| Task (|${\boldsymbol{T}}_{\boldsymbol{E}}$|) . | Technique (|${\boldsymbol{\tau}}_{\boldsymbol{E}}$|) . | Technology (|${\boldsymbol{\theta}}_{\boldsymbol{E}}$|) . | |
|---|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{2}\boldsymbol{E}}$| | |
| |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters. Exemplary tasks: ‘Spenser has a quasilinear utility function |$U\left({q}_1,{q}_2\right)=4{q}_1^{0.5}+{q}_2.$|For given prices and income, investigate how the utility maximum is obtained’ (Perloff, 2022, p. 116) | 1. Perform |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{E}}$| and 2. Check if the bundle obtained is non-negative. If yes, it is the optimal bundle. If not, the optimal bundle is a corner solution: |$\left({q}_1=\frac{Y}{p_1},{q}_2=0\right) or$| |$\left({q}_1=0,{q}_2=\frac{Y}{p_2}\ \right)$|. | |${\mathfrak{p}}_{1E}$|but if |${q}_1$|or |${q}_2$| is negative the bundle obtained is not feasible because one cannot consume negative amounts of a good. | |
| |${\boldsymbol{T}}_{\mathbf{3}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{3}\boldsymbol{E}}$| | |
| Derive the exponents of the Cobb Douglas utility function |$U\left({q}_1,{q}_2\right)=A{q}_1^a{q}_2^b$|. Exemplary task: Suppose that a consumer has a Cobb–Douglas utility function and buys two goods, |${q}_1$|and |${q}_2$|, with income of 200 euros per week. She/he spends 60 euros per week on good 1 and 140 euros per unit on good 2. What are the values of the exponents of her utility function? Using these values, what is the equation of her/his utility function? (Perloff, 2022, task 4.12) | 1. Calculate the budget shares |${s}_1=\frac{p_1{q}_1}{Y}$| and |${s}_2=\frac{p_2{q}_2}{Y}$|. 2. For a Cobb–Douglas function it holds that |${s}_1=a$| and |${s}_2$| = |$b$|. | 1. The MRS = MRT condition implies |${q}_1=\frac{aY}{p_1}$| and |${q}_2=\frac{bY}{p_2}$| for a Cobb Douglas utility function. 2. A budget share is the consumers expenditure on a good. E.g. |${p}_1{q}_1$| divided by her budget. | |
| Task (|${\boldsymbol{T}}_{\boldsymbol{E}}$|) . | Technique (|${\boldsymbol{\tau}}_{\boldsymbol{E}}$|) . | Technology (|${\boldsymbol{\theta}}_{\boldsymbol{E}}$|) . | |
|---|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{2}\boldsymbol{E}}$| | |
| |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters. Exemplary tasks: ‘Spenser has a quasilinear utility function |$U\left({q}_1,{q}_2\right)=4{q}_1^{0.5}+{q}_2.$|For given prices and income, investigate how the utility maximum is obtained’ (Perloff, 2022, p. 116) | 1. Perform |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{E}}$| and 2. Check if the bundle obtained is non-negative. If yes, it is the optimal bundle. If not, the optimal bundle is a corner solution: |$\left({q}_1=\frac{Y}{p_1},{q}_2=0\right) or$| |$\left({q}_1=0,{q}_2=\frac{Y}{p_2}\ \right)$|. | |${\mathfrak{p}}_{1E}$|but if |${q}_1$|or |${q}_2$| is negative the bundle obtained is not feasible because one cannot consume negative amounts of a good. | |
| |${\boldsymbol{T}}_{\mathbf{3}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{3}\boldsymbol{E}}$| | |
| Derive the exponents of the Cobb Douglas utility function |$U\left({q}_1,{q}_2\right)=A{q}_1^a{q}_2^b$|. Exemplary task: Suppose that a consumer has a Cobb–Douglas utility function and buys two goods, |${q}_1$|and |${q}_2$|, with income of 200 euros per week. She/he spends 60 euros per week on good 1 and 140 euros per unit on good 2. What are the values of the exponents of her utility function? Using these values, what is the equation of her/his utility function? (Perloff, 2022, task 4.12) | 1. Calculate the budget shares |${s}_1=\frac{p_1{q}_1}{Y}$| and |${s}_2=\frac{p_2{q}_2}{Y}$|. 2. For a Cobb–Douglas function it holds that |${s}_1=a$| and |${s}_2$| = |$b$|. | 1. The MRS = MRT condition implies |${q}_1=\frac{aY}{p_1}$| and |${q}_2=\frac{bY}{p_2}$| for a Cobb Douglas utility function. 2. A budget share is the consumers expenditure on a good. E.g. |${p}_1{q}_1$| divided by her budget. | |
| Task (|${\boldsymbol{T}}_{\boldsymbol{E}}$|) . | Technique (|${\boldsymbol{\tau}}_{\boldsymbol{E}}$|) . | Technology (|${\boldsymbol{\theta}}_{\boldsymbol{E}}$|) . | |
|---|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{2}\boldsymbol{E}}$| | |
| |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters. Exemplary tasks: ‘Spenser has a quasilinear utility function |$U\left({q}_1,{q}_2\right)=4{q}_1^{0.5}+{q}_2.$|For given prices and income, investigate how the utility maximum is obtained’ (Perloff, 2022, p. 116) | 1. Perform |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{E}}$| and 2. Check if the bundle obtained is non-negative. If yes, it is the optimal bundle. If not, the optimal bundle is a corner solution: |$\left({q}_1=\frac{Y}{p_1},{q}_2=0\right) or$| |$\left({q}_1=0,{q}_2=\frac{Y}{p_2}\ \right)$|. | |${\mathfrak{p}}_{1E}$|but if |${q}_1$|or |${q}_2$| is negative the bundle obtained is not feasible because one cannot consume negative amounts of a good. | |
| |${\boldsymbol{T}}_{\mathbf{3}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{3}\boldsymbol{E}}$| | |
| Derive the exponents of the Cobb Douglas utility function |$U\left({q}_1,{q}_2\right)=A{q}_1^a{q}_2^b$|. Exemplary task: Suppose that a consumer has a Cobb–Douglas utility function and buys two goods, |${q}_1$|and |${q}_2$|, with income of 200 euros per week. She/he spends 60 euros per week on good 1 and 140 euros per unit on good 2. What are the values of the exponents of her utility function? Using these values, what is the equation of her/his utility function? (Perloff, 2022, task 4.12) | 1. Calculate the budget shares |${s}_1=\frac{p_1{q}_1}{Y}$| and |${s}_2=\frac{p_2{q}_2}{Y}$|. 2. For a Cobb–Douglas function it holds that |${s}_1=a$| and |${s}_2$| = |$b$|. | 1. The MRS = MRT condition implies |${q}_1=\frac{aY}{p_1}$| and |${q}_2=\frac{bY}{p_2}$| for a Cobb Douglas utility function. 2. A budget share is the consumers expenditure on a good. E.g. |${p}_1{q}_1$| divided by her budget. | |
| Task (|${\boldsymbol{T}}_{\boldsymbol{E}}$|) . | Technique (|${\boldsymbol{\tau}}_{\boldsymbol{E}}$|) . | Technology (|${\boldsymbol{\theta}}_{\boldsymbol{E}}$|) . | |
|---|---|---|---|
| |${\boldsymbol{T}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{2}\boldsymbol{E}}$| | |
| |$\underset{q_1,{q}_2}{\max }U\left({q}_1,{q}_2\right)\ s.t.Y={p}_1{q}_1+{p}_2{q}_2$| where |$Y,{p}_1,{p}_2$| are positive parameters. Exemplary tasks: ‘Spenser has a quasilinear utility function |$U\left({q}_1,{q}_2\right)=4{q}_1^{0.5}+{q}_2.$|For given prices and income, investigate how the utility maximum is obtained’ (Perloff, 2022, p. 116) | 1. Perform |${\boldsymbol{\tau}}_{\mathbf{1}\boldsymbol{E}}$| and 2. Check if the bundle obtained is non-negative. If yes, it is the optimal bundle. If not, the optimal bundle is a corner solution: |$\left({q}_1=\frac{Y}{p_1},{q}_2=0\right) or$| |$\left({q}_1=0,{q}_2=\frac{Y}{p_2}\ \right)$|. | |${\mathfrak{p}}_{1E}$|but if |${q}_1$|or |${q}_2$| is negative the bundle obtained is not feasible because one cannot consume negative amounts of a good. | |
| |${\boldsymbol{T}}_{\mathbf{3}\boldsymbol{E}}$| | |${\boldsymbol{\tau}}_{\mathbf{2}\boldsymbol{E}}$| | |${\boldsymbol{\theta}}_{\mathbf{3}\boldsymbol{E}}$| | |
| Derive the exponents of the Cobb Douglas utility function |$U\left({q}_1,{q}_2\right)=A{q}_1^a{q}_2^b$|. Exemplary task: Suppose that a consumer has a Cobb–Douglas utility function and buys two goods, |${q}_1$|and |${q}_2$|, with income of 200 euros per week. She/he spends 60 euros per week on good 1 and 140 euros per unit on good 2. What are the values of the exponents of her utility function? Using these values, what is the equation of her/his utility function? (Perloff, 2022, task 4.12) | 1. Calculate the budget shares |${s}_1=\frac{p_1{q}_1}{Y}$| and |${s}_2=\frac{p_2{q}_2}{Y}$|. 2. For a Cobb–Douglas function it holds that |${s}_1=a$| and |${s}_2$| = |$b$|. | 1. The MRS = MRT condition implies |${q}_1=\frac{aY}{p_1}$| and |${q}_2=\frac{bY}{p_2}$| for a Cobb Douglas utility function. 2. A budget share is the consumers expenditure on a good. E.g. |${p}_1{q}_1$| divided by her budget. | |
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