Microeconomic Theory

Business

Quiz 10 :
Profit Maximization

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Quiz 10 :
Profit Maximization

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It is given that the cost function of the widget production is TC =0.25 q 2 where q = q A + q L. The two nations that demand widgets are nation- A and nation- L. The demand of widgets in nation- A is given as img and the demand of widgets in nation- L is given as img . Nation- A : The profit maximizing point will be attained at a point where MR=MC. The price is calculated from demand function as shown below: img The total revenue for widget in America is given below: img The marginal revenue is the derivative of TR. The derivation is shown below: img The marginal cost is derivative of total cost with respect to q as shown below: img Equate the marginal revenue to marginal cost. The calculation is shown below: img img …… (1) Nation- L Similarly, for maximizing profit in nation- L the firm should choose that output for which marginal revenue is equal to marginal cost. The price is calculated from demand function as shown below: img The total revenue is given below: img The marginal revenue is derivative of TR as shown below: img The marginal cost is derivative of total cost with respect to q. That is, img Evaluate the marginal revenue to marginal cost as shown below: img img …… (2) Equate equation (1) and (2). img Thus, the profit maximizing level of output in nation- A is 30 units. Substitute this value of output in nation- A 's inverse demand function. img Thus, the profit maximizing price in nation- A is $35 per unit. Substitute the value of output in equation (1) to get the optimal production level in nation- L. img Thus, the profit maximizing level of output in nation- L is 10 units. Substitute this value of output in nation- L 's inverse demand function. img Thus, the profit maximizing price in nation- L is $22.5 per unit. Conclusion: The firm must sell 30 units in nation- A and 10 units in nation- L to maximize profits. The price charged by the firm is $35 per unit in nation- A and $22.5 per unit in nation- L.

The production function for a calculator producing firm is given as img Here, img It is also given that: img (a) In order to calculate the total cost function for a calculator producing firm, first the value of l in terms of q is computed. The value of l is computed as shown below: img img img Thus, value of l is img . ……. (1) The total cost function represents the cost incurred by the firm in production of calculators. It is wage rate times the labor hours used to produce calculator. The value of l is taken from (1). The total cost function is calculated as shown below: img Therefore, the total cost function is img . (b) The profit function shows the difference between the total revenue earned by selling calculators and the cost incurred to produce them. The calculation of profit function is shown as follows: img Thus, the profit function for the firm given is img . (c) The level of output at which the firm maximizes its profit, gives the supply function for the firm. At given wage level and price, the supply function for the given firm is derived as shown below: img Differentiating with respect to q , img …… (2) Therefore, the supply function is given as img . (d) The demand function of labor for the firm can be calculated as follows: img Take the value of q from equation (2) img Thus, demand curve of labor for the firm is img . (e) The function obtained shows a positive relation between labor demanded and the price of calculator. The same relation holds for true for supply of calculators by the firm and price charged. This is so because if the price increases for the calculator, producer will be willing to increase its supply too. And hence, demand more labor, which is a factor of production. Both functions have inverse relationship with the wage rate. This is because with increase in wages, producer reduces their demand for labor and supply for calculator also decreases.

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