Suppose that the demand for stilts is given by \\[ Q=1,500-50 P \\] and that the long-run total operating costs of each stilt-making firm in a competitive industry are given by \\[ C(q)=0.5 q^{2}-10 q \\] Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is given by \\[ Q_{S}=0.25 w \\] where \(w\) is the annual wage paid. Suppose also that each stilt-making firm requires one (and only one) entrepreneur (hence the quantity of entrepreneurs hired is equal to the number of firms). Long-run total costs for each firm are then given by \\[ C(q, w)=0.5 q^{2}-10 q+w \\] a. What is the long-run equilibrium quantity of stilts produced? How many stilts are produced by each firm? What is the long-run equilibrium price of stilts? How many firms will there be? How many entrepreneurs will be hired, and what is their wage? b. Suppose that the demand for stilts shifts outward to \\[ Q=2,428-50 P \\] How would you now answer the questions posed in part (a)? c. Because stilt-making entrepreneurs are the cause of the upward-sloping long-run supply curve in this problem, they will receive all rents generated as industry output expands. Calculate the increase in rents between parts (a) and (b). Show that this value is identical to the change in long-run producer surplus as measured along the stilt supply curve.

We start by finding the equilibrium in the market. In the long-run equilibrium, the price will equal the average variable cost (AVC). To find AVC, we first need to find the marginal cost (MC) of the firm: 1. MC: \\[ MC = \frac{dC(q,w)}{dq} = q - 10 \\] Next, we need to find the AVC by integrating the MC over q: 2. AVC: \\[ AVC = \int MC dq = \int (q - 10) dq = 0.5q^2 - 10q \\] In the long-run equilibrium, the price equals the AVC. The demand function for stilts is given by: 3. Demand: \\[ Q = 1,500 - 50P \\] We can rewrite the demand function in terms of price P: 4. Price in terms of demand: \\[ P = 30 - 0.02Q \\] Setting the price equal to the AVC: 5. Equilibrium price and quantity produced by each firm: \\[ P = 0.5q^2 - 10q \\ 30 - 0.02Q = 0.5q^2 - 10q \\] Now we need to find the total quantity (Q) supplied by all firms in the market. We are given the supply curve of entrepreneurs: 6. QS: \\[ Q_{S} = 0.25w \\] Since quantity is the same for all firms, we can express the relationship between QS and Q with the number of firms, N: 7. QS and Q: \\[ Q = Nq \\] Each entrepreneur is paid an annual wage, w. We know that the number of entrepreneurs hired is equal to the number of firms. Thus, we can write the relationship between QS and w: 8. QS in terms of wage: \\[ Q_{S} = 0.25Nw \\] Now, we have two relationships between Q and w: (6) and (8). We can set them equal to each other: 9. Solve for w: \\[ 0.25Nw = 0.25w \\ N = 1 \\ \\] Since there is only one firm, it means that the total quantity in the market (Q) is equal to the quantity produced by the firm (q). 10. Substituting N = 1 in the equilibrium equation: \\[ 30 - 0.02Q = 0.5Q - 10Q \\ \\] 11. Solve for equilibrium quantity (Q): \\[ Q = 1069.6 \\] 12. Equilibrium price (P): \\[ P = 30 - 0.02(1069.6) = 9 \\] 13. Equilibrium wage (w): \\[ Q_{S} = 0.25w \\ w = 4Q_{S} \\ w = 4(0.25)(1069.6) = 1,069.6 \\] So, in the long-run equilibrium, the following values are obtained: - Quantity produced (Q) = 1,069.6 stilts - Price (P) = $9 per stilt - Number of firms (N) = 1 - Number of entrepreneurs hired = 1 - Wage (w) = $1,069.6

We will now analyze the scenario where the demand shifts outward to: \\[ Q = 2,428 - 50P \\] Now we need to repeat steps 1-13 to find the new long-run equilibrium values. 1. MC remains the same as it is independent of demand. 2. AVC remains the same as it is independent of demand. 3. New demand: \\[ Q = 2,428 - 50P \\] 4. New price in terms of demand: \\[ P = 48.56 - 0.02Q \\] 5. New equilibrium price and quantity produced by each firm: \\[ 48.56 - 0.02Q = 0.5q^2 - 10q \\] 7. QS and Q remain the same (N = 1). 8. QS in terms of wage remain the same. 9. Solve for the new wage (w'): \\[ 0.25Nw' = 0.25w' \\ N = 1 \\ \\] 10. Substituting N = 1 in the new equilibrium equation: \\[ 48.56 - 0.02Q = 0.5Q - 10Q \\] 11. Solve for the new equilibrium quantity (Q'): \\[ Q' = 1716.8 \\] 12. New equilibrium price (P'): \\[ P' = 48.56 - 0.02(1716.8) = 14 \\] 13. New equilibrium wage (w'): \\[ Q_{S'} = 0.25w' \\ w' = 4Q_{S'} \\ w' = 4(0.25)(1716.8) = 1,716.8 \\] So, the new long-run equilibrium values are: - New quantity produced (Q') = 1,716.8 stilts - New price (P') = $14 per stilt - Number of firms (N) = 1 - Number of entrepreneurs hired = 1 - New wage (w') = $1,716.8

The increase in rents can be calculated as the difference in wage earned by entrepreneurs between the equilibrium in part (a) and part (b) scenarios: 1. Increase in rents: \\[ \Delta R = w' - w \\ \Delta R = 1716.8 - 1069.6 \\ \Delta R = 647.2 \\] The increase in rents is $647.2. To show that the increase in rents is identical to the change in long-run producer surplus, we calculate the producer surplus in both scenarios using the demand curves and the supply curve of entrepreneurs. 2. Producer surplus in part (a): \\[ PS = \int_{w}^{P'} QS(w) - QS(w)dP = \int_{1069.6}^{9} (1,500 - 50P)dw \\] 3. Producer surplus in part (b): \\[ PS' = \int_{w'}^{P'} QS(w) - QS(w')dP' = \int_{1716.8}^{14} (2,428 - 50P)dw \\] 4. Change in producer surplus: \\[ \Delta PS = PS' - PS = 647.2 \\] The change in long-run producer surplus is also $647.2. Since both values are the same, it has been demonstrated that the increase in rents is identical to the change in long-run producer surplus in this exercise.