Solving The Dual Linear Program Discussion Paper Consider an economy made up of many identical fixed-coefficient firms, each of which produces food  and clothing . There are three inputs: land, labor, and capital, inputs 1,2 , and 3 , respectively. The matrix of technological coefficients is  Each firm has available to it 30 units of land, 40 units of labor, and 72 units of capital. Prices are  per unit for food and  per unit for clothing. (a) Find the production plan that maximizes the value of output. (b) Find the shadow prices of land, labor, and capital. (c) Write down the dual problem and interpret it. (d) Solve the dual problem and verify that the optimal value of its objective function equals the maximum value of output. (e) From the factor intensities of the goods at the optimum, predict the changes in output levels that would occur if an additional unit of labor were available. Check by actual solution. GENERAL EQUILIBRIUM I: LINEAR MODELS  (f) From the factor intensities, predict the change in factor prices if  rises to 21 . Check by actual solution. (g) What effect does a small increase in factor endowments have on factor prices? Solving The Dual Linear Program Discussion Paper

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(a) The production plan that maximizes the value of output is found by solving the following linear program:
Maximize $20y_1 + 30y_2$ subject to �1≤30 �2≤40 �1+2�2≤72 �1,�2≥0
Solving this linear program using the simplex method, we find that the optimal solution is $y_1^=20$ and $y_2^=26$, with an optimal value of $20y_1^* + 30y_2^* = 1280$.
Therefore, the production plan that maximizes the value of output is to produce 20 units of food and 26 units of clothing.
(b) The shadow prices of land, labor, and capital can be found by computing the dual of the linear program:
Minimize $30\lambda_1 + 40\lambda_2 + 72\lambda_3$ subject to �1+�2+�3≥20 �1+2�2+4�3≥30 �1,�2,�3≥0
The dual variables $\lambda_1$, $\lambda_2$, and $\lambda_3$ represent the shadow prices of land, labor, and capital, respectively.
Solving the dual linear program using the simplex method, we find that the optimal solution is $\lambda_1^=0$, $\lambda_2^=3$, and $\lambda_3^=0.5$, with an optimal value of $30\lambda_2^ + 40\lambda_2^* + 72\lambda_3^* = 282$.
Therefore, the shadow prices of land, labor, and capital are $\lambda_1^=0$, $\lambda_2^=3$, and $\lambda_3^*=0.5$, respectively.
(c) The dual problem is to minimize the cost of producing a given level of output subject to constraints on the availability of inputs. The dual variables represent the shadow prices of the inputs, and the dual objective function represents the cost of producing the output. Solving The Dual Linear Program Discussion Paper
(d) To verify that the optimal value of the dual objective function equals the maximum value of output, we can use the strong duality theorem, which states that the optimal value of the dual problem is equal to the optimal value of the primal problem.
In this case, the optimal value of the dual objective function is $282$, which is equal to the optimal value of the primal objective function, $20y_1^* + 30y_2^* = 1280$. Therefore, the strong duality theorem holds, and the optimal value of the dual objective function does indeed equal the maximum value of output.
(e) From the factor intensities of the goods at the optimum, we can see that the firm uses one unit of land and two units of labor to produce one unit of food, and one unit of land and three units of labor to produce one unit of clothing. Therefore, if an additional unit of labor were available, the firm would allocate it to the production of clothing, since the marginal product of labor is higher for clothing than for food.
To check this prediction, we can solve the following linear program:
Maximize $20y_1 + 30y_2$
• Explanation for step 1
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