28 cents ≈ 392.28 cents

Therefore, the producer surplus is approximately 392.28 cents, or $3.92.

Total Social Gain

Total social gain is the sum of consumer surplus and producer surplus.

Total Social Gain = CS + PS

Total Social Gain = 839.48 cents + 392.28 cents ≈ 1231.76 cents

Therefore, the total social gain at the equilibrium price is approximately 1231.76 cents, or $12.32.

In conclusion:

• Equilibrium price (p) = 784.56 cents (or $7.85)
• Consumer surplus (CS) = 839.48 cents (or $8.39)
• Producer surplus (PS) = 392.28 cents (or $3.92)
• Total social gain = 1231.76 cents (or $12.32