jii) Discuss if x. is a normal good to Mary's can Discuss the differences between the following terms'as used in ecor 2mics(use graph where iec ssary); i) Feasible allocation and Pareto efficient ellocation (2mks) ii) Shysiky and Hicksian substitution effect yii) Retuin to scale and Marginal product (3mks) (2mks) b) Mary has a utility function u(x_(1)x_(2))=x_(1)^1/xx_(2) i) Calculate the Marshallian demand sfarictions ii) Assuming that the initial price of x_(1) is given by P_(i)=4 and the initial income is given by m=160 Now essume that the price of x_(1) increase to p_(1)'=5 . Calculare the income and - substitution effect based on shisky equation. (8mks) -c) Determine the return;to scale of Britannia company production function is given as follows (6mks) . (2mks) (3mks)

a) i) Feasible allocation and Pareto efficient allocation:<br />- Feasible allocation: A feasible allocation is a combination of goods and services that can be produced and consumed given the available resources and technology. It represents a point on the production possibility frontier (PPF).<br />- Pareto efficient allocation: A Pareto efficient allocation is a feasible allocation where it is impossible to reallocate resources to make one individual better off without making another individual worse off. In other words, it is a point on the PPF where any movement would make at least one individual worse off.<br /><br />ii) Shysiky and Hicksian substitution effect:<br />- Shysiky effect: The Shysiky effect refers to the change in consumption resulting from a change in the relative prices of goods, holding real income constant. It is also known as the income effect.<br />- Hicksian substitution effect: The Hicksian substitution effect refers to the change in consumption resulting from a change in the relative prices of goods, holding utility constant. It is also known as the substitution effect.<br /><br />iii) Return to scale and Marginal product:<br />- Return to scale: Return to scale refers to the change in output resulting from a proportional increase in all inputs. If output increases by a larger proportion than the increase in inputs, it is called increasing returns to scale. If output increases by the same proportion as the increase in inputs, it is called constant returns to scale. If output increases by a smaller proportion than the increase in inputs, it is called decreasing returns to scale.<br />- Marginal product: The marginal product of an input is the additional output produced by an additional unit of that input, holding all other inputs constant.<br /><br />b) i) Calculate the Marshallian demand functions:<br />The Marshallian demand functions can be calculated by minimizing the cost of purchasing a given level of utility subject to the budget constraint. In this case, the utility function is given by $u(x_{1}x_{2})=x_{1}^{1/x}x_{2}$. The budget constraint is given by $P_{1}x_{1} + P_{2}x_{2} = m$, where $P_{1}$ and $P_{2}$ are the prices of $x_{1}$ and $x_{2}$, respectively, and $m$ is the income. The Marshallian demand functions can be found by solving the Lagrangian function:<br /><br />$L(x_{1}, x_{2}, \lambda) = x_{1}^{1/x}x_{2} - \lambda(P_{1}x_{1} + P_{2}x_{2} - m)$<br /><br />ii) Calculate the income and substitution effect based on the Shysiky equation:<br />The Shysiky equation is given by:<br /><br />$E = \frac{P_{1}x_{1}}{P_{1}x_{1} + P_{2}x_{2}}$<br /><br />where $E$ is the expenditure on $x_{1}$. The income effect is the change in expenditure on $x_{1}$ due to a change in income, holding prices constant. The substitution effect is the change in expenditure on $x_{1}$ due to a change in the relative prices of $x_{1}$ and $x_{2}$, holding income constant.<br /><br />c) Determine the return to scale of the Britannia company's production function:<br />The return to scale can be determined by examining the behavior of the production function as all inputs are increased proportionally. If the production function exhibits increasing returns to scale, the output will increase by a larger proportion than the increase in inputs. If the production function exhibits constant returns to scale, the output will increase by the same proportion as the increase in inputs. If the production function exhibits decreasing returns to scale, the output will increase by a smaller proportion than the increase in inputs.