Question

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.

Short answer

#Question# In a competitive market for stilts, the average variable cost (AVC) for producing stilts equals marginal costs (MC), and firms use entrepreneurs who are paid an annual wage. The demand function for stilts is Q = 1,500 - 50P, and the cost function of a firm is given as C(q,w) = 0.5q^3 - 10q^2 + w, where q is the level of output and w is the wage paid to entrepreneurs. The supply curve of entrepreneurs is QS = 0.25w. Suppose that the demand for stilts shifts outwards, and the new demand function is Q = 2,428 - 50P. Calculate the increase in rents between the two scenarios and show that it is identical to the change in long-run producer surplus. #Answer# The increase in rents between the two scenarios is $647.2, and it is identical to the change in long-run producer surplus.

Step-by-step solution

a. Find the long-run equilibrium quantity, price, number of firms, entrepreneurs hired, and their wage.

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

b. New scenario with the demand shifts outward

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

c. Increase in rents between parts (a) and (b)

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.

Key concepts

Market Demand and Supply

Market demand and supply are fundamental concepts in economics that describe how much of a product is desired by buyers and how much is provided by sellers at various price levels. In a competitive market, demand typically decreases as prices rise, since consumers tend to purchase less of a good that costs more. Conversely, as prices decrease, demand usually increases, as the product becomes more affordable.

In the case of stilts, the demand function is given as \( Q = 1,500 - 50P \), where \( Q \) represents the quantity demanded and \( P \) is the price. This linear equation suggests that for every unit increase in price, the quantity demanded falls by 50 units. When market conditions change, this demand curve can shift, affecting the long-term equilibrium price and quantity.

Supply, on the other hand, is determined by the willingness of firms to provide a particular quantity at various price points. In our exercise, the supply curve for entrepreneurs, who are necessary for producing stilts, is given by \( Q_{S} = 0.25w \). Supply becomes fundamental in setting the stage for long-run equilibrium, where the quantity supplied matches the quantity demanded at a given price.

Competitive Industry

A competitive industry is characterized by many small firms that compete against each other, offering similar products with no individual firm able to influence market prices. In such an industry, firms are ‘price takers’ because the market dictates the price based on the collective supply and demand.

Each firm aims to minimize its costs to survive, especially over the long run when economic profits tend to zero. Here, stilt-making firms face long-run total operating costs expressed as \( C(q, w) = 0.5q^2 - 10q + w \). This cost also includes the wage paid to the entrepreneur, representing the firm’s crucial input.

In the long-run equilibrium for a competitive industry, all firms operate where their economic profits are zero, meaning that prices are equal to average costs over time. Firms that can't cover their costs will exit the market, while surviving firms will operate at optimal efficiency, a condition where they produce at the minimal average cost and make no supernormal profit.

Entrepreneurial Talent in Economics

Entrepreneurial talent refers to the ability of individuals to innovate, bring together resources, and push for business success. In our case, it's the entrepreneurs who spearhead stilt-making firms. Their talent is scarce, hence the upward-sloping supply curve for their services, \( Q_{S} = 0.25w \), implying that more talent, and thus more entrepreneurs, can only be attracted by higher wages.

Each stilt-making firm requires one entrepreneur to operate effectively. They are the cornerstone in determining how many firms exist, as each firm translates to the hiring of one unique entrepreneur. This scarcity leads to competitive wages, and as demand increases (more firms are needed due to higher demand for stilts), entrepreneurs capture additional rents.

When the demand for stilts increases, the economic rent (or surplus) earned by these entrepreneurs also rises, bridging the gap between what they were making before and the new, higher wage. This change reflects the added value brought by increased demand and is a testament to how vital entrepreneurial talent is to economic shifts and market dynamics.