Quiz 13: General Equilibrium and Welfare

Business

The production possibility frontier shows different combinations of two goods which can be produces with given technology and resources. a. The production possibility frontier for guns (x) and butter (y) is given below: img This is an equation of a parabola. The production possibility frontier for the guns and butter given below: img Figure 1 On X axis the units of guns is shown and on Y axis the units of butter is shown. PPF curve shows the combination of guns and butter that can be produced with in given resources. b. If individuals always choose consumption bundles in which y = 2x so the production should be in this ratio only. Put value of y in the production possibility equation as shown below: img Put value of x in the equation of y as shown below: img If individuals always prefer consumption bundles in which y = 2x, then 10 units of x and 20 units of y will be produced. The rate of product transformation (RPT) between two outputs is the negative of the slope of the production possibility frontier of those outputs. The formula for the RPT is given below: img The production possibility frontier is given below: img Differentiate both sides with respect to x as shown below: img RPT is calculated at img and img as shown below: img img At a point where x = 10 and y = 20, the rate of product transformation (RPT) is img . At optimality the price ration is same as RPT Hence, the price ratio that will cause production to take place at this point is img . d. The solution point is shown in the figure below by the point A. img Figure 2 On X axis the units of guns is shown and on Y axis the units of butter is shown. PPF curve shows the combination of guns and butter that can be produced with in given resources. Point A shows the combination where, x = 10 and y = 20.

A utility function represents individual's preferences for consuming bundle of goods and services, it calculates individual's preferences in numerical terms. Utility function assign a numerical value to a particular bundle according to its preferences.  As per the question, Person S and J each have 10 hours of labour for the production of ice-cream or chicken. The quantity of ice-cream and chicken is represented by x and y respectively. The utility function of Person S is: img The utility function of Person J is: img The production function of ice-cream and chicken are: img where, l is labor employed a. In order to calculate the price ratio, img , first profit is maximized for both ice-cream and chicken. The profit of ice-cream is maximized. The value of x is put in the given situation as shown below: img In this the total revenue earned from ice-cream is subtracted from total cost incurred. Now, it is differentiated with respect to labor, to maximize the profit. img The value of p x is img . …… (1) The profit of chicken is maximized. The value of y is put in the given situation as shown below: img In this the total revenue earned from chicken is subtracted from total cost incurred. Now, it is differentiated with respect to labor, to maximize the profit. img The value of p y is img . …… (2) Now, the price ratio is calculated by using values from (1) and (2) as shown below: img Dividing these two equations, the following is derived: img Thus, the price ratio is img . b. First, the quantity demanded by Person S is calculated. In order to calculate the quantity demanded of x and y , the utility function of Person S is maximized subject to his budget constraint. The total number of labor hours is 10. The values of price of ice-cream and chicken are taken from (1) and (2). The budget constraint of Person S is shown below: img In the above equation, x 1 is ice-cream demanded by Person S y 1 is chicken demanded by Person S The condition of utility maximization is shown below: img Subject to img . Now, using Lagrangian, the following is obtained as shown below: img img …… (3) img …… (4) img Now, dividing equations (3) and (4), the following is derived: img Therefore, the value of y 1 is img . …… (5) The value obtained of y 1 in (5) is substituted in the budget constraint in following manner as shown: img The value of y 1 is calculated as shown below: img Thus, for Person S x 1 is img and y 1 is img . Similarly, the quantity demanded for Person J is calculated. The budget constraint for Person J is shown below: img In the above equation, x 1 is ice-cream demanded by Person J y 1 is chicken demanded by Person J The condition of utility maximization is shown below: img Subject to img . Now, using Lagrangian, the following is obtained as shown below: img img …… (6) img …… (7) img Now, dividing equations (6) and (7), the following is derived: img Therefore, the value of y 1 is img . …… (8) The value obtained of y 1 in (8) is substituted in the budget constraint in following manner as shown: img The value of y 1 is calculated as shown below: img Thus, for Person S x 1 is img and y 1 is img . c. It is given that the production function of ice-cream and chicken are: img where, l = labor employed To calculate the number of hours spent on the production of ice-cream, first total number of ice-cream produced is calculated. The total number of ice-cream produced is computed as follows: img Now, the labor allocated to production of ice-cream is calculated as follows: img Thus, the labor devoted to the production of ice-cream is img . To calculate the number of hours spent on the production of chicken, first total number of chicken produced is calculated. The total number of chicken produced is computed as follows: img Now, the labor allocated to production of Chicken is calculated as follows: img Thus, the labor devoted to the production of Chicken is img .

a) In this case, labor has a limit of 100. The PPF would look like below. img The corresponding PPF graph is given below in Figure 1. img Figure 1 b) In this case, land has a limit of 150. The PPF would look like below. img The corresponding PPF graph is given below in Figure 2. img Figure 2 c) The combined PPF (ABC) is shown below in Figure 3. img Figure 3 d) The combined PPF is concave because of increasing rate of product transformation. In the case of labor, RPT is 1 and in the case of land, RPT is 2. e) The relative price is shown by curve DBE in Figure 4. The slope of curve represents the relative price. Before B, slope is 1 and after B, slope is 2. img Figure 4 f) The individual can buy 4 units of food by paying 5 units of cloth. So, the relative price is 5/4. g) We have seen in Figure 4, the way the relative price changes at point B. The relative price rises instantaneously from 1 to 2. So, the production would remain the same whatever it is at point B. h) The PPF for capital is given in equation below and sketched in Figure 5. img img Figure 5

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