Local reference praxeologies |${\protect\mathfrak{p}}_{iE}=\left[{T}_{iE},{\tau}_{iE},{\theta}_{iE},{\varTheta}_E\right],i\in \left\{1,2,3\right\}$| from the microeconomics textbook
| Task (|${\boldsymbol{T}}_{\boldsymbol{E}}$|) . | Technique (|${\boldsymbol{\tau}}_{\boldsymbol{E}}$|) . | Technology (|${\boldsymbol{\theta}}_{\boldsymbol{E}}$|) . | |
|---|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{1}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{1}\boldsymbol{E}}$| | |
| |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters when there exists an interior solution. Exemplary tasks outlined in Section 6.2 | |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{EG}}\right)$| | 1. Draw the budget line |$Y={p}_1{q}_1+{p}_2{q}_2$| 2. Draw an indifference curve (IC), |$U\left({q}_1,{q}_2\right)=\overline{q}$|, on which utility is fixed at some number |$\overline{q}.$| 3. Highest IC rule: Find the highest such IC (farthest from the origin) that satisfies |${p}_1{q}_1+{p}_2{q}_2\le Y$| for at least one bundle (|${q}_1,{q}_2$|). Graphically, this implies (tangency rule) shifting the IC up until it just touches the budget constraint. | Assumption of the model (A1)–(A4) 1. As more is preferred to less a consumer (weakly) prefers bundle (|${q}_i,{q}_j$|) to bundle (|${q}_i,{q}_k$|) for any |${q}_j\ge{q}_k.$| This ensures the IC are convex to the origin and that higher IC are ‘better’. 2. Bundles for which |${p}_1{q}_1+{p}_2{q}_2>Y$|are not feasible, hence |${p}_1{q}_1+{p}_2{q}_2\le Y$|must hold for the optimal bundle. 3. Point 1 and 2 ensure that the highest IC rule holds. 4. The last step ensures the consumer cannot purchase negative quantities of a good. 5. When the IC are strictly convex and differentiable. The MRS = MRT holds at a unique point of tangency (|$\mathrm{MRS}=-\frac{U_1}{U_2}=-\frac{p_1}{p_2}=\mathrm{MRT}$|) |
| |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{EL}}\right)$| | 1. Set up the function |$L\left({q}_1,{q}_2,\lambda \right)\!=\!U\left({q}_1,{q}_2\right)+ \lambda \left(Y-{p}_1{q}_1-{p}_2{q}_2\right)$| 2. Solve for |${q}_1,{q}_2$| from: (1) |$\frac{\partial L}{\partial{q}_1}=\frac{\partial U}{\partial{q}_1}-\lambda{p}_1={U}_1-\lambda{p}_1=0$| (2) |$\frac{\partial L}{\partial{q}_2}={U}_2-\lambda{p}_2=0$| (3) |$\frac{\partial L}{\partial \lambda }=Y-{p}_1{q}_1-{p}_2{q}_2=0$| | ||
| |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{ES}}\right)$| | 1. If the IC are strictly convex and differentiable, calculate the slope of the IC, |$\mathrm{MRS}=-\frac{d{q}_2}{d{q}_1}=-\frac{U_1}{U_2}$|, and equate it with the slope of the budget line |$\mathrm{MRT}-\frac{p_1}{p_2}.$| (Tangency rule.) Solve for |${q}_1$| and |${q}_2$|. For some functions |${q}_1$| and |${q}_2$| can be directly taken from a table. 2. If the IC is not strictly convex and differentiable, the optimal bundle must be obtained by other techniques which we do not detail here. 3. Plug in |${q}_1$| and |${q}_2$|from Step 1 into the budget constraint |$Y={p}_1{q}_1+{p}_2{q}_2$| to obtain the optimal bundle. | ||
| Task (|${\boldsymbol{T}}_{\boldsymbol{E}}$|) . | Technique (|${\boldsymbol{\tau}}_{\boldsymbol{E}}$|) . | Technology (|${\boldsymbol{\theta}}_{\boldsymbol{E}}$|) . | |
|---|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{1}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{1}\boldsymbol{E}}$| | |
| |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters when there exists an interior solution. Exemplary tasks outlined in Section 6.2 | |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{EG}}\right)$| | 1. Draw the budget line |$Y={p}_1{q}_1+{p}_2{q}_2$| 2. Draw an indifference curve (IC), |$U\left({q}_1,{q}_2\right)=\overline{q}$|, on which utility is fixed at some number |$\overline{q}.$| 3. Highest IC rule: Find the highest such IC (farthest from the origin) that satisfies |${p}_1{q}_1+{p}_2{q}_2\le Y$| for at least one bundle (|${q}_1,{q}_2$|). Graphically, this implies (tangency rule) shifting the IC up until it just touches the budget constraint. | Assumption of the model (A1)–(A4) 1. As more is preferred to less a consumer (weakly) prefers bundle (|${q}_i,{q}_j$|) to bundle (|${q}_i,{q}_k$|) for any |${q}_j\ge{q}_k.$| This ensures the IC are convex to the origin and that higher IC are ‘better’. 2. Bundles for which |${p}_1{q}_1+{p}_2{q}_2>Y$|are not feasible, hence |${p}_1{q}_1+{p}_2{q}_2\le Y$|must hold for the optimal bundle. 3. Point 1 and 2 ensure that the highest IC rule holds. 4. The last step ensures the consumer cannot purchase negative quantities of a good. 5. When the IC are strictly convex and differentiable. The MRS = MRT holds at a unique point of tangency (|$\mathrm{MRS}=-\frac{U_1}{U_2}=-\frac{p_1}{p_2}=\mathrm{MRT}$|) |
| |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{EL}}\right)$| | 1. Set up the function |$L\left({q}_1,{q}_2,\lambda \right)\!=\!U\left({q}_1,{q}_2\right)+ \lambda \left(Y-{p}_1{q}_1-{p}_2{q}_2\right)$| 2. Solve for |${q}_1,{q}_2$| from: (1) |$\frac{\partial L}{\partial{q}_1}=\frac{\partial U}{\partial{q}_1}-\lambda{p}_1={U}_1-\lambda{p}_1=0$| (2) |$\frac{\partial L}{\partial{q}_2}={U}_2-\lambda{p}_2=0$| (3) |$\frac{\partial L}{\partial \lambda }=Y-{p}_1{q}_1-{p}_2{q}_2=0$| | ||
| |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{ES}}\right)$| | 1. If the IC are strictly convex and differentiable, calculate the slope of the IC, |$\mathrm{MRS}=-\frac{d{q}_2}{d{q}_1}=-\frac{U_1}{U_2}$|, and equate it with the slope of the budget line |$\mathrm{MRT}-\frac{p_1}{p_2}.$| (Tangency rule.) Solve for |${q}_1$| and |${q}_2$|. For some functions |${q}_1$| and |${q}_2$| can be directly taken from a table. 2. If the IC is not strictly convex and differentiable, the optimal bundle must be obtained by other techniques which we do not detail here. 3. Plug in |${q}_1$| and |${q}_2$|from Step 1 into the budget constraint |$Y={p}_1{q}_1+{p}_2{q}_2$| to obtain the optimal bundle. | ||
(Continued)
Local reference praxeologies |${\protect\mathfrak{p}}_{iE}=\left[{T}_{iE},{\tau}_{iE},{\theta}_{iE},{\varTheta}_E\right],i\in \left\{1,2,3\right\}$| from the microeconomics textbook
| Task (|${\boldsymbol{T}}_{\boldsymbol{E}}$|) . | Technique (|${\boldsymbol{\tau}}_{\boldsymbol{E}}$|) . | Technology (|${\boldsymbol{\theta}}_{\boldsymbol{E}}$|) . | |
|---|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{1}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{1}\boldsymbol{E}}$| | |
| |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters when there exists an interior solution. Exemplary tasks outlined in Section 6.2 | |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{EG}}\right)$| | 1. Draw the budget line |$Y={p}_1{q}_1+{p}_2{q}_2$| 2. Draw an indifference curve (IC), |$U\left({q}_1,{q}_2\right)=\overline{q}$|, on which utility is fixed at some number |$\overline{q}.$| 3. Highest IC rule: Find the highest such IC (farthest from the origin) that satisfies |${p}_1{q}_1+{p}_2{q}_2\le Y$| for at least one bundle (|${q}_1,{q}_2$|). Graphically, this implies (tangency rule) shifting the IC up until it just touches the budget constraint. | Assumption of the model (A1)–(A4) 1. As more is preferred to less a consumer (weakly) prefers bundle (|${q}_i,{q}_j$|) to bundle (|${q}_i,{q}_k$|) for any |${q}_j\ge{q}_k.$| This ensures the IC are convex to the origin and that higher IC are ‘better’. 2. Bundles for which |${p}_1{q}_1+{p}_2{q}_2>Y$|are not feasible, hence |${p}_1{q}_1+{p}_2{q}_2\le Y$|must hold for the optimal bundle. 3. Point 1 and 2 ensure that the highest IC rule holds. 4. The last step ensures the consumer cannot purchase negative quantities of a good. 5. When the IC are strictly convex and differentiable. The MRS = MRT holds at a unique point of tangency (|$\mathrm{MRS}=-\frac{U_1}{U_2}=-\frac{p_1}{p_2}=\mathrm{MRT}$|) |
| |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{EL}}\right)$| | 1. Set up the function |$L\left({q}_1,{q}_2,\lambda \right)\!=\!U\left({q}_1,{q}_2\right)+ \lambda \left(Y-{p}_1{q}_1-{p}_2{q}_2\right)$| 2. Solve for |${q}_1,{q}_2$| from: (1) |$\frac{\partial L}{\partial{q}_1}=\frac{\partial U}{\partial{q}_1}-\lambda{p}_1={U}_1-\lambda{p}_1=0$| (2) |$\frac{\partial L}{\partial{q}_2}={U}_2-\lambda{p}_2=0$| (3) |$\frac{\partial L}{\partial \lambda }=Y-{p}_1{q}_1-{p}_2{q}_2=0$| | ||
| |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{ES}}\right)$| | 1. If the IC are strictly convex and differentiable, calculate the slope of the IC, |$\mathrm{MRS}=-\frac{d{q}_2}{d{q}_1}=-\frac{U_1}{U_2}$|, and equate it with the slope of the budget line |$\mathrm{MRT}-\frac{p_1}{p_2}.$| (Tangency rule.) Solve for |${q}_1$| and |${q}_2$|. For some functions |${q}_1$| and |${q}_2$| can be directly taken from a table. 2. If the IC is not strictly convex and differentiable, the optimal bundle must be obtained by other techniques which we do not detail here. 3. Plug in |${q}_1$| and |${q}_2$|from Step 1 into the budget constraint |$Y={p}_1{q}_1+{p}_2{q}_2$| to obtain the optimal bundle. | ||
| Task (|${\boldsymbol{T}}_{\boldsymbol{E}}$|) . | Technique (|${\boldsymbol{\tau}}_{\boldsymbol{E}}$|) . | Technology (|${\boldsymbol{\theta}}_{\boldsymbol{E}}$|) . | |
|---|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{1}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{1}\boldsymbol{E}}$| | |
| |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters when there exists an interior solution. Exemplary tasks outlined in Section 6.2 | |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{EG}}\right)$| | 1. Draw the budget line |$Y={p}_1{q}_1+{p}_2{q}_2$| 2. Draw an indifference curve (IC), |$U\left({q}_1,{q}_2\right)=\overline{q}$|, on which utility is fixed at some number |$\overline{q}.$| 3. Highest IC rule: Find the highest such IC (farthest from the origin) that satisfies |${p}_1{q}_1+{p}_2{q}_2\le Y$| for at least one bundle (|${q}_1,{q}_2$|). Graphically, this implies (tangency rule) shifting the IC up until it just touches the budget constraint. | Assumption of the model (A1)–(A4) 1. As more is preferred to less a consumer (weakly) prefers bundle (|${q}_i,{q}_j$|) to bundle (|${q}_i,{q}_k$|) for any |${q}_j\ge{q}_k.$| This ensures the IC are convex to the origin and that higher IC are ‘better’. 2. Bundles for which |${p}_1{q}_1+{p}_2{q}_2>Y$|are not feasible, hence |${p}_1{q}_1+{p}_2{q}_2\le Y$|must hold for the optimal bundle. 3. Point 1 and 2 ensure that the highest IC rule holds. 4. The last step ensures the consumer cannot purchase negative quantities of a good. 5. When the IC are strictly convex and differentiable. The MRS = MRT holds at a unique point of tangency (|$\mathrm{MRS}=-\frac{U_1}{U_2}=-\frac{p_1}{p_2}=\mathrm{MRT}$|) |
| |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{EL}}\right)$| | 1. Set up the function |$L\left({q}_1,{q}_2,\lambda \right)\!=\!U\left({q}_1,{q}_2\right)+ \lambda \left(Y-{p}_1{q}_1-{p}_2{q}_2\right)$| 2. Solve for |${q}_1,{q}_2$| from: (1) |$\frac{\partial L}{\partial{q}_1}=\frac{\partial U}{\partial{q}_1}-\lambda{p}_1={U}_1-\lambda{p}_1=0$| (2) |$\frac{\partial L}{\partial{q}_2}={U}_2-\lambda{p}_2=0$| (3) |$\frac{\partial L}{\partial \lambda }=Y-{p}_1{q}_1-{p}_2{q}_2=0$| | ||
| |$\left({\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{ES}}\right)$| | 1. If the IC are strictly convex and differentiable, calculate the slope of the IC, |$\mathrm{MRS}=-\frac{d{q}_2}{d{q}_1}=-\frac{U_1}{U_2}$|, and equate it with the slope of the budget line |$\mathrm{MRT}-\frac{p_1}{p_2}.$| (Tangency rule.) Solve for |${q}_1$| and |${q}_2$|. For some functions |${q}_1$| and |${q}_2$| can be directly taken from a table. 2. If the IC is not strictly convex and differentiable, the optimal bundle must be obtained by other techniques which we do not detail here. 3. Plug in |${q}_1$| and |${q}_2$|from Step 1 into the budget constraint |$Y={p}_1{q}_1+{p}_2{q}_2$| to obtain the optimal bundle. | ||
(Continued)
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