assume the utility function from two goods x and y given by uxy 2x xy 4y suppose that x and y ar

a. The consumer maximization problem is to maximize the utility function U(x,y) = 2x + xy + 4y s

Assume the utility function from two goods x and y given by...

Assume the utility function for two goods (x and y) is given by: U(x,y) = 2x + xy + 4y. Suppose that x and y are purchased in the market at a constant unit price of birr 3 and birr 2 respectively, and the consumer has a money income of birr 104, which he spends on the two goods. a. Formulate the consumer maximization problem. b. Compute and interpret MRS(x,y) at x = 2 and y = 8. c. Use the Lagrangian method to find the values of x and y that maximize utility.

Transcript

00:01 So here we have an optimization problem, right? and we start off with a utility function over x plus y, which is given like this.

00:08 We are also given that the price of x is equal to three, the price of y is equal to two, and the income of the consumer is 104.

00:17 And so the way that we would set this up is we would want to maximize utility subject to the budget constraint.

00:25 And the budget constraint would be 104 is equal to three x plus two y, right? this is my total spending.

00:32 And the idea is that your spending cannot increase your income.

00:37 So for b, the marginal rate of substitution is equal to the derivative, what we usually think of as the marginal product of x over the marginal product of y, sometimes with a negative sign.

00:51 So we need the marginal products.

00:53 And those are the partial derivatives, which are gonna be two plus y, the marginal product.

00:59 Oh, i shouldn't be saying, i should be saying the marginal utility, not the marginal product.

01:02 This is a utility problem, not a cost or a profit problem.

01:06 So let's say marginal utility, which is du dy would be equal to x plus four.

01:15 So there's my marginal utilities.

01:17 And therefore my marginal rate of substitution would be minus two plus y over x plus four.

01:26 And so my marginal rate of substitution at two eight would be equal to minus 10 over six is equal to minus five over three, right? that would be my marginal rate of substitution.

01:43 And now for c, my lagrangian is defined as my objective constraint plus x, y plus four y plus my budget constraint, which is 100 minus, sorry, three x minus two y, i believe is that my prices are correct...

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