Question

The production function for a firm in the business of calculator assembly is given by \\[ q=2 V L \\] where \(q\) is finished calculator output and \(L\) represents hours of labor input. The firm is a price taker for both calculators (which sell for \(P\) ) and workers (which can be hired at a wage rate of \(w \text { per hour })\) a. What is the supply function for assembled calculators \([q=f(P, w)] ?\) b. Explain both algebraically and graphically why this supply function is homogeneous of degree zero in \(P\) and \(w\) and why profits are homogeneous of degree one in these vari ables. c. Show explicitly how changes in \(w\) shift the supply curve for this firm.

Short answer

Answer: A change in the wage rate (w) affects the supply curve for this firm because the supply function is given by q = f(P, w) = (w/P)L. An increase in the wage rate (w) results in a rightward shift of the supply curve, while a decrease in the wage rate (w) will result in a leftward shift of the supply curve.

Step-by-step solution

Determine the supply function for assembled calculators

We are given the production function \(q=2VL\). The firm is a price taker for both calculators (which sell for \(P\)) and workers (which can be hired at a wage rate of \(w\) per hour). To find the supply function, we must determine how the firm's output quantity (\(q\)) varies as a function of the price of calculators (\(P\)) and the wage rate \((w)\). The firm's profit function can be represented as: \\[ \pi = P q - w L \\] Substituting the production function \(q=2VL\): \\[ \pi = P (2 V L) - w L \\] Since the firm aims to maximize its profit, we will take the first-order condition with respect to labor input \(L\) and set it equal to 0: \\[ \frac{\partial\pi}{\partial L} = 2PV - w = 0 \\] Now, we can express \(V\) in terms of \(P\) and \(w\): \\[ V = \frac{w}{2P} \\] Substitute this expression for \(V\) back into the production function \(q=2VL\): \\[ q = 2\left( \frac{w}{2P} \right) L \\] \\[ q = \frac{w}{P}L \\] So the supply function for assembled calculators is given by: \\[ q = f(P, w) = \frac{w}{P}L \\]

Analyze the homogeneity of the supply function algebraically and graphically

To examine the homogeneity of the supply function with respect to variables \(P\) and \(w\), consider a positive scalar \(k\). We will calculate \(f(kP, kw)\) and compare it to \(f(P,w)\): \\[ f(kP, kw) = \frac{kw}{kP}L \\] We observe that \(k\) cancels out in the numerator and denominator, resulting in: \\[ f(kP, kw) = \frac{w}{P}L = f(P, w) \\] This demonstrates that the supply function is homogeneous of degree zero in \(P\) and \(w\). Graphically, this means that if we scale both the price of calculators and the wage rate simultaneously, the supply curve does not change its position. Regarding the profit function, we can see that it is homogeneous of degree one in variables \(P\) and \(w\). Consider the profit function with the scalar \(k\): \\[ \pi(kP, kw, L) = kP(2V L) - kw L = k(P(2V L) - w L) = k\pi(P, w, L) \\] This equation shows that the profit function is homogeneous of degree one in \(P\) and \(w\). This means that if the price of calculators and the wage rate both increase by a factor of \(k\), the firm's profit will also increase by the same factor.

Demonstrate how changes in \(w\) shift the supply curve for this firm

Having the supply function: \\[ q = f(P, w) = \frac{w}{P}L \\] We can now analyze how a change in the wage rate \(w\) affects the supply curve. An increase in the wage rate \(w\) will cause the fraction \(\frac{w}{P}\) to increase, assuming that the price \(P\) remains constant. Consequently, the output quantity \(q\) will also increase, resulting in a shift of the supply curve to the right. Conversely, a decrease in the wage rate \(w\) will cause the fraction \(\frac{w}{P}\) to decrease, leading to a decrease in the output quantity \(q\), and the supply curve will shift to the left. In conclusion, an increase in the wage rate will shift the supply curve to the right, while a decrease in the wage rate will shift the supply curve to the left.

Key concepts

Supply Function

A supply function in economics describes how the quantity of a product that a firm is willing to produce and sell is influenced by various factors, predominantly price. For the calculator assembly firm in our example, the production function given is \( q = 2VL \), indicating that the output quantity \( q \) depends on two factors: labor \( L \) and a factor \( V \). The supply function, in this case, can be derived from the profit-maximizing conditions of the firm.

The profit function for the firm is \( \pi = Pq - wL \), where \( \pi \) represents profit, \( P \) represents the price of calculators, and \( w \) the wage rate. To find the optimal supply function, we determine the point where profits are maximized by setting the derivative of the profit function with respect to labor \( L \) to zero. This provides the condition \( 2PV - w = 0 \), allowing us to solve for \( V \) and substitute back into the production function to derive the supply function \( q = \left( \frac{w}{P} \right)L \).

• This function shows that the quantity supplied \( q \) is directly proportional to the wage \( w \) and inversely proportional to the price \( P \).
• When wage or price changes, the quantity supplied adjusts in response.

Homogeneity in Economics

Homogeneity in economics refers to functions that demonstrate consistent scaling behavior as inputs are altered simultaneously. It’s a vital property for understanding how a firm's supply and profit react to proportional changes in all input prices.

In the given example, the supply function \( q = \frac{w}{P}L \) is homogeneous of degree zero in terms of the price \( P \) and wage \( w \). This means if all inputs, namely the price of calculators and the wage rate, are scaled by a constant factor \( k \), the supply function remains unchanged: \( f(kP, kw) = f(P, w) \). This stability implies that simultaneous proportional changes in input prices do not affect the output level.

• Homogeneity of degree zero indicates no change in supply despite uniform scaling of input prices.
• Graphically, the supply curve remains stationary, demonstrating its resilience to proportional price changes.

The profit function \( \pi = Pq - wL \) is homogeneous of degree one. If both the price of calculators and the wage rate increase by a factor \( k \), the profit increases by the same factor: \( \pi(kP, kw, L) = k\pi(P, w, L) \). This indicates that profits scale proportionally with changes in input costs and revenue prices.

Profit Maximization

Profit maximization is a core objective for firms, where they attempt to produce the quantity of output that yields the highest possible profit. For the calculator assembly firm, the profit maximization problem revolves around optimizing labor \( L \), given the fixed sale price of calculators \( P \) and the variable wage rate \( w \).

By setting the profit function \( \pi = Pq - wL \) and taking the derivative with respect to labor \( L \), we find the condition for profit maximization: \( 2PV - w = 0 \). This derivative measures how a small change in labor input affects profit. Setting it to zero finds the point of maximum profit, balancing the additional revenue from one more unit of labor against the cost of that labor.

• Profit maximization involves adjusting inputs (like labor) to where additional costs equal additional revenues.
• This ensures that every input unit contributes positively to profit, maximizing economic efficiency.

Understanding profit maximization helps in realizing how firms decide on production quantities based on external factors such as cost and price changes.