Solving The Dual Linear Program Discussion Paper

Consider an economy made up of many identical fixed-coefficient firms, each of which produces food $y_{1}$ and clothing $y_{2}$. There are three inputs: land, labor, and capital, inputs 1,2 , and 3 , respectively. The matrix of technological coefficients is $A=⎝⎛ 111 124 ⎠⎞ $ Each firm has available to it 30 units of land, 40 units of labor, and 72 units of capital. Prices are $$20$ per unit for food and $$30$ 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 $535$ (f) From the factor intensities, predict the change in factor prices if $p_{1}$ 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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### Step-by-step

**Step 1/1**

*=20$ and $y_2^*=26$, with an optimal value of $20y_1^* + 30y_2^* = 1280$.

*=0$, $\lambda_2^*=3$, and $\lambda_3^

*=0.5$, with an optimal value of $30\lambda_2^*+ 40\lambda_2^* + 72\lambda_3^* = 282$.

*=0$, $\lambda_2^*=3$, and $\lambda_3^*=0.5$, respectively.

- Explanation for step 1