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(\boldsymbol{q}, \boldsymbol{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-sioping 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

Equilibrium stilts produced initially: 1000 stilts, each firm produces 20, at P=10, with 50 firms and entrepreneurs. After shift: 1478 stilts, each produces 20, at P=19, with 74 firms and entrepreneurs. Rent increased by 1478.

Step-by-step solution

Understand the Demand and Supply

Given the demand function for stilts \(Q = 1500 - 50P\) and the long-run cost for firms \(C(q, w) = 0.5q^2 - 10q + w\), where \(w\) is the wage paid to entrepreneurs. Each firm requires one entrepreneur and produces \(q\) stilts.

Calculate the Long-Run Equilibrium Price and Quantity

To find equilibrium, set up the condition where supply meets demand. The supply side involves the entrepreneurs represented by \(Q_s = 0.25w\), reflecting the firm count since one firm hires one entrepreneur. We need the marginal cost \(MC\), requiring differentiation of \(C(q, w)\), yielding \(MC = q - 10\). Equate marginal cost with price derived from demand: \(Q = 1500 - 50MC\).

Determine Firm and Industry Output

Given \(MC = P\), set \(MC = q - 10\) using equilibrium. Substitute \(P = q - 10\) into demand: \(Q = 1500 - 50(q - 10)\) to solve for both \(q\) and total output \(Q\). Given identical firms, \(q = Q/n\), derive how many firms (\(n\)) based on industry demand and individual \(q\).

Determine Equilibrium Wage and Entrepreneur Count

Find \(w\) using \(Q_s = 0.25w\), where \(Q_s\) is the equilibrium output, hence \(w = 4Q_s\). Determine the number of firms and entrepreneurs, knowing each firm hires one entrepreneur.

Analyze Shifted Demand Scenario

With new demand \(Q = 2428 - 50P\), repeat steps for \(a\) using new demand parameters for \(MC\), \(P\), and \(Q\). Recalculate quantities, price, and wages, reflecting effects of demand shift.

Calculate Rent Increase

Compare entrepreneur wages pre- and post-demand shift to calculate changed rents. Identify producer surplus using area between the supply curve and horizontal mechanism of price change.

Key concepts

Demand and Supply Analysis

The foundational concept of demand and supply involves understanding how the quantity demanded and supplied determine market equilibrium. In this scenario, we analyze the demand for stilts, represented by the equation \(Q = 1500 - 50P\), indicating that as the price \(P\) increases, the quantity demanded \(Q\) decreases. This inverse relationship is typical in consumer behavior. On the other side, supply focuses on the availability of entrepreneurs, where their supply is represented by \(Q_s = 0.25w\), showing that more entrepreneurs (and thus, firms) enter the market as wages \(w\) increase.

This exercise requires setting demand equal to supply to find equilibrium conditions. By comparing these fundamental equations, we can identify the market equilibrium, where supply meets demand, determining the equilibrium price and quantity in the long run. Understanding these dynamics is paramount in predicting how external changes, like a shift in demand, can affect these equilibrium points.

Cost Functions

Cost functions in economics describe how total production costs change with different levels of output. For the stilt-making firm, the total cost function is given by \(C(q, w) = 0.5q^2 - 10q + w\). This formula combines variable costs related to output \(q\) and a fixed wage cost \(w\) to hire entrepreneurs. The inclusion of wage \(w\) shows the cost of entrepreneurial talent as a critical factor affecting total costs.

To determine the firm's optimal production level, we derive the marginal cost (MC) from this function. This involves differentiating the total cost function with respect to \(q\), leading to \(MC = q - 10\). In competitive markets, firms equate marginal cost with price to ensure efficient production levels. Thus, understanding and calculating MC helps businesses set prices that cover costs while remaining competitive.

Entrepreneurial Supply

The supply of entrepreneurs is unique because it directly affects the overall supply of stilts. In this scenario, each stilt-making firm requires one entrepreneur, aligning the number of firms with the number of entrepreneurs supplied. The supply function \(Q_s = 0.25w\) illustrates that more entrepreneurs are available when wages \(w\) are higher, suggesting that wages need to increase to attract the necessary entrepreneurial talent.

This relationship also implies a critical impact on market dynamics, as the scarcity of entrepreneurial talent creates an upward slope in the long-run supply curve. In economics, such upward-sloping supply curves highlight how increases in demand can lead to price increases, affecting wage levels and ultimately influencing the extent of industry expansion.

Producer Surplus

Producer surplus is a measure of the benefits producers receive from selling goods at market prices above their minimum acceptable prices. In this context, as entrepreneurs experience a wage increase due to demand shifts, they capture additional economic rents or producer surplus. To calculate this change, we look at the area above the supply curve but below the price line, capturing the difference brought about by the shift in market conditions.

When demand for stilts rises from \(Q = 1500 - 50P\) to \(Q = 2428 - 50P\), we need to reassess producer surplus. This involves recalculating equilibrium conditions with the shifted demand to see how much additional producer surplus is engendered. Essentially, calculating the increase in the producer surplus helps us evaluate how beneficial the market changes are for producers, particularly in competitive scenarios where supply dynamics like scarcity and wages play essential roles.