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The figure above shoes a Cobb-Douglas utility function of the form U(x,y)=x^\alpha y^\beta. The budget line is represented by I = P_x\times x + P_y\times y. The optimum point of consumption is where the indifference curve and the budget constraint are tangent.

The slope of the indifference curve is given by \frac{\Delta y}{\Delta x}. By rearranging, we can write \frac{\partial U}{\partial y} \Delta y = - \frac{\partial U}{\partial x}\Delta x. This can be further re-written as:

(1)   \begin{equation*} MRS= \frac{\Delta y} {\Delta x} = - \dfrac{\dfrac{\partial U}{\partial x}}{\dfrac{\partial U}{\partial y}} \end{equation*}

(2)   \begin{equation*} \nonumber \boxed{MRS = - \frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = - \frac{MU_x}{MU_y}} \end{equation*}

 

Practice Questions by Economics Tutor

  1. Suppose that a consumer has a preference relation on R2 represented by the quasilinear utility function U(x) = \sqrt{x_1} + x_2 has income and faces prices and
  1. Derive the (Marshallian) demand
  2. Derive the indirect utility
  3. What is the marginal utility of income?
  4. Derive the expenditure
  5. Derive the Hicksian demand functions.

Cindy consumes two goods . Her preferences are represented by . The price of good  is denoted by , the price of good  is denoted by , and her income is denoted by 𝑚. Suppose Cindy gets a weekly allowance of £4. Suppose both goods are £2 per unit.

  • Graph her budget constraint, and write down the equation of her budget line. In the same graph show Cindy’s indifference curves corresponding to the utility levels 𝑢, and . Solve for Cindy’s optimal consumption bundle and indicate it in your
  • Derive her demand functions for good 1 and good 2 as functions of her income, , price of good 1, , and price of good 2, . Is good 1 normal or inferior? Ordinary or Giffen? Are goods 1 and 2 substitutes or complements?
  • Suppose the price of good 1 rises to £4. How much would her parents have to increase her allowance in order to leave her exactly as well off as she was originally?

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